help-27

[lime plaza]

Ok so u write it as

(-infinity , 0]

So the brackets () mean the value is not included and [] means the value is included

[normal bolt]

0 is included and -infinity is excluded

[lime plaza]

Yes

Do you know the value of -infinity?

[full cloak]

you never include infinity as they aren't really numbers, but just an idea

that applies to negative infinity as well

[normal bolt]

well -infinity is a number that keep increasing in -ve side

[lime plaza]

Exactly

That is why we don't include it

We don't know what that is

So did you understand how to find domain and range

[normal bolt]

but our function is giving negative values for any -ve no. We should do [-infinity, 0]

[lime plaza]

Tell me the value of -infinity

The exact value

[normal bolt]

can't

[lime plaza]

Then we cannot include it

Infinity and -infinity are always () excluded

[normal bolt]

the reason being is we can't tell its value?

[lime plaza]

Uh yes

[normal bolt]

can we write that interval as [0 , -infinity)

[lime plaza]

-infinity just tells us that the value keeps going more negative

So basically

Do u know the number line

[normal bolt]

yup

[lime plaza]

THat is ok but number line

In number line it's NEGATIVE then ZERO then POSITIVE

[normal bolt]

- values , 0 +values

[lime plaza]

Hence negative to 0 then 0 to positive

Yes

That is why I wrote (-infinity,0]

It goes according to the number line

But if you want to write it the other way you can

[normal bolt]

got it

[lime plaza]

Ok so try 2nd question

[normal bolt]

so domain is all the input values to define f(X) and range is set of values of input from domain

[lime plaza]

Yes range is output you get by inputting values from domain

[normal bolt]

let me work 2nd

[lime plaza]

k

[normal bolt]

domain = R
range = [3,3]

@lime plaza

[lime plaza]

uh

No

Domain is wrong

x=5

Tell me

[normal bolt]

negative root

[lime plaza]

is that defined

[normal bolt]

no

[lime plaza]

Then?

How is domain=R

[normal bolt]

no its not

[lime plaza]

Then tell me domain

[normal bolt]

domain = {x : -3 <= x <= 3}
range = [3,3]

both are same thing

ig

well

i got it

domain = 0
range = 0

[lime plaza]

Ok domain is [-3,3]

[normal bolt]

is this correct domain = {x : -3 <= x <= 3}

[lime plaza]

You can write it as [-3,3]

[normal bolt]

fine

range = {-3,..,0,..3}

[lime plaza]

ok tell me what number you enter

to get -3

Don't use {} btw they represent a single element

[normal bolt]

not possible

[lime plaza]

Then why do you include that

[normal bolt]

it is possible for x because x^2 = (-3)^2 = 0

[lime plaza]

huh?

$f(x)=\sqrt{9-x^2}$

[woven radish]

devthemasked

[lime plaza]

What x will you enter

[normal bolt]

f(-3) = √(9 - (-3)^2)
=> 0

here ^

[lime plaza]

Yes so how did you get -3 as output

[normal bolt]

we can't get

[lime plaza]

Yea then what is your range

[normal bolt]

[0, 3]

[lime plaza]

Uh wait

I think you are

[normal bolt]

domain = [-3,3]
range = [0,3]

[lime plaza]

Uh I'm not entirely sure

About range

Ok it's fine

[lime plaza]

Ok

[normal bolt]

ok

clear

😄

What is the purpose of function in life?

How it is helpful?

[lime plaza]

No idea

Just do math cause math is fun

[normal bolt]

that is indeed

What problem did mathematicians trying to solve that they had to invent functions?

[lime plaza]

I don't know

[normal bolt]

fine

what is the meaning of this f(x) = {x, x>=0 , -x, x < 0 in modulus function

@lime plaza

[lime plaza]

Ok so

Modulus opens with a sign

Modulus has to be always positive

Hence when |x| x is positive that means |x| is same as x

But in |x| if x is negative then it would be |-x| but modulus is always positive

So it opens with a negative sign

|-x| will be -(-x)

[normal bolt]

so |-3| = -3?

[lime plaza]

no

|-3|= -(-3)

Modulus always positive

If number inside is negative it needs to make it positive

[normal bolt]

from where the extra - come?

[lime plaza]

Modulus opens with a negative sign

[lime plaza]

That is why it says -x when x<0

Modulus can open with a + or with a - it depends on the number inside

[normal bolt]

so if f(-5) = |-5| = 5?

[lime plaza]

yes

if f(x)= |x| then yea

f(-5)= 5

[normal bolt]

f(-x) = |-x| = -(-x)

[lime plaza]

Yes

[normal bolt]

i didn't understand why the extra -ve sign

[lime plaza]

Do you know that modulus range is positive?

[normal bolt]

no

[lime plaza]

Ok modulus range is always positive and 0 is included

f(x)= |x|

f(-2) = 2

[normal bolt]

and modulus domain is real number?

[lime plaza]

f(2) = 2

Both are same

Modulus is always positive

So if number is negative we multiply with a - to make it positive

[normal bolt]

it means it range is [0, infinity)

[lime plaza]

no

[0,infinity)

yes

[normal bolt]

that was bracket mistake

[lime plaza]

{} means a single element

[normal bolt]

yeah

[lime plaza]

Ok so u understood or

[normal bolt]

we use in defining set

how to understand its graph

like feel it

[lime plaza]

It's graph is a slope

Like

Straight lines*

Draw 2 symmetric straight lines from 0

at angle 45

Or I will show

[lime plaza]

yes

[normal bolt]

why is it like this

why that line is going in -ve x-axis

[lime plaza]

Domain

Can be negative

I don't know much about graphs

[normal bolt]

its alright

i understood that

thanks

[lime plaza]

:)

if done .close

[devout snow]

@normal bolt Has your question been resolved?

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[normal bolt]

What does this highlighted box part want to mean? I can't see the things on the fig 12.1 as mentioned in text.

[normal bolt]

basically i want someone to explain read box text

[ripe flax]

for the first one

[pure flower]

what is this from

[ripe flax]

its basically explaining how to get a gradient from a curve

[normal bolt]

from my textbook

[ripe flax]

so yeah u could basically draw a tangent line to a specific point and that would be it

but it wont be that exact would it

[pure flower]

why is there physics here lol

[ripe flax]

so we start by drawing something we know is exact first, a line intersecting the curve twice

its not the same as the tangent line

but move the points of intersection closer together

does it look like a bit more like the tangent line

so just do that and that and that

[normal bolt]

intersection of that chord?

[ripe flax]

yes

just move one of the points closer and closer to the point u want

but we want to be exact we’re mathematicians

lemme draw a graph hold on

[normal bolt]

yes please, i wanna understand everything basically or fundamentally

[ripe flax]

say a simple graph y=x^2

[pure flower]

if you have zero calculus fundamentals, you might wanna watch the 3b1b series on it

[normal bolt]

parabola

[ripe flax]

u have 2 points u can intersect with a line easily

[normal bolt]

how?

[ripe flax]

say one point is fixed: (x,x^2)

its called a secant line

[normal bolt]

in (x,x^2) where is x^2

[ripe flax]

it dont look like a tangent line yet but we will make it look like one

something like this

because y=x^2

sub y in for x^2, so (x,y) becomes (x,x^2)

[normal bolt]

oh

[ripe flax]

and say we have another point

but this time its not the same point

[normal bolt]

and h is the distance on x-axis from fpoint x?

[ripe flax]

h is the distance from the point u want to any point u place on the graph

[normal bolt]

ok

[ripe flax]

h could be 100 or 0.4 and we would still have the same, if ur original point is fixed

[normal bolt]

any real number on x-axis?

[ripe flax]

heres a cool thing we can do to make this line a tangent line tho

[ripe flax]

theoretically yes but lets make h positive for now

[normal bolt]

the secant line?

[ripe flax]

yes

[normal bolt]

okay

[pure flower]

do you know limits?

[ripe flax]

if we move the points closer to our original point

h gets smaller

[normal bolt]

move the second point along the parabola curve to the first point?

[ripe flax]

and the approximation for the tangent line is better

but we could do better

we could make h closer and closer and closer

so close that there is so little distance between them

like 0.000000000001

but we disregard that, since its so small its not gonna change shit

[normal bolt]

that is approximately 0 by rounding off?

[ripe flax]

yes

so as h approaches 0

so the point is moving towards the original point

it slowly becomes more of a tangent line

[normal bolt]

h is going toward 0 distnace or reaching to 0 distance constantly??

[ripe flax]

therefore we can find the gradient as h approaches to 0

[ripe flax]

reaching

it never gets to 0, otherwise the 2 points overlap

and we cant get a line from that

but can u see that

if h is so small so small

we can disregard that value altogether

cuz its ultimately not gonna make a difference

[normal bolt]

what does it mean

[ripe flax]

essentially the gap is there but we dont care

[ripe flax]

we can round it off to 0

[normal bolt]

do we round off actually?

[ripe flax]

so in proper notation we would do something like this

[pure flower]

this might sound wishy washy but dont let that dissaude you

[pure flower]

nah, we have good rigor for it

[pure flower]

it's called the epsilon-delta definiton of limits, and it's a nightmare to use lmao, but yeah the rigor exists

[ripe flax]

lim h->0 means as h gets so close to 0 that we could just say its 0 we dont care

the right hand side is just the gradient function

(x+h)^2 -x^2 is just the rise

h is the run

rise/run

[normal bolt]

wait let me absorb this 2 min and ask you doubt if i get

[pure flower]

okay slight nitpicky connection here, we'd never say it's 'basically zero we don't care', we say it's an infinitesimal, and we still have to deal with the infinitesimal

just remember that

[ripe flax]

fair nitpick

[pure flower]

pitfall prevention

[ripe flax]

just not sure everyone can grasp the idea of infinity thats all

[normal bolt]

what do you mean by gradient function?

[pure flower]

the slope.

they're british, presumably

[ripe flax]

gradient means slope

[ripe flax]

correct

so slope is rise/run

[pure flower]

mfw to call the slope the gradient so i sound like i know multivariable calculus 😔

[ripe flax]

nah the brits jsut call it that

i stay with it cuz it sounds cool

[normal bolt]

how do you write 2nd point coordinate?

[pure flower]

it's $\left(x+h, f(x+h)\right)$

[ripe flax]

if the x distance between the 2 points is h, if one point is x, would it make sense for the other point to be x+h

[woven radish]

Percy

[pure flower]

whoopsie daisy

[ripe flax]

essentially yes

funny thing most of the time we use y instead of f(x)

[pure flower]

well it's f(x+h) in the definition sooo

[normal bolt]

and y axis is (X+h)^2 why

[ripe flax]

because recall that f(x) = x^2

[pure flower]

the function is $f(x)=x^2$ no?

[woven radish]

Percy

[pure flower]

okay ill stick to nitpicks here i like how you're dealing with the derivatives 101 class llol /gen

[ripe flax]

for the first point its (x,x^2) just sub y in

for the second point however, its (x+h, and something else

that something, is because y=x^2, would it make sense to the y value to be y=(x+h)^2

im just adding some value to the x

[normal bolt]

got it

[ripe flax]

nice

[normal bolt]

when we say h -> 0 do we mean the distance of h is changing towards 0 or is it going towards (0,0) which is origin

[ripe flax]

the distance of h is changing towards 0

[normal bolt]

but it makes sense that h can't approach (0,0) but any point on parabola can

[ripe flax]

heres the thing

we dc about the origin

because say f(x)=x^2 and g(x) = x^2 +1

they look the same but g(x) does not pass thru the origin

[normal bolt]

why g(x) does not pass thru the origin

[ripe flax]

recall that the origin is (0,0)

if you substitute x=0 to g(x), (0)^2+1 =1

the y value does not equal 0

therefore it does not intersect the origin

[normal bolt]

oohh

got it now

[ripe flax]

so yeah not every parabola crosses the origin

[normal bolt]

how does it changes by only adding +1

[ripe flax]

by adding 1 to f(x)

can u see every y value has shifted by 1

so say a point is (2,4)

now its transformed to (2,5)

because we added a 1 on top

[normal bolt]

oh

what about a cubic polynomial like this

[ripe flax]

same thing

if u add 1, the y value would shift all by 1

[normal bolt]

will that limit concept work with f(x) = x^3 too?

[ripe flax]

yes

[pure flower]

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

[ripe flax]

but u need to evaluate the limit first

[pure flower]

please

[ripe flax]

on the right we have say like ((x+h)^2 -x^2) all over a measly little h

we know that h is basically 0

but we cant divide by 0

[normal bolt]

then?

[ripe flax]

so we try to expand the numerator first see if we can factorise anything

[pure flower]

In fancy terms $\frac{d}{dx} f(c) = \lim_{h \to 0} \frac{f(c+h) - f(c)}{h}$, which is the (kinda) formal definition of the derivative.

[ripe flax]

yes

[pure flower]

fuck

not infty lmao

[woven radish]

Percy

[ripe flax]

ah close enough

[pure flower]

major whoopsie daisy but we're so back

[ripe flax]

infinitely small more like

expand the numerator out u get x^2 +2xh + h^2 -x^2

do u see anything nice

the x^2s cancel each other

so we’re left with 2xh + h^2 left

since the denominator is an h

[pure flower]

yes it's actually $\lim_{h \to \lim_{c \to \infty} \left(\frac{1}{c}\right)}$ lmao

[woven radish]

Percy

[ripe flax]

we can factor the h out

h(2x+h) all over h

so the 2 h’s cancel

[pure flower]

ill leave now

[ripe flax]

we have 2x+h left

but since i said h is basically 0 in this scenario

[normal bolt]

@ripe flax explain me the x^3 thing

[ripe flax]

would it be sensible to substitute as h=0

[ripe flax]

i’ll send a pic hold on

[woven radish]

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

Mar 26, 2025, 10:43 AM

[ripe flax]

,rotate

[ripe flax]

hope this helps a bit

[normal bolt]

yeah more clear

so limit process is all about finding the gradient?

[ripe flax]

yes

u can even do it with x^4, hell even sinx

[normal bolt]

is that l'hospital is also limit rule?

[ripe flax]

tbh i havent learnt hospital yet

[normal bolt]

i also didn't

[ripe flax]

i learn it in 5 months time cuz my curriculum is scuffed

i learned all of basic integration before hospital lol

[normal bolt]

is this called fundamental theorm of calculus?

[ripe flax]

its called differentiation from first principles in my country

idk about urs

but yes its the first thing u learn in calculus

[winter patrol]

,tex .ftc

1 sec

[normal bolt]

f(x) = x^2 takes very close values to 0 and f(x) = x^2 =+ 1 does not?

not this ss but you got it

ok i close this

[ripe flax]

ok

[normal bolt]

.cloaw

.cloaw

lol

.close

[devout snow]

Channel closed

Closed by @normal bolt

Use .reopen if this was a mistake.

[pure flower]

no

[winter patrol]

,tex .FTC1

[woven radish]

ℝαμOmeganato5

[winter patrol]

,tex .FTC2

[woven radish]

ℝαμOmeganato5

[pure flower]

ftc is about integrals from riemann sums and limits of antiderivatives

[pure flower]

is this ftc?

i think its more like leibniz

isn't ftc 2 $\frac{d}{dx} \int^x_a f(t) dt = f(x)$

[woven radish]

Percy

[pure flower]

like the simple version

[winter patrol]

yeh, the one above is more general

[pure flower]

hmm

yeah just checking

[pure flower]

are these general factoid commands?

[winter patrol]

custom, from riemann and snow

[pure flower]

oooh

fun

[devout snow]

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[pale thistle]

broooooooooooooooooooo

[pale thistle]

i have a big doubt

[iron kindle]

ok

[pure flower]

very good

[lime plaza]

!da2a

[devout snow]

No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for everyone if you just ask your question! See https://dontasktoask.com/

@pale thistle Has your question been resolved?

[devout snow]

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• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
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• 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.

[solid hull]

what is this question asking?

[topaz axle]

true or false

likely

[solid hull]

are we attempting to show equivalence between the two statements?

[sullen island]

the whole thing is one statement

[solid hull]

can I clarify what the tribar means in context

as in what would we want to show (it's a T/F q)

[sullen island]

if that helps you

I know what it means, it's not like I need clarification

[solid hull]

sorry I guess I wasn't clear I was asking what it means in this context

lol

[sand dove]

it's equivalence/congruence modulo 2

[solid hull]

so do you apply modulo 2 to both 17 and 5 and check if they are equal?

[sand dove]

$a\equiv b \mod n$ means $a-b = kn, k\in \bZ$

[woven radish]

rafilou is not not born in 2003

[sand dove]

if by "applying modulo 2" you mean computing the remainder of division by 2

sure

otherwise

computing a-b and seeing if it's a multiple of 2 doesn't hurt

[solid hull]

alright, I'll try that thanks

.close

[devout snow]

Channel closed

Closed by @solid hull

Use .reopen if this was a mistake.

[devout snow]

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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.

[primal hollow]

Limit $\lim_{x\to\a} f(x^2)$

[woven radish]

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

[primal hollow]

um sorry I don't really know how to use latex

[runic prawn]

is f continuous

$\lim_{x\to a} f(x^2)$

[primal hollow]

anyhow I don't understand the general epsilon delta proof of the limit of x² when x aproaches a

[woven radish]

martingale

[primal hollow]

yes thanks

[primal hollow]

yes

[runic prawn]

or do just mean $\lim_{x\to a} x^2$

[woven radish]

martingale

[primal hollow]

yes

that's how it's witten in my notes

[runic prawn]

let $\epsilon > 0 $

find $$\delta > 0$$ such that $$|x- a| < \delta \implies |x^2 - a^2| < \epsilon $$

[woven radish]

martingale

[primal hollow]

without a right zo biger than a

[runic prawn]

ok what part of the proof do u not understand

[runic prawn]

?

[primal hollow]

at a certain point you get |x-a| * |x+a| and then you try to find some value so you can solve it and that part I don't understand

[runic prawn]

|x-a| * |x+a| <= |x-a| *( |x-a| + 2|a| )

[primal hollow]

||x|-|a|| les than or eaqual to |x-a| les than 1

[primal hollow]

that's not how my reacher did it but I'm not sure

he reasons that |x-a| must me les than 1 at some point

[runic prawn]

do u agree this true for all x

remember a is fixed

[primal hollow]

I don't know

[runic prawn]

this is a choice btw it doesn't have to be <1

[primal hollow]

I know

[runic prawn]

so no

[primal hollow]

no than

[runic prawn]

|x+a| = |(x-a) + 2a|

then triange ineq

[primal hollow]

I haven o idea what that is

[runic prawn]

triangle inequality?

[primal hollow]

he arives at 2|a| -1 < |x| < 2|a|+1

[runic prawn]

for all p, q

|p + q| <= |p| + |q|

wait u must know the triange inequality

[primal hollow]

oh yes he did use that

[runic prawn]

ok let's forget about him for a sec

[primal hollow]

ok

[runic prawn]

do dat here

[runic prawn]

u get this

|x-a| * |x+a| <= |x-a| *( |x-a| + 2|a| )

now we pick delta

[primal hollow]

but how is this the same

[runic prawn]

p = x-a, q = 2a

[primal hollow]

ah

[runic prawn]

wdym

x-a + 2a = x+a

ru following

[primal hollow]

yes

[runic prawn]

ru sure

[primal hollow]

um not really

[primal hollow]

yes about the previous step I'm sure

[runic prawn]

ok do u agree with this

[primal hollow]

yes

[runic prawn]

apply triangle ineq on that what do u get

|(x-a) + 2a| <= ... ?

[primal hollow]

|x-a| + |2a|

[runic prawn]

yes

[primal hollow]

wich is eqaual to just |x-a|+ 2|a|

[runic prawn]

so |x+a| <= |x-a|+ 2|a|

[primal hollow]

yes

that I understand

[runic prawn]

so
|x^2 - a^2| = |x-a| * |x+a| <= |x-a| *( |x-a| + 2|a| )

[primal hollow]

you just replace x+a with this

ok I think I get it

[runic prawn]

i bound it above yes

[runic prawn]

so |x^2 - a^2| <= |x-a| *( |x-a| + 2|a| ) for all x

[primal hollow]

yes but not when x = a right?

[runic prawn]

no even then

then just 0<=0

[primal hollow]

oh then why do you need to exempt a from your base region

[runic prawn]

oh for the limt defn

umm

[primal hollow]

yes

[runic prawn]

here it works at x=a because x^2 is continous but in general it won't

regardless of definition

it's just a fact that

|x^2 - a^2| <= |x-a| *( |x-a| + 2|a| ) for all x

[primal hollow]

ok

[runic prawn]

but if x is such that |x - a| < 1, i.e. x in the interval (a-1, a+1),

|x^2 - a^2| <= |x-a| * (1 + 2|a|)

mb

[primal hollow]

um I don't get this

the second bit

the first I get

[runic prawn]

basically use |x - a| < 1 in the second |x-a|

like if
|x - a| < 1 then
|x - a| + 2|a| < 1 + 2|a|
yes?

[primal hollow]

yes

[runic prawn]

ok multiply |x-a| on both sides

[runic prawn]

u get this

[primal hollow]

that's the same no?

[runic prawn]

same as what

[primal hollow]

we had before

nothing changed

[runic prawn]

u gotta think abt for a bit

[primal hollow]

I meaen this step didn't heppan right?

happen*

[runic prawn]

wym

don't think abt steps, think abt what is true

[primal hollow]

if you got |x²-a²| <= |x-a| *(1+2|a|) and you multiplie both sides by |x-a|

how can you get |x²-a²| <= |x-a| *(1+2|a|) again

[runic prawn]

i meant multplity that here

[primal hollow]

what?

[runic prawn]

what hwat

[primal hollow]

I don't understand what you mean

[primal hollow]

with this

[primal hollow]

and this

[runic prawn]

bruh ok

if |x - a| < 1 then
|x - a| + 2|a| < 1 + 2|a| so
|x - a| * (|x - a| + 2|a|) < |x - a| * (1 + 2|a|)

[primal hollow]

I'm sorry I don't understand

[runic prawn]

if p < q and r > 0 then pr < qr

p = |x - a| + 2|a|
q = 1 + 2|a|
r = |x-a|

[primal hollow]

and where comes p from

[runic prawn]

gtg good luck

[primal hollow]

np thanks

bye

[devout snow]

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[fast rose]

I saw this question earlier but I don’t understand the methodology. How do you divide by x^2 while it runs from 0 to 1?

[ripe flax]

goddamn it

[fast rose]

@lime plaza sorry for the ping, u did this earlier, can u help me if possible?

[ripe flax]

partial fractions my ass

or is it an arctan integral

[fast rose]

Ye I tried too but the nominator isn’t relevant ..

[lime plaza]

Divide it by x^2

[fast rose]

But it runs from 0

[lime plaza]

Who posted this question earlier

[ripe flax]

personally i’d factorise x^4 -1 into something like (x^2+1)^2 - 2x^2

[fast rose]

U can’t divide by 0 can u

[ripe flax]

then u can cancel one of the x^2 +1s

[lime plaza]

forget that there are limits

[ripe flax]

then ur left with 1/(1-x^2)

which is a diff of squares

1/(1+x)(1-x)

[safe knoll]

||its not x^4 -1||

[ripe flax]

then do partial from there

[ripe flax]

thats why i included the -2x^2

[nova canopy]

guys i think im failing maths

[ripe flax]

if it were x^4 -1 i’d say (x^2+1)(x^2-1)

[safe knoll]

ah , i didnt see , srry

[tawny thorn]

you can use the beta function

[ripe flax]

so yea just do partial fractions for 1/(1+x)(1-x)

[tawny thorn]

or if you didnt know it

factor the x² up

[lime plaza]

So @fast rose

Forget there is a integratino from 0 1

divide by x^2 and take x-(1/x) as t

[ripe flax]

theoretically

x^4 + 1 is a diff of 2 sq

[lime plaza]

You can get answer in 4 steps

[ripe flax]

using i

(x^2+i)(x^2-i)

[lime plaza]

Why do you want to complicate

[ripe flax]

and u know what i is: e^pi/2

[ripe flax]

idk

[lime plaza]

Don't

[ripe flax]

😭

[lime plaza]

It works by just dividing by x^2 and taking x-(1/x) as t

yes

[fast rose]

That’s my point how do you divide by x when it runs from x=0

Even when u got it all

[lime plaza]

I just told you

[tawny thorn]

put x-1/x = y

[lime plaza]

Forget there are 1 to 0

Treat it like an indefinite integral

[tawny thorn]

(1+1/x²)dx=dy

[lime plaza]

x^2

[tawny thorn]

then change the limits of the integrals

[ripe flax]

i thought we use u for sub

[tawny thorn]

you can use anything

the last integral can be solved easly

this is the final result

[ripe flax]

inverse tan (-inf) is pi/2?

oh yeah its an asymptote

[tawny thorn]

no its -pi/2

btw

[ripe flax]

shit repeats every pi dont it

[tawny thorn]

there is an other way to solve this integral

by using the Beta function for Euler

but no need

@fast rose i hope that i helped you 👍

[fast rose]

Thanks for the solution and tips y’all, this gonna take a while to to sink in

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[sweet ridge]

What is the least multiple of 2016 whose sum of digits is 2016 ?

[sweet ridge]

number has > 224 digits

and I got upper bound

for number

[pseudo basin]

and what's your upper bound then

[sweet ridge]

which was 225 digit number all 9s except last 4 which would be 8496

so (221 9's)8496

[pseudo basin]

hmm

can we go lower i wonder

,w factorize 2016

[woven radish]

Results provided by WolframAlpha

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

Requested by ann_dec

[sweet ridge]

we probably can

all hail wolfram almighty

[pseudo basin]

let's see

999999 is divisible by 3^2 and 7

so we can move a low digit up by six spots without affecting either digit sum or divisibility as long as we do not touch the last two digits

this improves the upper bound somewhat but does not give much info as to whether we can do even better...

[devout snow]

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[sweet ridge]

<@&286206848099549185>

[devout snow]

@sweet ridge Has your question been resolved?

[sweet ridge]

<@&286206848099549185>

HELP ME DAWGS

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[digital egret]

for $||r||^2$

[woven radish]

allarkvarkk

[digital egret]

do i just do like $x^2+y^2+z^2$?

[woven radish]

allarkvarkk

[digital egret]

like it doesnt have the sub 1 2 or infinity so idk which to do

does that just mean magnitude?

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[sullen island]

what about it ?

[fading hollow]

Is the answer to 1b (1+p)/6 or (4-p)/3 or are they both wrong?

[sullen island]

the answer doesn't depend on p

[fading hollow]

Ok

[sullen island]

how did you compute the probabilities P(X_1 = 1/2/3) ?

@fading hollow

[fading hollow]

I did these two different ways

Sorry I will need to come back in about an hour

[sullen island]

yeah idk how you're getting both of them

you should just get 1/3 for all

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[umbral torrent]

Given AB = BA , A and B both orthoonally diagonalizable, Proof AB is also orthogonally diagonalizable, $(AB)^T = (Q_A^TD_AQ_AQ_B^TD_BQ_D)^T=(Q_B^TD_B^TQ_BQ_A^TD_A^TQ_A)=Q_B^TDQ_BQ_A^TD_AQ_A= BA = AB$
So $(AB)^T=AB$, AB is symmetrical so AB is orthogonal diagonalizable

[woven radish]

Brett

[umbral torrent]

Is this good enough proof

[devout snow]

@umbral torrent Has your question been resolved?

[umbral torrent]

<@&286206848099549185>

[wicked rover]

@umbral torrent yes

[umbral torrent]

Thank you very much!

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[wicked rover]

np

[devout snow]

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[ripe flax]

firstly we can see a log graph

do u know what the generic log graph is shaped like

ok

lets find the asymptotes of a log base 9 x graph first

before doing it with the a and stuff

would u agree that x cannot be 0

now lets add an “a” inside the function

if an a is added inside the function, the whole graph moves left by “a” spaces

so essentially, the vertical asymptote(x=0) is now changed to x=-a

there is a new x intercept as well

if x=1, y=0

that is the x intercept

now move it left a units

then u get the new x intercept (1-a,0)

there is also a y intercept

we gotta make x=0 for it to work

y=log9 (a)

so y is log9 of a

new y intercept (0,log9 a)

so the newly transformed graph has equation y=log9 (x+a), with asymptotes x=-a, x intercepts(1-a,0) and y intercept (0,log9 a)

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[ripe grove]

im lost on q18

[ripe grove]

wait its just cos rule

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[sterile musk]

Why is there a 1/n Infront of the stigma?

[polar chasm]

*sigma

[supple knot]

just a definition

[sterile musk]

Oh my bad

[acoustic leaf]

it makes the definition independent of the number of data points

[polar chasm]

my german is far from perfect, but it seems to be a formula for variance. What the formula does is that it calculates how far each of those points is from the mean and squares it (that's the (x-xmean)^2 part) and then takes the average of all those

and to calculate the average, you need to sum them all up and then divide it by the number of them

[sterile musk]

Ohhh, so is it like calculating the average, and dividing it by the amount of numbers I put in?

[polar chasm]

so it's basically the average square of distance between the mean and the data points

[polar chasm]

yep

it's averaging (x-x̄)^2

for all x

[sterile musk]

Ahh okay I think I get it now. Thanks a lot 🙏🙏

. close

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[agile cypress]

how do i avoid making mistake here

[agile cypress]

2+1+x=6

x+3=6

i divided x+3=6 with 3

when i should just move it to right side

[ripe flax]

u should subtract both sides by 3

so only x would be in the left hand side

[ripe flax]

if u wanna avoid doing that again,

[agile cypress]

why cant i divide it?

[ripe flax]

remember only divide when is more than 1 x involved

[acoustic leaf]

you have to think about what operation you are undoing

• subtraction is the opposite of addition, so to cancel out an addition you have to subtract
• division is the opposite of multiplication, so to cancel out a multiplication you have to divide

[lost shale]

cuz the 3 is pos, it would switch to -3

• will become - - will become +
  x will become / / will become x

[agile cypress]

okay ty

[lost shale]

what grade in?

just asken

[agile cypress]

im not on school anymore

[lost shale]

....

r u trollen?

[agile cypress]

i stopped studying math after grade 9

nahh

[lost shale]

ok

u can close it now

[agile cypress]

.close

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[shut yarrow]

W

[agile cypress]

.reopen

[devout snow]

✅

[agile cypress]

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[wise scarab]

So, I was scrolling back to the start of my ODE book to re-read some content I glossed over since my class switched books

[wise scarab]

and I was wondering if anyone knew what it means for $\phi_0=Id$ and what that circle operator between $\phi_t$ and $\phi_s$ means

[woven radish]

00100000

[wise scarab]

is it like, some sort of convolution?

ah, it should be $\varphi$ whoops.

[woven radish]

00100000

[sullen island]

each phi_t is a function itself

Id is the identity function, the function that does nothing to its input

[wise scarab]

wait... does Id mean it's an identity?

ohhhh

[sullen island]

and circ is composition

[wise scarab]

ok I just realized that lmao

[wise scarab]

oh, I feel so foolish for not realizing that

that makes a lot of sense

thanks!

I feel like usually, the smaller circles are used for composition lol

I thought the bigger circle was something different

I guess now my follow-up question is, is there any more intuitive way of understanding what a "flow" is meant to signify? I have very very little physics experience, so I have absolutely no intuition for any of these things that sort of seem like they come from physics

hmmm... maybe I need to think in linear algebra terms to get some intuition. is there a nice easy matrix that's a flow?

well uhhhhh

the matrix $\phi_0$ would ofc have to be the identity matrix

[woven radish]

00100000

[wise scarab]

but the weird property of... $\phi_{t+s}=\phi_t \circ \phi_s$

how would one make that work in a matrix......

[woven radish]

00100000

[wise scarab]

idt the matrix where $a_{ij}=t$ when $i=j$ and 0 otherwise works lol

[woven radish]

00100000

[sullen island]

well typically the flows you look at in DEs essentially involve all possible trajectories of the DE

and phi_t essentially means "run this DE for t seconds"

[wise scarab]

OH WAIT IM AN IDIOT the textbook has an immediate example 💀

I'm gonna read that

thanks for the help!

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[arctic pulsar]

my eyes hurt hel

[arctic pulsar]

ik they need to have the same intercepted arc to be equal but theres to much angles i cant rlly tell

also im lost in the stone world like senkuu on this one

[ripe flax]

do u know the basic circle theorems

[arctic pulsar]

i mean some yeah but its hard to memorize them all

[ripe flax]

do u wanna do the second one first

[arctic pulsar]

also i finshed this one

[ripe flax]

looks a bit easier

[arctic pulsar]

oh

Yeahxd

[ripe flax]

ok

AC and AD are tangents to the circle

whlile CO and DO are the radiuses

[arctic pulsar]

ye

[ripe flax]

there is one theorem involving them

which is?

this ones prob the easiest to remember

[arctic pulsar]

oh

[ripe flax]

ok write this down then

if a point involves a tangent and a radius, the angle is always 90 degrees

in this case C and D are connected to tangents and radiuses as well

so their angle should be 90 degrees

correct?

[arctic pulsar]

yes

[ripe flax]

ok

now look at quadrilateral ACOD

it looks like a kite cuz it is

[arctic pulsar]

[ripe flax]

recall that angles in a quadrilateral add up to 360

but angle c and angle d are taken with 90 for each

so would it be safe to assume that angle a and angle o add up to 180

[arctic pulsar]

yes

[ripe flax]

lets name angle a x and angle o y

cuz we’re gonna do some stuff to it

[arctic pulsar]

ok

[ripe flax]

now look at quadrilateral BCOD

it forms a rhombus, doesnt it

[arctic pulsar]

yes

[ripe flax]

so angle o (which we called y) should be equal to angle b

ok not sure if you’ve heard this before

but an angle at the centre is twice the angle at the circumference

aka the reflex angle o and the normal angle b

[arctic pulsar]

yea i dont know that

[ripe flax]

since angle o is y

would it make sense for the reflex angle part of o to be 360-y

and since the normal angle b is at the circumeference

the circle theorem states the reflex angle seen is twice the normal angle b

“ an angle at the centre is twice the angle at the circumference”

so, we generate an equation for y.

360-y (the reflex angle o) = 2y (angle b, which we previously proved b=o hence the y)

arrange the y terms to one side

then we have 3y=360

y=120 which is your normal angle o

that should be your answer

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[arctic pulsar]

its right thanks

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[young sedge]

I have a question regarding this task, I am supposed to solve it using partial integration but I just loop around between this and cos(nx)cos(mx)

For the first one that is

[devout snow]

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

[young sedge]

I end up with this, which if I partially integrate it again I just end up with the starting term again…

[lunar harbor]

missing chain rule here

but do ibp on this again - You should notice something

Also, you're missing your dx, so fix that

[young sedge]

okay

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[median patio]

i got x=2 and x=4 but how to find the last one

[drifting mauve]

feels very like lambert function-esque

[versed vault]

It’s a calculator question

Just use your graphing calculator

[median patio]

is there no way to solve it manually

[versed vault]

why would you lol

[visual hazel]

numerical methods

[versed vault]

You have a calculator

[median patio]

well i have a test tmrw

[drifting mauve]

yeah just use a calculator

[median patio]

and she kinda hinted at

this kind of question

[drifting mauve]

too much of a hassle to do it by hand

[versed vault]

New ap exam wouldn’t ever ask

Cords of intersection

It’s a purely algebraic question

So they wouldn’t ask it

It’s the same reason why you don’t have to simplify 1+1 on the exam

It’s an non-calculus question

[median patio]

oh

[versed vault]

You might have to find the x values for bounds, but that would be it

Not to say a teacher wouldn’t put it on an exam

But for the actual exam

They don’t ask anything too algebraically intensive

[median patio]

ty

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[hexed holly]

What formula does my teacher use to get this answer?

[distant helm]

https://en.wikipedia.org/wiki/Quadratic_equation#Discriminant

Be sure to pay more attention to the lesson

[fierce relic]

Root 1 = (-b+rootD)/2a
Root 2 = (-b - rootD)/2a
D = b^2 - 4ac = 0 (in this case)
So Root(1) = Root(2) = -b/2a

[hexed holly]

thanks

[fierce relic]

thou art welcome

[hexed holly]

wait im still confused

what does it mean

[hexed holly]

this is a quadratic right?

[thorn cypress]

yes

[hexed holly]

are we solving it

[thorn cypress]

if you have a quadratic of the form $ax^2 + bx + c$, the discriminant is $b^2 - 4ac$ (it's the part of the quadratic formula that is square rooted)

[woven radish]

AMysteriousStranger

[hexed holly]

oh okay

like in here?

[thorn cypress]

yes

and bc it is square rooted, if it is >0 there are 2 distinct real roots

[hexed holly]

how does using the discriminant only solve it

[thorn cypress]

if it is =0 there is a repeated real root

and if it is <0 there are no real roots

[thorn cypress]

it doesnt solve it, it just tells you how many real roots there are

[hexed holly]

why do i need that to find a solution

[thorn cypress]

your question here just says determine how many solutions, not what the solutions actually are

[hexed holly]

oh okay

how do i know for future how many it will have at each number

[thorn cypress]

if the discriminant is <0, there are no real roots, if it =0, there is a repeated real root, and if it is >0 then there are 2

[hexed holly]

is there only 1 real root at 0? or infinite?

[thorn cypress]

there is one real root, the discriminant would be zero if the quadratic was something like $(x+1)^2$

so x would just equal -1

[woven radish]

AMysteriousStranger

[hexed holly]

okay!

thank you so much!!

[thorn cypress]

np

[hexed holly]

.close

[devout snow]

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[wispy geyser]

did i do everything right here

[wispy geyser]

for the answer would it be squared?

[fierce relic]

What is "in"?

[wispy geyser]

heres the question if u need it im on part c

[wispy geyser]

iches

inches

[fierce relic]

Of course, it will be A square inches or A inch^2

[glossy dew]

you could've done half of base * height to get a perfect 32

since it's a right triangle

but this is ok too i guess

[wispy geyser]

yeah i see what u mean

im jus following a vid my teacher sent and she did exactly this

[fierce relic]

Yeah, and it's pretty easy to identify it as a right angled triangle

[glossy dew]

yes, squared inches

[wispy geyser]

okay got it

[fierce relic]

I think 31.91 inch^2 is incorrect

[wispy geyser]

omg

[fierce relic]

the correct answer is 32

[wispy geyser]

😭

[glossy dew]

close enough

[fierce relic]

no

the actual answer is not a decimal, so the decimal here is wrong

[glossy dew]

i mean

that's alright

[wispy geyser]

i was copying this

[glossy dew]

it's difficult when you are forced to use herons formula

and not the easy way

[wispy geyser]

true yea

[glossy dew]

actual area is 21.6506

spot on

[devout snow]

@wispy geyser Has your question been resolved?

[devout snow]

Channel closed

Closed due to timeout

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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.

[inner ibex]

is j or k the one going up

[ripe flax]

k

goes up

[river pebble]

why is organic chem tutor tripping

[ripe flax]

its related to z

[river pebble]

cuz this is not right lol

[inner ibex]

oh ok

thankss guys

is this right

for 2

this is kind of confusing

why is it 3d now

does AB = 0B - OA

still apply

when its 3d

[glossy dew]

it is