precalculus

[white rapids]

@winter comet I need help

[safe basin]

Dunno if this is what ur asking, as the above said it’s kind of general but if u define an isomorphism of vector spaces $\varphi : \mathbb{C} \mapsto \mathbb{R}^2 $ via $a+bi \mapsto \begin{pmatrix} a \ b \ \end{pmatrix}$, and a basis of {1,i}, then if $V = \mathbb{C}$ there’s smth called the dual V* which is the set of linear forms of V, ie linear maps $V \rightarrow \mathbb{C}$

[winter comet]

can you take the limit as x approaches -infinity and the limit as x approaches infinity?

[white rapids]

They didnt give me any insight on limits

Considering i havent learned it yet

[obsidian monolith]

chipotle

[winter comet]

oh aight

um

[safe basin]

LMAO

[winter comet]

can you...look at a graph?

[white rapids]

Wdym

[paper tartan]

I liked it!

more FA in preuni channels

[winter comet]

i mean it doesn't necesarily show you slant asymptotes but at least you could see if its unbounded n stuff...

hmm

[white rapids]

Graphing this would be a nightmare

Could I do anything with the exponents?

[safe basin]

It started with explaining why lin alg and diff eqs are related

[paper tartan]

[safe basin]

I didn’t even see the channel i was in till like a minute ago lmfao

[paper tartan]

happens

[winter comet]

you could note that the higher power in the numerator means the numerator grows more than the denominator...

[paper tartan]

higher!

[winter comet]

i really don't know the logistics without limits 💀

[winter comet]

XD

[paper tartan]

VV, do you roam every preuni channel + calculus?

I've never seen you anywhere else it seems

[winter comet]

i don't know anything else, really 😂

[white rapids]

What are limits

[paper tartan]

ah, fair enough

[winter comet]

higher by chance do you know how to do this without limits 💀

i forgo

[white rapids]

I have no idea

[winter comet]

bruh

[white rapids]

I dont know what limits are though

I guess as x goes to infinity and negative infinity

🤷‍♂️

[winter comet]

its like a tool to find what things approach i guess :l

hmm

[white rapids]

Can you show an example

[paper tartan]

what're the options they give?

[white rapids]

For both infinity and negative infinity?

[paper tartan]

yeah

[white rapids]

Options: Infinity, 0, 3 & negative infinity

[winter comet]

oh maybe

[paper tartan]

right

[winter comet]

plug in bigger numbers

[white rapids]

I wanna say it's both 3 but I dont know the reason behind it

[winter comet]

plug in big numbers for x

[paper tartan]

think about what the division of these two polynomials means

in the numerator, we have a cubic

in the denominator, we have a linear function

[white rapids]

yes

[paper tartan]

cubics outgrow linear functions

by a lot

[winter comet]

by a non-constant amount...

ig?

[paper tartan]

as x goes to infinity, we only have to pay attention to the first term of each polynomial

[paper tartan]

yes

[paper tartan]

because everything else becomes tiny compared to the leading terms

you can formalize this with limits, but if you haven’t seen them before, this is the best we can do

[white rapids]

Sooo

Do I do long division

[paper tartan]

let’s do x -> infinity first

[paper tartan]

no need

[white rapids]

Okey

[paper tartan]

it’s easier than that

let’s do some simplifications to help us think about this here

[winter comet]

i mean you could technically envision that a cubic function divided by a linear function would relatively just be an ugly quadratic...

[safe basin]

Does it not mirror the end behavior of the quotient of the leading terms?

[paper tartan]

we have 6x^3 - x^2 + 7 in the numerator, and 2x + 5 in the denominator

[paper tartan]

yes, it does

[safe basin]

Yea

[white rapids]

I dont know the logic behind this

[paper tartan]

I’m not sure if you’ve seen this trick before, but try factoring x^3 out of the numerator and x out of the denominator

just try it

[white rapids]

Alright

[winter comet]

basically doing limits at this point 🗿

[paper tartan]

shhh

[white rapids]

So I 6x^2/2 = 3x^2

[paper tartan]

that doesn’t look right

when we factor x^3 out of the numerator, we should still have three terms there

[white rapids]

I thought you meant with the highest leading terms

[paper tartan]

ehh, we’ll get there

for now, just factor those out like I said

what do we get?

[white rapids]

(6x^2 - x + 7/x)/ 2 + 5/x

?

[paper tartan]

right, but you’re missing the terms factored out

keep em there

we’ll need them

[paper tartan]

oh wait no

you factored x out of the numerator

we should factor x^3

the reason will become apparent soon enough

[white rapids]

The numerator is all together

[paper tartan]

yes…

[white rapids]

Yes

[winter comet]

divide the entire numerator by x^3, and divide the entire denominator by x^3

[white rapids]

Ohhh

Ok i see now

Give me just a sec

[paper tartan]

we probably don’t want the denominator to have a x^3 factored out

[winter comet]

uh

[white rapids]

(6 - x + 7/x^3) /

Im very confused here

[paper tartan]

ah, this is not quite it

do you remember how to factor?

[white rapids]

You cant factor out x^3 from the entire numerator

With a constant like 7

[paper tartan]

you can

just divide 7 by x^3 too

you can always do this factoring trick

there’s nothing stopping you

[white rapids]

Alright...

[paper tartan]

I’m still waiting to show you the magic lol

[white rapids]

Just gibe me a second

[paper tartan]

[white rapids]

I dont know what im doing here

The denominator is x^3 (2/x^2 + 5/x^3)?

[paper tartan]

I am so sorry, but I can’t read this

[white rapids]

Numerator i think

[white rapids]

Apologies

[paper tartan]

it should be x^3(6 - 1/x + 7/x^3)

oh wait

you said denom

my bad

[white rapids]

🤷‍♂️

[paper tartan]

I wanted a factor of x to be taken out of the denom

not x^3

this can work, but it’s harder

[white rapids]

I mean i

Am i suppose to factor out the same factors from both numerator and denominator?

[paper tartan]

you do not have to

in fact, it’s often better not to

because doing things equally often gets you nowhere

[white rapids]

@winter comet

[winter comet]

yo

[white rapids]

Uhhh

[winter comet]

uh

[white rapids]

Yo how would you do this

[paper tartan]

what I wanted was x^3(6 - 1/x + 7/x^3)/x(2 + 5/x)

[winter comet]

i would use the infinity rule of limits 😭

[paper tartan]

I’m trying to get there!

[winter comet]

higher knows the logistics better lol

[winter comet]

i know 😭

[white rapids]

Alr lets just stick to this i guess

[paper tartan]

okay

now, we have a x^3 in the numerator, and an x in the denominator

what can we do with that?

immediately

[white rapids]

x^2

[paper tartan]

nice

[white rapids]

Quotient rule from exponent rules

yes

[paper tartan]

we cancel, so we have a x^2 in the numerator

good good

okay, here’s where the magic begins

[white rapids]

Yeh

[paper tartan]

think about what happens if we take a really really big number in the place of x

[white rapids]

It becomes a big number

[paper tartan]

how so?

[white rapids]

Because...

It just gets bigger

If we're squaring it

[paper tartan]

what happens to the other terms?

we currently have $\frac{x^2(6 - \frac{1}{x} + \frac{7}{x^3})}{2 + \frac{5}{x}}$

[white rapids]

-1/x

[paper tartan]

ah

why

[obsidian monolith]

higher!

[paper tartan]

right

if we choose x to be a really really big number

what happens to -1/x?

[white rapids]

I mean if we look at -1/x but itself it'll just become smaller

Because x is replacing the denominator

[paper tartan]

right!

so if we take x to be infinitely large, what happens to -1/x?

if it keeps getting smaller and smaller, as we take larger and larger values of x, what happens in the limit as x goes to infinity?

[white rapids]

It gets closer to negative

negative infinity

[paper tartan]

not quite

1/10000 is a very small number

but 1/100000000000000000000000 is even smaller

so what about 1/10000000000000000000000000000000000000000000000000000000000000000?

it's a really small number, right?

[winter comet]

you need a few more zeros

thats too big :<

[white rapids]

Yeah

[safe basin]

just a few

[paper tartan]

so as x gets larger, 1/x becomes very very close to 0

in the limit, 1/x is 0

does that make sense?

if we choose x to be infinitely large, 1/x must be infinitely small

infinitely small, i.e. 0

[white rapids]

I think so ye

[paper tartan]

right, so in the limit, we have $\frac{x^2(6 - 0 + \frac{7}{x^3})}{2 + \frac{5}{x}}$

[obsidian monolith]

higher!

[paper tartan]

I just replaced the 1/x with a 0

what about the 7/x^3 and 5/x terms?

what should they be?

[white rapids]

Uhh

For 7/x^3,

[paper tartan]

what happens if we pick a super large x?

[white rapids]

I mean for both 7/x^3 and 5/x it should be like 1/x right?

[paper tartan]

so what do they become?

[white rapids]

It just gets closer to 0

[paper tartan]

yes!

[white rapids]

It's a bit confusing to comprehend

[paper tartan]

$\frac{x^2(6 - 0 + 0)}{2 + 0}$

[obsidian monolith]

higher!

[paper tartan]

limits can be a bit confusing to think about at first haha

you need to think about what happens at infinity

it can be hard!

[paper tartan]

so this is where we stand, agreed?

[white rapids]

I thought you could do this mathematically

[paper tartan]

you can!

[white rapids]

ye

[paper tartan]

I want to emphasize the intuition here though

we're just thinking about what happens when x gets larger and larger

now we see that our earlier work with factoring is paying off

if we didn't factor, we couldn't conclude that 1/x, 7/x^3, or 5/x would go to 0 as x goes to infinity

the fact that we did factor showed us explicitly that it really was only the leading terms that mattered as we took larger and larger x!

[white rapids]

Ye

[paper tartan]

everything else gets outpaced, so to speak

[paper tartan]

now, we simplify this to $\frac{6x^2}{2}$

[obsidian monolith]

higher!

[paper tartan]

which is just 3x^2, yeah?

[white rapids]

Yep

[paper tartan]

so that entire function at the start, that division of polynomials which was super ugly

has the exact same limiting behaviour as the function 3x^2, when x goes to infinity

at infinity, there is no difference between your original function and 3x^2

isn't that amazing?

[white rapids]

Uhhh

I'll be honest

[paper tartan]

ahaha

[white rapids]

Im still a little confused

[paper tartan]

it's alright if you didn't follow all of it!

do tell me which parts you were confused by

I can try to reexplain

[white rapids]

So from here,

If x^2 is factored out of the numerator, and everything else is in the parentheses

Shouldnt as x increases, y increases? Because x^2 is being multiplied with everything inside

[paper tartan]

that's correct!

[white rapids]

Right

[paper tartan]

I didn't actually yet finish the problem!

[white rapids]

Oh

[paper tartan]

you're completing it now

by making that observation

[white rapids]

🤷‍♂️

[paper tartan]

at infinity, our original function behaves precisely like 3x^2

what does y = 3x^2 do as x goes to infinity?

[white rapids]

Y increases

[paper tartan]

mhm, does it ever stop?

[white rapids]

Nope

[paper tartan]

so what is the end behaviour of our original function?

what value does it achieve?

it can't be finite, because it just keeps increasing

[white rapids]

as x increases, y also increases

[paper tartan]

so it must be...

[white rapids]

infinite

[paper tartan]

ding ding ding!

a winner is you!

[white rapids]

Uhhh

What if y is negative infinity

How would the happen if we just looked at 3x^2

If we plug in a bigger negative number the negative sign just cancels out

[paper tartan]

correct

[white rapids]

Oh crap

Other way around as x goes to negative infinity

[paper tartan]

if x is negative, x^3 is also negative

[white rapids]

sorry

[paper tartan]

do you agree with this?

[white rapids]

Yeah

[paper tartan]

so our heuristic is this

we've already seen that as x gets larger, the leading terms of the numerator and denominator dominate the other terms

so we should focus our attention on those leading terms

[white rapids]

Yes

[paper tartan]

we've also just seen that x^3 is negative if x is negative, so that means that as the numerator and denominators grow more negative (as x gets larger), their negative signs should always cancel out

[white rapids]

Right

[paper tartan]

we've also just seen that the numerator actually dominates the denominator, since cubics beat linear functions in a race to infinity

so our hope (and what we should think the answer is) is that the limit as x goes to negative infinity is also positive infinity

does that make any sense?

I dunno if I'm just yapping here

[white rapids]

I think other way around

[paper tartan]

what do you think?

hello VV

[white rapids]

Ohh wait

Nvm I see it now

[paper tartan]

[winter comet]

the other way around XD

[winter comet]

yoo wsp

[white rapids]

Really?

[winter comet]

no i mean

[paper tartan]

@white rapids alright, we can do the same analysis as before

you can go through it if you'd like, but we'd end up with 3x^2 all the same

[winter comet]

he said "our hope... is that the limit as x goes to negative infinity is also positive infinity"
the other way around means our hope is that the limit as x goes to positive infinity is also negative infinity i mean it just doesnt make sense to me so i was laughing 💀

[paper tartan]

so I'll just cut to the chase lol

[white rapids]

Ok

[paper tartan]

if x is negative infinity, what is y = 3x^2?

[white rapids]

Uh

Undefined

[paper tartan]

hmm

what if x was a super big negative number, like x = -99999999

[white rapids]

y = 3x^2
√x^2 = √negative infinity

[paper tartan]

what would the sign of y be?

[white rapids]

Oh crap

[paper tartan]

there's no need to invoke square roots!

[white rapids]

Sorry I thought you said when y = negative inf

So i just plugged it

[paper tartan]

ah

no worries

[white rapids]

Gets closer to infinity

[paper tartan]

!

so at x = negative infinity, y = ??

[white rapids]

infinity

[paper tartan]

which is exactly what we thought it'd be!

so that is the answer to your problem

both at x = inf and x = -inf, y = infinity

there's just one more thing I want you to take away from all this

[paper tartan]

did you notice how we got this after doing all that factoring, cancelling, and limit behaviour shenanigans?

[white rapids]

Yea

[paper tartan]

take a close look at your original function too

[white rapids]

Honestly we just had to look at the biggest leading terms

[paper tartan]

exactly!

if we ignore everything but the leading terms of the original function

we get the same thing

[white rapids]

Right

[paper tartan]

that's a very important fact you'll use a lot when you do calculus

the growth of polynomial functions is dictated solely by their leading terms, and their coefficients

hello chipotle

[safe basin]

That was very entertaining

Bros graduated out of this channel now

[paper tartan]

whoops

@white rapids take some more time to think about this stuff!

math only gets more fun from here on out

[paper tartan]

did I do a bad job

[safe basin]

No it was really good

[paper tartan]

I felt like I didn't explain too well idk

[safe basin]

Nah he filled in the gaps properly so you def got your point across

[paper tartan]

😌

[white rapids]

The factoring was a little extra

[paper tartan]

it might be time for me to retire from this channel too

[paper tartan]

the reason I wanted you to do that

is because you cannot argue by intuition when you formalize this using limits

that factoring step becomes necessary

[safe basin]

I like checking these cause it’s fun to randomly answer stuff

[paper tartan]

with limits, there's you don't really argue "the leading terms outgrow everyone else"

you kind of have to use limit laws + some well known limits, like what happens when x goes to infinity for 1/x

[winter comet]

i like checking because sometimes i find stuff that i do not know

or it lets me review stuff im...not too good at

n stuff

[winter comet]

and then i start making things up

[paper tartan]

mhm

[winter comet]

nah jk 😂

[paper tartan]

sometimes, a bit of imagination is all that's needed!

[winter comet]

fr

[safe basin]

Fair

[paper tartan]

the preuni channels are a bit too stressful for me

I feel like I suck at explaining these concepts

and I'm being watched by the world or smth

I shall stick to my #linear-algebra

[winter comet]

bro is pressured

[safe basin]

Linear algebra is so goated

[winter comet]

linear algebra doesnt look bad

lol

i mean it looks bad but

maybe fun

maybe

[safe basin]

I’m trynna learn diff geo rn

[winter comet]

differential geometry?

[safe basin]

Yeah

[winter comet]

bruh

[paper tartan]

my goal is to get to DG!

[safe basin]

It’s so interesting

[winter comet]

geometry itself is

hard

for me

💀

[paper tartan]

DG/DT/AT/FA/math phys all look so interesting

I wanna study it all

oh, and GR/CM

in physics

[safe basin]

If you know linear algebra and calc 3, and a bit of general topology dg basically picks up from there

[winter comet]

differential geometry / data topology / astro topology / father algebra ?

[paper tartan]

I'm studying multivariate analysis via Spivak and point set topology via Lee rn

[safe basin]

Most regular calc 3 courses don’t cover differential forms from what I’ve seen

[paper tartan]

lmfao

[winter comet]

my best guesses

[safe basin]

Yeah Slovak is good

Slovak

[winter comet]

😂

[safe basin]

Spivak

[winter comet]

ah yes, slovak

my favorite

[safe basin]

LMAO

[paper tartan]

differential geometry, differential topology, algebraic topology, functional analysis, mathematical physics, general relativity, classical mechanics

it's a lot

[winter comet]

DUDE i could have guessed that im so dumb

except maybe not functional analysis

[paper tartan]

I'm almost certainly never going to learn it all, but I also haven't decided my focuses too much yet

hello Mqnic!

[hallow tide]

nice precalculus discussion on go d

[winter comet]

dude i literally said algebra for A, why did i do that 💀

[paper tartan]

fellow Battle Cats player

[hallow tide]

hello higher

[paper tartan]

there was functional analysis earlier

you missed it

[hallow tide]

amazing

[paper tartan]

math and physics at the moment

[hallow tide]

yoS

will get there eventually

[paper tartan]

that pfp picture 😵‍💫

[safe basin]

W

[hallow tide]

dots

[hallow tide]

indeed a W

[paper tartan]

what are you majoring in Mqnic

[hallow tide]

i have no idea!

[paper tartan]

you're not an UG?

[hallow tide]

no

[paper tartan]

ah

[hallow tide]

😔

[paper tartan]

I could've sworn you were

[hallow tide]

[paper tartan]

wait

I need to join that server

okay done

now, I will stop talking here and let business proceed as usual

this has gone very off topic

[safe basin]

LMAO

I just imagine someone that’s gonna take precal next year randomly landing on one of these discussions and thinking that’s what it is

[winter comet]

no math has been done to confuse

only words 😔

[safe basin]

Lmfao

[winter comet]

also if anyone wants to head to calculus channel i have problem probably simple seperable differential equation but idk how to set it up ;-;

unless everyone gone 😔

[proven night]

Hello

How do i solve this equation without graphing

Photomath is showing only graphing solution

[chrome ether]

well don't rely on photomath

[proven night]

I turned log2(x) into log(x)/log(2), but it didn't help at all

[proven night]

Yes, sometimes it doesn't solve a problem, instead it draws a graph

[safe basin]

expand left hand side

[proven night]

It's gonna be log(x(x^2 + 1)) = log(x) + log(x^2 + 1), right?

[willow skiff]

yes

[willow skiff]

ay, you're getting somewhere

remember if the logs are to the same base, then log a = log b implies a = b

So you can bring the exponent inside the logarithm with (1 - 1/log(2)) log(x)

or actually wait

[willow skiff]

just guess from this what x is

hint: x is a positive integer

[proven night]

Okay

[willow skiff]

rearranging gives you log(10) log(x) = log(2) log(x^3 + x), log(10) = 1

[chrome ether]

yea sometimes you do have to suss out solutions "by inspection"

[willow skiff]

by inspection yeah

honestly WA helped me realise you can do it by inspection

[proven night]

I decided to expand left hand side and then change the base of right hand side logarithm

Than i let the numbers to a common factor

I understand that the numerator is equal to 0

I dont know how to factor the numerator

[summer ruin]

instead of doing that, factor the expression before that

[proven night]

Which expression?

[summer ruin]

ah nvm, I don't think that brings you anywhere anyway

$\log (x) + \log (x^2+1) - \frac{\log (x)}{\log (2)} = \left( 1 - \frac{1}{\log (2)} \right) \log (x) + \log (x^2+1)$

[obsidian monolith]

Transparent Elemental

[willow skiff]

sorry to have to interrupt you but yeah honestly you'd be better if you just considered this

inspection

[proven night]

what is (1 - 1/log(2))? Is it log(x)/log(2)?

[summer ruin]

I just factored common factor log(x)

[proven night]

Got it

Okay

But i still dont understand what i should do next

[thick herald]

i was learning limts .. but i got confused a bit in epsilon dela defination.. can someone explain siplified

[summer ruin]

no matter how small error tolerance you choose, there will always be a neighborhood of inputs for which the error between function value and it's limit is smaller than your error tolerance

[thick herald]

ok .. so how can i write it ? this is just a statement

[summer ruin]

isn't this enough to understand the definition in a simplified manner?

[thick herald]

uhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhhh

uhh i was actually tring to find proof of Sum Law there they wanted toproove this i dint understnd this

[summer ruin]

I just explained the same concept in a human language

which is what you asked for, it's simpler than reading symbols

[safe basin]

which part are u confused on

[thick herald]

yha.. i got what u ment but i am confused in how to use it

[summer ruin]

what is h(x) here? the sum of two functions?

[thick herald]

yes

M is limit of one and L of other

[summer ruin]

well then you know that $h(x) = f(x) + g(x)$ and $|f(x)-L| < \varepsilon$ and $|g(x)-M| < \varepsilon$, also $|h(x) - (L+M)| = |f(x) + g(x) - L - M|$

[obsidian monolith]

Transparent Elemental

[thick herald]

i dint get |f(x) - L |<E and things after that 😦

[summer ruin]

that is what is meant by L is the limit of f(x)

(under the same initial assumptions you had in your screenshot)

[thick herald]

but L is limt when f(x) approces c

[summer ruin]

yes

you probably meant x approaches c

[thick herald]

yes

so what happens when we subtract limit from function ?

it gives us nothing

[summer ruin]

you know that two things are less than epsilon and you're trying to prove another thing is also less than epsilon

do you know what is triangle inequality?

[thick herald]

what is epsilon .. the nieghbouring values of f(c) in range of f(c- delta/c+delta)?

[thick herald]

uh no

[summer ruin]

|a+b| <= |a| + |b|

[thick herald]

yes

[summer ruin]

doesn't |f(x) - L + g(x) - M| look familiar

[thick herald]

no they seem same (function- limit)

[summer ruin]

it's a sum of two things

[thick herald]

but why are we subtractimg and summing them ?

[summer ruin]

we're trying to prove |h(x) - (L+M)| < epsilon

[thick herald]

i dint uderstand this

[summer ruin]

here

[thick herald]

yes but i did not understand what it means

[summer ruin]

the error tolerance is epsilon, the limit is L+M, the delta-neighborhood are all points x that are delta distance away from c

|x-c| is the distance between x and c if you didn't know that

[thick herald]

whats x ? x are infinite values

[summer ruin]

any real number delta-distance away from c

[thick herald]

isint distance delta then ?

[summer ruin]

that's what I said

[thick herald]

x-c is not delta

[summer ruin]

it isn't

[thick herald]

?

[thick herald]

.

[summer ruin]

the absolute value of a difference of two things is the distance between these two things

where am I saying that x - c is delta?

all I'm saying is the distance between x and c is called delta

[thick herald]

but x is not a point

[summer ruin]

it is a point

[thick herald]

??

[summer ruin]

you consider all points x that are delta distance away from c

why wouldn't each individual x be a point?

[thick herald]

oh yes .. good point

[summer ruin]

there could be many points delta distance away from c, that's fine

[thick herald]

so x < or = c+ delta

or c-delta

[summer ruin]

c - delta < x < c + delta

[thick herald]

it can also be equal to one value

[summer ruin]

we don't require that

[thick herald]

k

[summer ruin]

if we were, then it could've been that the definition failed when x = c for some functions we wished it should hold

especially when the function is not defined at c

[thick herald]

uh yha

then ?

[summer ruin]

given such and such conditions on x, we want to show that |f(x) - L + g(x) - M| < epsilon

[summer ruin]

it's easy to do using this inequality

[thick herald]

wait .. lemme understand this .

i can say |f(x) - L|< epsilon and |g(x)-M| < epsilon but not when tehy are added ..

ohh got it

we need to prove .. it now i get it ..

THANKS A LOT

[stiff sluice]

,, |a|^2 -2|ab| + |b|^2 \leq a^2 -2ab + b^2

[obsidian monolith]

938c2cc0dcc05f2b68c4287040cfcf71

[stiff sluice]

how do I show that this holds for a and b in R

[halcyon yew]

well |a|^2 = a^2 and |b|^2 = b^2

[stiff sluice]

indeed

[halcyon yew]

and then -|2ab| is negative

so either it's equal to or less than -2ab

[stiff sluice]

yes

-2|ab| <= -2ab

2|ab| >= 2ab

okay thank you panda, as always

[rain drum]

hi. I am new to the channel

[paper tartan]

hello!

do you have a question?

[rain drum]

not right now but i have recently started relations and functions so if i would hv one i would definately be asking

anyways...some tips while getting this area covered?

[winter comet]

could we have just skipped the entire step by finding t when e^(-kt) = 0.05 ? because after 95% or 0.95 is gone, 0.05 is left? or is this just a coincidence

😭

like no matter how much material there is, it would take the same amount of time to decay by 95%...or is that not true

:l

nah but yeah they just multiply 0.05 by 10, and then divide by 10. no matter what the initial amount is, its just gonna be 0.05. i swear they just tryna confuse me ;-;

[ebon ridge]

How would one go about determining an answer?

[winter comet]

those are the only two options?

:l

[ebon ridge]

Yeah, I'm not sure how arrive at a conclusion though

[winter comet]

💀

[shadow summit]

Wait, I don’t like this question.

[ebon ridge]

😭

[winter comet]

let me see

i can think of a case where neither of them are true

:l

probably

[shadow summit]

Yeah it’s b

[winter comet]

how u know

[ebon ridge]

Please teach us 🙏😭

[river drift]

if we have $\lim_{x \to a} \frac{f(x)}{g(x)}$ and we know $\lim_{x \to a} g(x) = 0$ then the sided limits are $\pm \infty$ unless $f(x)$ also goes to 0

[obsidian monolith]

cloud

[ebon ridge]

Thanks

I'm gonna try to make sense of this

[rain scroll]

$\trig$

[obsidian monolith]

Deathly_Dunce

$\trig$
```Compilation error:```! Undefined control sequence.
<recently read> \trig 
                      
l.49 $\trig
           $
The control sequence at the end of the top line
of your error message was never \def'ed. If you have
misspelled it (e.g., `\hobx'), type `I' and the correct
spelling (e.g., `I\hbox'). Otherwise just continue,
and I'll forget about whatever was undefined.```

[rain scroll]

Can i use the pinned message on my test

[winter comet]

is your test open internet?

[rain scroll]

No ima print it on a paper and bring that

[winter comet]

if it allows a cheat sheet

then i dont see why not

[rain scroll]

Heheh of corse

[winter comet]

lole

[shadow summit]

There’s a few ways. The way I did it was using algebra.

let limit x-> 0 (f(x))=L

let limit x -> (x)=M

L/M=1

L=M

Therefor:

,,\lim_{x\to 0}f(x)=\lim_{x\to0}x

[obsidian monolith]

TheLord26

[shadow summit]

,,\lim_{x\to 0}f(x)=0

[obsidian monolith]

TheLord26

[elfin patrol]

I have an 86% in precalculus right now, if I somehow fail my final exam worth 15%, will I still finish with a passing mark?

[vapid plaza]

Depends on

• how many marks you consider as pass and
• how many marks you get in the final (from 0% to almost passing)

[elfin patrol]

50% is a passing mark

[shadow summit]

Just do 0.86•0.85+x•0.15

[vapid plaza]

Then even if you get zero in final you can still pass 😅

[shadow summit]

Yeah

[elfin patrol]

Oh thank god

So then I should be okay in all my other classes

Wait

No

[shadow summit]

But why would you fail if you got 86%?

[elfin patrol]

Biology

I'm so bad at exams

Biology is worth 20% but I have a 90 something

[shadow summit]

Well try your best

[elfin patrol]

I have 4 final exams and I'm lowkey stressing

[shadow summit]

Dw

Are they important exams? Do they affect what university you get into?

[elfin patrol]

They are the final exams so they're all worth 15% of my grade, with the exception of biology which is 20% because it's a grade 12 course

I'm looking at probably a finishing mark of 85% if I wanna be competitive

[shadow summit]

Ig start studying then

[elfin patrol]

I need a 70% average to be considered

I just looked and got two different answers for the acceptance rate

Which was 65% and 91% which are greatly different

My grades are all 86% and higher

3 classes in the low 90s and 2 classes in the high 80s

[shadow summit]

Just study.

I’m studying for my finals in ~4 months

[elfin patrol]

My finals start next Thursday

Final on Thursday, none on Friday due to the fact it's an indigenous recognition day, and then the last three from Monday to Wednesday

I can't believe it's almost summer already

[teal osprey]

can @everyone anyone teach me continuity

[worn tendon]

lets all(211972 people) teach this guy continuity

[quartz inlet]

💀

[elder sky]

can some1

teach limits

and stuf

[agile sparrow]

I'm down

[winter comet]

everyone, all at once, all together

[noble basalt]

what's the primitive integral of sin³(x)?

got a test tomorrow and I just started studying XD

[winter comet]

try int(1-cos^2(x))sin(x) dx

[winter comet]

what for?\

[noble basalt]

some basic integral stuff and trigonometry

[noble basalt]

gotcha

[winter comet]

they'd make you integrate sin^3(x) tho?

[noble basalt]

yea

[winter comet]

weird

[noble basalt]

also, what do I do with xe^2x?

*integrating again

[round current]

yo guys, how do i answer this?

it says im wrong with the answer

i think im on the last step

[uncut mulch]

be careful as there are values where the expression is undefined

[round current]

ohh

[winter comet]

integration by parts

[round current]

so it will be like this?

-4 is undefined

[noble basalt]

sounds good, how do I do that though?

[round current]

thanks

it was right

[mellow leaf]

it's like the product rule backwards. But your sure its going to be on the test, if you never heard about it in school?

[noble basalt]

I'm not too familiar with the terms, I'm getting math classes in another language

but I do know product rule

[mellow leaf]

oh which language?

[noble basalt]

Dutch

[mellow leaf]

welp have you heard about partial integration? Its a synonym

[noble basalt]

I vaguely remember, yes

just checked, need to know that as well for my test 💀

[mellow leaf]

ohhhhh

then you need to know substitution as well?

[noble basalt]

Yea

I might be cooked

[mellow leaf]

well it's just the product rule backwards, so you could learn the basics today, there are some great videos about this topic

[noble basalt]

I'll have a look, ty

[mellow leaf]

no problem

[jovial hare]

how to prove that the recurrent series is convergent an+1 = 2an/(n+2), a1 = 2?

[viscid thistle]

can u put that in latex?

[jovial hare]

Idk how but there is a pic. Asking for the first one.

[tepid quarry]

,rotate

[proven night]

How to solve the equation 3^x + x = 11

[proven night]

Without trial and error method

Just pure algebra

Maybe i should apply W Lambert function?

[whole void]

but the LHS is monotonously increasing so u should only expect one solution so why not guess and check?

[round current]

i dont know where i messed up

wait holdup

it has to be greater than 0 right

[proven night]

Okay

[round current]

nvm i have no clue where i messed up still

[shadow summit]

Yes

[buoyant elm]

8/5 is wrong

wait

[buoyant elm]

it never goes below zero

[buoyant elm]

try all real number

[round current]

thanks

it worked

[viscid thistle]

!da2a

[tender quest]

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/

[teal osprey]

@everyone if x approaches infinity, then how can its limit be infintely close to zero

[river drift]

context needed

also please don't attempt mass pings

[teal osprey]

ok, but can you explain, i just came across it, it was written somewhere

lim_(x->∞) f(x) ≈ 0

[teal osprey]

@cloud pls

[teal osprey]

@river drift

[river drift]

which function is f(x) ?

[teal osprey]

it is being used to show limit of x

[river drift]

if you just want an example of a function that does that then consider [ \lim_{x \to \infty} \frac 1x = 0 ]

[obsidian monolith]

cloud

[river drift]

,, \lim_{x \to \infty} \frac 1x = 0

[obsidian monolith]

cloud

[river drift]

weird

[teal osprey]

I actually want to understand theory or cocnept behind limit of x being 0

[river drift]

consider this example: if we divide by a very large number, we get a very small number

[teal osprey]

yeah, i get it, thanks

1/infinty is zero, so approaching infinty will mean somewhat really closer to zero

so, is there anyone who knows about continuity and limits in deep, to teach a person, who just started studying about limits

[teal osprey]

<@&1022875670466023534>

@cunning prawn

[cunning prawn]

Hello

[teal osprey]

@cunning prawn there are different channels for your need, if you want to share memes and facts or discuss random math topics, go to #discussion , #serious-discussion -2 or #math-discussion

if you have doubts for specific topics go to #precalculus #prealg-and-algebra #geometry-and-trigonometry and #competition-math

@cunning prawn ?

[mellow leaf]

Hello I came across some exercises and one stood out to me

does anyone know why they solved it like that? It seems a bit overcomplicated, because I solved it like this and it seems fine

[willow skiff]

That's very weird

Did they use your choice of u and dv somewhere else on the page? Maybe it's to show the choice of u and c doesn't matter

u and dv

[mellow leaf]

No they didn't

but you could be right they may just show of some tricks to solve it in other ways

[elder sky]

some1 teach limiyts'

[chrome ether]

!da2a

[tender quest]

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/

[chrome ether]

you're gonna have to be way more specific

the saying goes "garbage in garbage out", the answer you get is only gonna be as good as the question you ask

[elder sky]

👍

bro

idk what it is

which is why i asked a gerneal

[chrome ether]

google "limits", do some reading first

not everything is gonna be handed to you

[elder sky]

👍

mb sir

[errant girder]

i need pdf for calculus books

basic calculus

differentiation

and stuff

[fading monolith]

Spivak or Apostol

[winter comet]

😭

[fading monolith]

?

[winter comet]

cough nothing

[fading monolith]

Mmmm

If you have some recommendations too just said them

[crystal plaza]

I mean in fairness, Spivak would not be my recommendation for someone who wants "basic calculus"

[fading monolith]

Thats a better explanation than just an emoji

[winter comet]

I am not tryna dissuade anyone, the emoji was just to convey my emotion for what you just said 💀

[fading monolith]

Then just express with words and not just an emoji

Is annoying

[crystal plaza]

For basic calculus, probably Stewart?

Though this seems like more of a #book-recommendations or #calculus question than something for #precalculus

[shadow summit]

Thomas Calculus is also pretty good.

[dire aspen]

Spivak’s manifolds is a great beginner one, all time favourite

[paper tartan]

as someone reading Calculus on Manifolds at the moment, I can confirm that it is, indeed, a very solid introduction to basic calculus.

[solid hollow]

stewart calculus was very good for self studying in my case

[shadow ether]

pauls online notes do be carryin

[simple moss]

what does ur name mean

[viscid thistle]

Arabic

[flat cape]

Solve for this with the steps

Find a and b

[viscid thistle]

use SOHCAHTOA

[flat cape]

I do not know what sin cos tan really represrnt

[simple moss]

I don't know the terms in English, But here sin30 = a/10

and cos30 = b/10

[flat cape]

So what's value of a

@simple moss

[simple moss]

10*sin30

= 5 i think

[flat cape]

Got it

[simple moss]

I speak Arabic and idk what it means 💀😭

[viscid thistle]

it says

"Rachit is cooler"

[simple moss]

lowkey i think it does

[viscid thistle]

im getting no commissions at all 😭

[simple moss]

wait lemme give u one

lemme find the pic

[viscid thistle]

what pic

[simple moss]

I made this integral cuz it looks funny

[viscid thistle]

calculus

how tf

am i supposed to solve it

[daring tapir]

doesnt look bad

[viscid thistle]

its unsolvable

[daring tapir]

its easily solvable

[simple moss]

yeah its surprisingly easy

[viscid thistle]

whats the

[simple moss]

the answer is one term

[daring tapir]

facts lmao

[viscid thistle]

x^xlogx

[daring tapir]

nah

[simple moss]

finally someone who finds this integral funny 😭

[daring tapir]

do you know IBP?

[simple moss]

its x^x

[daring tapir]

xD

[viscid thistle]

what

[viscid thistle]

yea

thats just 1

[daring tapir]

well yea then you cant do it

differentiate x^x and you arrive at this and just say you found it by hit and trial

[viscid thistle]

lol

[daring tapir]

no

[viscid thistle]

but did i ask

[simple moss]

whats ibp

integration by parts?

[daring tapir]

integration by parts

[simple moss]

oh

[viscid thistle]

yea

[daring tapir]

shoulda

[simple moss]

i don't do math in English so its kinda confusing

[viscid thistle]

i do math in math

[simple moss]

real

relatable

[daring tapir]

i do maths in binary

[simple moss]

bro is on another level

[daring tapir]

and sometimes morse code

[viscid thistle]

01010101010101

[simple moss]

I do it in braille

😎

[daring tapir]

amateurs
i do it in sign language

[viscid thistle]

01111001 01101111 01110101 01110010 00100000 01101101 01101111 01101101

[viscid thistle]

I do it in air

[simple moss]

woah this is kinda offensive ngl

[viscid thistle]

lol

did u

?

[daring tapir]

,w 01111001 01101111 01110101 01110010 00100000 01101101 01101111 01101101

[viscid thistle]

lmfao

[daring tapir]

lmao

,w 01111001 01101111 01110101 01110010 00100000 01101101 01101111 01101101 from binary

[viscid thistle]

nice dm

cmd

[simple moss]

I didn't translate it but I sensed bad aura

[viscid thistle]

I smell it

[simple moss]

this might be u

[viscid thistle]

anyone wants nitro

[daring tapir]

my man wwent through that much effort for this

[viscid thistle]

my cigar

[simple moss]

since when is discord generous

or is it planting nitro addiction seeds

[viscid thistle]

they want u to get nitro for 2 weeks

then u get used to nitro

[daring tapir]

you really hate privacy huh 💀 bro just showed half his frnd list

[viscid thistle]

then u crave nitro

[viscid thistle]

do i care?

no

[daring tapir]

🤡 have fun being doxxed

[viscid thistle]

who gonna dox me?

you

?

no one can

[simple moss]

I don't think anyone can be doxxed from a friend list 😭🙏👺

[viscid thistle]

he isnt good at tech i think

[daring tapir]

nah im too laxy

someone else probably

lmao

[daring tapir]

🤡 who knows?

i was being sarcastic anyways

[simple moss]

real

[simple moss]

wait I thought of a way

someone can msg one of thr friends asking for info

[viscid thistle]

lmao

they arent even my friend

[simple moss]

but this requires thr friend being an opp secretly

[viscid thistle]

its just dms

[daring tapir]

but its like how stupid will the fnrds be lmao

[simple moss]

oh welp this might make it easier 😈

[simple moss]

idk 💀

[viscid thistle]

geniuses in disguise

when it comes to proving shit wrong that they couldnt get

#calculus

[simple moss]

shhhh no one saw

[viscid thistle]

well I did

[daring tapir]

does it even matter

lmao

[viscid thistle]

yeah

[simple moss]

saw what? I can't see anything

[viscid thistle]

no

someday I will have enough and something bad will happen... 😈

[simple moss]

I mean we both are cats

we should fw eachother

[viscid thistle]

fr

[simple moss]

not in a sexual manner but u get me

[simple moss]

🗣

[viscid thistle]

why

[simple moss]

after rethinking it i don't think it matters

in hindsight tho 😔

[viscid thistle]

lol

[dire aspen]

me 😉

[viscid thistle]

add

[crystal plaza]

It's "kharshuuf", it means artichoke in Arabic

[simple moss]

OHHH

[nova obsidian]

Does anyone know why the answer is PTO? what does it even mean

^ trig

[uncut mulch]

please turn over

answer is supposedly on the back

[tiny pier]

is this <c written over there?

[nova obsidian]

Angle c = 40 degrees

[nova obsidian]

Huh?

[nova obsidian]

Ohhh

I don't see anything else though

It's ok, tysm anyway!

[errant girder]

wheres my nitro

[viscid thistle]

Can anyone solve this question

[proven night]

Hello

I really need help with this equation

I don't know how to solve the underlined equation

Maybe i have done something wrong at the very begining, so i got this "bad" equation

But i think my actions are pretty reasonable

[stable knot]

you want to solve $3^x+4^x-6^x=1$?

[obsidian monolith]

ƒ(Why am. I here)=I don't know

[proven night]

Yes

But i stopped at underlined equation

[stable knot]

well, one solution should be obvious

do you need to solve it analytically?

[proven night]

Algrbraically, without guessing numbers

[stable knot]

oh

so so can't say that x=1

*0

hmm, I think this function is monotonic

so there's only one soln

[proven night]

How to solve it without guessing? I did it but got only one of two sultions

[stable knot]

,w 3^x+4^x-6^x =1 graph

[proven night]

There are two solutions

[stable knot]

how

the graph shows only 1

[proven night]

Try plugging in 1 and 0 into the equation

Both satisfy it

[stable knot]

right

[obsidian monolith]

Results provided by WolframAlpha

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

Requested by physicsrocks

[stable knot]

so you need help to solve $2^x-3^x+1=0$

[obsidian monolith]

ƒ(Why am. I here)=I don't know

[proven night]

Right

[stable knot]

well, can you use calculus ?

[stable knot]

maybe AM>GM?

to find the range of values $2^x-3^x$ takes

[obsidian monolith]

ƒ(Why am. I here)=I don't know

[stable knot]

you just need to show it can reach -1 at one point

[proven night]

What exactly should be used

[stable knot]

actually AM>GM may work better

[proven night]

By the way i dont think it's something difficult because it's just a casual equation from insta

[proven night]

I dont know what it is

[stable knot]

Arithmatic mean > geometric mean

[proven night]

I havent studied it yet

[stable knot]

ah

then I'm not too sure

sorry

you could use graphical methods?

[proven night]

It was not mentioned at all, so i guess we can

But it's kinda lazy way of solving

[stable knot]

hmm, I see a way to use IVT here ( Intermmediate value theorm)

so essentially we find the local maxima

[proven night]

I suppose i had done something wrong at the beginning so i got this strange equation

[stable knot]

not really

IVT works here( sort of)

so we know there's a local maxima

that you can obtain using calculus

you also know that the function is negative at some point

and it is continuous

so it must assume y=1 at atleast one point in between

this is for the right hand side of the curve

you can use similar logic for the LHS

[proven night]

What is IVT?

[stable knot]

Intermediate value theorm

[hallow tide]

2^x - 3^x + 1 = 0 is not really an exponential equation that can be solved algebraically

all you do is notice that x = 1 is a solution and realize that it's the only one

[proven night]

And 0

[hallow tide]

no, not 0

i'm talking about 2^x - 3^x + 1 = 0

0 is a solution in the original problem you posted but not the equation that was discussed after

[stable knot]

hey, sorry for misleading you, as it turns out, not all the ideas I typed out are feasible