precalculus

[hidden lagoon]

Try finding the equation for the normal cubic here

then all you need to do is flip the x’s and y’s

[barren jewel]

I am so confused 👁️ 👄👁️

[hidden lagoon]

about what?

[barren jewel]

The Graph

[hidden lagoon]

Inverse functions are flipped over the line y=x

The green graph is the inverse of what you’re trying to find

[barren jewel]

Oh

So all I need to do is the find out the inverse and inverse it

[hidden lagoon]

yep

[barren jewel]

And how do I find the inverse? (I’ve forgot some stuff in precalc)

[hidden lagoon]

Switch the x’s and y’s

[barren jewel]

I think I found the function

Idk how

[hidden lagoon]

what is it?

[obsidian monolith]

Sharp

[hidden lagoon]

that’s right

[sour phoenix]

please i need help🙏

[hexed totem]

can you rewrite what is written?

[sour phoenix]

integral (dx)/a+b(tanx)

[golden cairn]

How do I do b?

[willow bear]

you need the consonant-vowel pattern to be CVCVCVC exactly

i.e. the consonants have to go in slots 1, 3, 5 and 7 while the vowels go in slots 2, 4 and 6

no other way to make them alternate

[golden cairn]

Yeah but how do I figure out how many ways there are to arrange them?

[willow bear]

did you do part a

[golden cairn]

Yes and c

[willow bear]

what was your answer for part a

[golden cairn]

5040

[willow bear]

right, 7!

[golden cairn]

Yes

[willow bear]

so it is almost the same here

you are arranging the consonants amongst themselves (there are 4 of them)

[sour phoenix]

@hexed totem i sent the question

[willow bear]

and also the vowels amongst themselves, independently of the consonants (there are 3 vowels)

[golden cairn]

Yeah so what do I do?

[willow bear]

well how many ways are there to place the consonants

you are definitely able to answer me this

[golden cairn]

4?

[willow bear]

no, not 4.

the answer to part a, where you were arranging 7 things, was not 7.

[golden cairn]

4!

[willow bear]

that's right.

and how many ways for the vowels?

[golden cairn]

3!

[willow bear]

right

so how many ways to make a CVCVCVC pattern?

[golden cairn]

144?

[willow bear]

would have been better as a computation than as a value

4! * 3! is what i wanted to hear

but you are correct

[golden cairn]

Why does the textbook say 288?

The answer key

[daring tapir]

Uh @willow bear ig the problem doesnt expect us to use all the letters or some shit then?

[golden cairn]

I got 144 for c

But I don’t understand why it is 288 for b

[willow bear]

?!

what the hell lmao

[golden cairn]

The answer key says 288

[daring tapir]

Yea well if it is 144 for c and then it has to be 144 for b too, the ans key is wrong

[golden cairn]

For question b

[willow bear]

yeah the answer key seems wrong

[golden cairn]

Okay

[willow bear]

for this problem in particular

[daring tapir]

Ann i have a question
Since you have the combinatorics role nd you said you are pursuing a masters,
I assume you are takin some combi courses
What do they tchr in combi at your level?

[willow bear]

wrong time and place for that

[daring tapir]

Ahem ;-;

[willow bear]

im tired and i dont want to answer this rn

[daring tapir]

Okyy thats fine

[golden cairn]

How do I simplify this?

I don’t know what n! means

Never mind

I got the answer

[dire swallow]

(n-1)! = n!-n so it is equla (2n!-n)/(n) I hope it is correct 🙂

[golden cairn]

I got n+1/n-1

In simplified form

[dire swallow]

isn't it equal = 2n!-1

[daring tapir]

Well nope

[daring tapir]

Well n!= n*(n-1)*....*2*1

[dire swallow]

n! = 123..*n

[daring tapir]

Yep

[dire swallow]

sure my bad

I got it

(n-1)! = n!/n

[golden cairn]

This one is confusing me

I tried everything and I just don’t even know how to do this

[dire swallow]

(x!*(x+1) + x!/x )/x! = x+1+1/x=x`2+x+1

[golden cairn]

The answer key says (x^2+x+1)/x

[dire swallow]

My bad I modified (x+1)! = x*(x+1)

when (x-1)! = x!/x

[young gust]

i have a quick question about partial fraction decomposition, and ill use this problem as an example:

(x^2 - x + 2) / (x^3 - 2x^2 + x)

why isn't A/x + B/(x-1) + C/(x-1) the correct form?

from what I understand, it should be written as A/x + B/(x-1) + C/(x-1)^2 but i dont really get why thats the case so if someone could explain thatd be nice

[river drift]

that's what happens generally when you have repeated factors

[willow skiff]

You can just combine the B and C terms if it were like that

Also the LCM of your denominators needs to be degree 3

I guess you should think of that term as (Px + Q)/(x - 1)^2

Where the deg of the numerator is one less than the denominator

Then Px + Q = R(x - 1) + S and it follows

[willow bear]

curious: how do you claim not to know what n! means even though you were able to use factorials just fine in an earlier problem

[zenith pivot]

yo can someone help solve these precal problems? https://docs.google.com/document/d/13z7_yAqtYYXr4tfzUpL8mGXVsKN9Eb1Qc2jIyXM7T2s/edit?usp=sharing

https://docs.google.com/document/d/13z7_yAqtYYXr4tfzUpL8mGXVsKN9Eb1Qc2jIyXM7T2s/edit?usp=sharing

mb i think I forgot to change access

[lime parrot]

@golden cairn

^this comes up a lot in Calculus 2 when you do infinite series

[daring tapir]

I mean they might have been taught a different notation? Or maybe no notation at all yet just expected to write it all out

[winter comet]

kind of defeats the point of factorial

[willow bear]

they answered my questions with the exclamation point notation

[daring tapir]

Oh shit lmao-

[daring tapir]

Idk we were formally told about the exclamation mark notations 2-3 hours after we began combi, it might be their first class too
But yea as ann pointed out this assumption of mine is wrong

[winter comet]

combi?

[daring tapir]

Combinatorics

[winter comet]

oh

heard of it, dunno really what it is ;-;

[daring tapir]

Welll all the "find the number of way.... " kinda thingies come under it for one

[willow bear]

to put it somewhat tongue in cheek, combinatorics is the science of counting

namely that of calculating the cardinalities of finite sets

[winter comet]

i can do that! 1, 2, 3, 4, 5, 6, jk XD

[willow bear]

counting != enumerating the naturals in order

[winter comet]

is it more theoretical than applicational?

[winter comet]

XD

[willow bear]

@daring tapir to answer your question from last night re: what is taught in uni-level combinatorics, the course im taking right now deals with certain combinatorial optimization problems (e.g. what's the highest number of subsets of {1, ..., n} you can make such that any two of them have intersection consisting of at least k elements)

[willow bear]

i mean idk

you can apply basic combinatorics to find some probability shit

like if you're calculating card game probabilities

[winter comet]

oh yeah thats true

[willow bear]

of any variety

[winter comet]

that reminds me, what are the applications of sequences and series (from calc 2)?

i'm kind of confused why it seems to be the prerequisite for a lot of higher math classes, at least at the colleges ive seen

cuz like i dont know when it even appears again

[daring tapir]

Damn that sounds insanely fun

[winter comet]

i guess i should have asked this in the calc section 💀

whatvr

[willow bear]

i mean sequences of all kinds appear damn near everywhere

• if you're studying analysis you're gonna have to deal with limits in more exotic settings

and series are used a lot in numerical methods too, such as for calculating shit like e^x or sin(x) on computers

[chrome ether]

many calc 2 classes do a terrible job of actually teaching much in the way of substance for series and their motivations

[daring tapir]

*most

[chrome ether]

true

[willow skiff]

a lot

but maybe it's a bit unrealistic to give tons of sequence and series examples without going into analysis

Anyways you should know that the harmonic series exists and it diverges

so that you know that just cause a sequence tends to 0, the summation doesn't have to converge

And you should obviously know the geometric series by now

And maybe some other series would be cool to do series tricks on, like telescoping

but that wouldn't be in a remit of a normal calc course

[late karma]

I havent learned calculus yet but why does the secant disappear at the tangent? (we're learning calc)

[willow bear]

disappear in what sense?

as in this is in geogebra or a similar graphing utility, and you're asking why the green line disappears if you drag the red point to coincide with the blue point? @late karma

[late karma]

Geogebra

Disappear algebraically i think

[willow bear]

the slope of that thing is $\frac{f(a+h)-f(a)}{h}$, and if you take $h$ to be 0, then that fraction becomes $\frac{0}{0}$

[obsidian monolith]

|Ann⟩

[willow bear]

since you are learning calculus: the \textbf{limit} of that fraction as $h$ goes to 0 is called the \textbf{derivative} of $f$ at $a$, written $f'(a)$.

[obsidian monolith]

|Ann⟩

[late karma]

Ohhh alr thanks

[golden cairn]

I was first confused while doing the question, but then I figured it out. In that question n! Just meant n(n-1)

[willow bear]

n! does not mean n(n-1).

[golden cairn]

I wrote it as that and got the correct answer in the end

I just multiplied the expanded it and got n^2-1

And then I did the factoring after adding the other (n-1) with n^2-1.

And I did the same with the denominator

And I ended up getting the same answer as the textbook answer key

[golden cairn]

In a similar question like that, my teacher wrote n! As n(n-2)(n-1)! And in the denominator there was (n-2)!

[daring tapir]

though that is not what you did

[willow bear]

you mean n(n-1) * (n-2)!.

and the (n-2)! cancelled out, yes.

but saying 'the n! meant n(n-1)' is just wrong

[golden cairn]

No I wrote it is that I meant to say

[golden cairn]

I did do that

In the question I sent, If I cancelled out the n(n-1) in the numerator and denominator. I was getting the wrong answer

I don’t understand these questions too well

Because the teacher made it an independent unit which meant she just gave us questions and we had to find out how to do them online

[willow bear]

theres basically exactly one trick here and it is unfolding the factorial

like n! = n * (n-1)! = n(n-1) * (n-2)! = n(n-1)(n-2) * (n-3)! = ...

aside from this, it is just run of the mill fraction manipulation bullshit

[daring tapir]

you didnt say that you did that though

[golden cairn]

Ok

[golden cairn]

Well I got (n-1)(n+1)/(n-1)(n-1) and then I just cancelled

[willow bear]

do you have your work written out somewhere

or did you do it all in your head

[golden cairn]

I dont have my work with me right now

But this is what I did on paper

[willow bear]

first, you cannot write fractions like this without putting parentheses around their numerators and denominators

second, already the first line looks wrong

[golden cairn]

Well that’s not how I wrote it on paper. I just don’t know how to make fractions on my phone

And even though you are saying it looks wrong, I still got the right answer so I don’t know

Maybe you could show me the correct way to do it? And then send me a picture

[willow bear]

the original expression is $\frac{(n+1)! + (n-1)!}{n!}$, yes?

[obsidian monolith]

|Ann⟩

[willow bear]

@golden cairn

[golden cairn]

Wait

T

This is

The expression

@willow bear

[golden cairn]

Isn’t that what I did?

[willow bear]

your first line reads to me as $\frac{n(n-1) + (n-1)!}{n(n-1) - (n-1)!}$

[obsidian monolith]

|Ann⟩

[golden cairn]

Yeah well I forgot to write the ! After n(n-1)

[willow bear]

ok so you were sloppy and careless

[golden cairn]

Yeah

[willow bear]

and in so doing downgraded your work from "correct" to "made a positive even number of mistakes and ended up at the right answer"

[golden cairn]

It was a mistake

I know even though it doesn’t change the answer I should have wrote it, I just forgot to.

[golden cairn]

Thanks, I was confused about that question but now I get it.

[onyx dirge]

hii, anyone down to help me with part d? 😭

[willow bear]

have you done c?

[onyx dirge]

yes

[willow bear]

show what you got for part c

[willow bear]

missing parentheses around the whole thing & the 0 and 1 are positioned a bit weird, but ok.

you also did part b, right? if yes, show me what you got for part b.

@onyx dirge

[willow bear]

your notation has several issues

but ok

notice that $6 - 3x^2 \sqrt{x^3+x} - \sqrt{x^3 + x} = 6 - (3x^2+1)\sqrt{x^3+x}$.

[obsidian monolith]

|Ann⟩

[willow bear]

this means that you can (and should) integrate your function for d using your answer for part b, and also your knowledge of how to integrate the constant function 6.

[onyx dirge]

ohh alright, I did not infact notice that.

[golden cairn]

What’s the answer to this?

I got (x+3)(x+2)/(x)(x+1)!

I don’t have an answer key for this question

I have a feeling it’s wrong

Or (x+3)(x+2)/x!

[compact stump]

That is the correct simplification

[golden cairn]

Okay

[jovial lodge]

Is this a level?

[winter comet]

wdym?

[jovial lodge]

It’s a uk thing

Like A-Level

[river drift]

for us students it's roughly analogous to ap classes

[winter comet]

oh

lol

[jovial lodge]

Analogous idk what that even means that’s how I know u good at maths 😭

[winter comet]

it's calculus

[winter comet]

analogous just means like relatable to or similar to

[winter comet]

somehow good at english = good at math 😂 also yes cloud gud at math

[jovial lodge]

Damn i wish I was good at either

Need to be good at maths cus i got an exam in like 1.5 weeks

[winter comet]

what exam

[charred epoch]

I'm so confused

If 5 is the base of the exponent, and the resulting number is 5.25, how is the exponent equal to 1/4?

[hushed sphinx]

It's not $\log_5(5\tfrac14)$ -- it's $\log_5(5^{1/4})$.

[obsidian monolith]

Troposphere

[charred epoch]

Oh

[hushed sphinx]

The typography could be better, though..

[charred epoch]

Thanks

[hearty bison]

i haven't learned this. is $(x+n)!=(x+n)(x+(n-1))\dots$? or is there some other way to interpret this

[obsidian monolith]

syzygy

[hearty bison]

there might be a better way to write out what i mean lol it looks a bit weird

[hushed sphinx]

Yes.

[hearty bison]

okay thank you

[hushed sphinx]

(The factors continue all the way down to 1, of course, they don't stop at x or x+1).

[hearty bison]

does this assume that all numbers involved are natural numbers

[hushed sphinx]

Yes, the ! notation is generally used only for natural numbers.

[hearty bison]

okay thanks 😁

[hushed sphinx]

Once in a blue moon you may see it abused as
$$ \frac{(x+n)!}{x!} = (x+n)(x+n-1)\cdots(x+2)(x+1) $$
even if $x$ is not an integer, but that is not really standard.

[obsidian monolith]

Troposphere

[hearty bison]

I understand that a lot less lol I might be missing something

[hushed sphinx]

Well, suppose both $x$ and $n$ are natural numbers. Then,
$$ \frac{(x+n)!}{x!} = \frac{(x+n)(x+n-1)\cdots(x+1)x(x-1)\cdots 3\cdot 2\cdot 1}{x(x-1)\cdots3\cdot2\cdot1} $$
so all the factors in the denominators cancel out with the tail end of the numerator.

[obsidian monolith]

Troposphere

[hearty bison]

ohhh

i see it now

[hearty bison]

so this uses that idea and applies it to non-integers?

[hushed sphinx]

Yeah. It is, as I said, somewhat nonstandard.

[hearty bison]

Okay 👍

[viscid thistle]

Is this right or am I tweaking

[chrome ether]

that looks right

[viscid thistle]

https://cdn.discordapp.com/attachments/1214256094189199380/1242321085173596254/IMG_6809.jpg?ex=664d692b&is=664c17ab&hm=3e899885e7204fb1bafd680c28cb1ee36dac26cdde7da2eb8cf4f69736ec0271&

Then isnt my textbook wrong

https://cdn.discordapp.com/attachments/1214256094189199380/1242321182452093008/IMG_6810.jpg?ex=664d6943&is=664c17c3&hm=ed01b31d336daf585cd0275ea660d84c16a8fed7832af1872f9754e9f1406639&

[chrome ether]

hmm

I don’t remember much about this kinda stuff lmao but that does still look sus

would the Ws be replaced with Es?

how does your book define those

[viscid thistle]

I think my answer sheet on the book is just wrong

[chrome ether]

nice lol

[viscid thistle]

there is no way west can = east

how does that make sense

[chrome ether]

it happens lmao

especially in these “textbooks” that people just shit out on the daily

there are bound to be errors when they can’t be arsed to write solutions to everything

[warm thunder]

Public education

[willow skiff]

Yeah unfortunately typos in questions are all too common

As long as you understand how to do it by resolving the x- and y-components of each vector, then finding the total x- and y-displacements

I wouldn't worry

[narrow briar]

hi! how can i calculate the domain of the function f(x)= sqrt(cosx -3sinx) ? thanks

[pine geyser]

hello, what have you tried?

or what do you think?

[hot basin]

cosx-3sinx>=0

make it look like x-3y>=0

-3y>=-x

you will divide -

so the >= will change to <=

y <= x/3

sin<=cos/3

idk if i got it right but i tried.

i might be wrong

i think im wrong

idk

wait

im wrong

i remember now

cosx-3sinx>=0

cosx - 3sinx = 0

cosx = 3sinx

cosx/sinx = 3

cosx/sinx = cotx

cotx = 3

i still dont know if im right😂

someone help

[hidden lagoon]

this is what cosx - 3sinx looks like idk how you’re supposed to write the domain

[hot basin]

ikr

cytos fr

-infi to infi?

[pine geyser]

cotx>=3

[hot basin]

right cotx>= ty

[hot basin]

w

[pine geyser]

then you write the domain based on cot-1(3)

[hot basin]

w

w Alieen

[pine geyser]

the problem is probably sqrt(3)

[hot basin]

now i feel confidence

thanks man

[hot basin]

the graph lol

its bouncing like free throw

[pine geyser]

First try to solve this question the method we shared here, and then try to solve your problem yourself!

[sour phoenix]

Hello please I need help

[winter comet]

._.

...bruh

oh maybe

nah bruh how you do that 💀

partial fraction decomposition doesn't work, it doesnt factor

u sub doesnt work

integration by parts doesnt seem like it would work

well actually lemme try

[dire aspen]

U can write x+1 as 1/2*(2x+4) - 1 and split it

[sour phoenix]

Where did that answer come from😅😅😅😅😩😩😩😩

[winter comet]

wha?

how does that do anything

[hushed sphinx]

It's #calculus anyway.

[winter comet]

tru

do u know how to do it

it looks like pain 💀

[hushed sphinx]

It might work to find a substitution that makes the denominator into u²+constant.

[winter comet]

hmm

u=x+2?

OH

hmm

i dont think that worked

brep

theoretically, if you factored x^2 + 4x + 6 taking complex numbers into account and then did partial fraction decomposition could you get the answer

or would that not work

i'm not sure since its messing around with the imaginary plane lol

prob not

oh yeah and then u have integral of complex number that aint goin well 💀

nvm

that just doesnt make sense

[hushed sphinx]

After the substitution you'd have to split (u-1)/(u²+2) into u/(u²+2) and 1/(u²+2) and integrate each separately. The first is a further substitution t=u²; the second can be scaled to a well-known trigonometric solution.

[winter comet]

ohh wait how do you integrate 1/(u^2 + 2) ?

do you do that weird inverse tan(x/a) thing?

isn't that a/(x^2 + a^2) tho

the derivative of inverse tan(x/a)

hm

[hushed sphinx]

The a in the numerator there is just a constant that you can lift out of the integral.

[winter comet]

💀💀i'm dumb XD

i see now

[charred epoch]

Why is y=log(x) the inverse of y=10^x?

[winter comet]

log(x) is log_10(x)

log base 10 of x

it just defaults to base 10

[charred epoch]

It was briefly mentioned in class that logarithmic functions are inverses of exponential functions

[winter comet]

what part of it do you not understand

like do you not understand why the inverse is an exponential function, or do you not understand where the 10 came from, or....what

[charred epoch]

Sorry. I don't understand why logarithmic functions are the inverse of exponential functions.

[winter comet]

alr

log_b(a) = y means that a = b^y
does that make sense?

or nah

[charred epoch]

I feel like I'm pretty close to understanding.

[hushed sphinx]

"inverse of this exponential function" is pretty often how "logarithm" is defined.

[charred epoch]

I know how to set up a logarithm, and that log is used to find the exponent you have to raise a base to in order to get a specific value.

[winter comet]

maybe this help?

[dire aspen]

Try it yourself.

[winter comet]

i wrote it down and do not see how it help lel

[dire aspen]

cuz you can cheese the numerator into looking something like the derivative of the denominator by doing it that way

[dire aspen]

U must’ve done something wrong then because from there it’s just a u sub and arctan thing

[winter comet]

?

[dire aspen]

Why is the denominator of the integral on the second line 2x+4?

[winter comet]

because you have to multiply by du/(2x+4)

[dire aspen]

I think you split the integral incorrectly, should be

(x+2)/(x^2+4x+6) - 1/(x^2+4x+6)

[winter comet]

oh i thought you said 1/2 * (2x+4) - 1

oh you just meant (x+2) - 1 in numerator?

[dire aspen]

I did that’s what u turn x+1 into

[winter comet]

i don't see

[dire aspen]

Don see what?

[winter comet]

why you would do that instead of (x+2)-1

[dire aspen]

cuz u can’t just change the numerator liek that

[winter comet]

you...can?

x+1 = (x+2)-1

and then you can split it into two fractions

is 1/2 * (2x+4) - 1 different from that?

[dire aspen]

Um

Then I’d think you’d probably have to write it as -1 + 1 first

[winter comet]

i mean you don't have to 😂 but that would be the idea

is ur method just the same or

[dire aspen]

Anyways

Doesn’t matter after u split it then it’s easy

[winter comet]

yeh

if you know the generalized arctangent

derivative

which i should know better than i do

i memorized the regular arctangent but not the other one ;-;

[dire aspen]

I’ve never used the other one before lol

[winter comet]

yeah

come to think of it i don't think i've really ever used the derivatives of any inverse trig functions besides tangent

derivatives or antiderivatives

lol

[viscid thistle]

this channel is precalc btw

[daring tapir]

(In some places basic integrals are introduced in precalc tho)

[viscid thistle]

this is based on the US precalc curriculum so I don't think it's introduced in precalc

and I don't think that integral is basic either

[dire aspen]

india moment where you learn differential geometry and graduate from harvard at the age of 6

[daring tapir]

lmao first of all, thats just objectively wrong, secondly i have heard of some precalc classes in the us touching basic integrals towards the end

[dire aspen]

yep i was being sarcastic lol

[smoky panther]

when the FUVK did (y) get added to derivatives

[willow bear]

... what do you mean

[proven night]

Can anyone help me out

[willow skiff]

what's the original question? this alone makes no sense

!xy

[tender quest]

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

[proven night]

Find the minimum value of "a" so that the equation can be solved

[echo sinew]

You need the RHS to be between -1 and 1 (end points included).

You should know why

[proven night]

What is RHS

[echo sinew]

Right-hand side (of the equation)

[proven night]

So i should solve inequality?

[echo sinew]

Yes

$$-1 \leq 2a^2+5a-6 \leq 1$$

[obsidian monolith]

Alberto Z.

[echo sinew]

Which is equivalent to $$2a^2+5a-6\geq -1 \quad \wedge \quad 2a^2+5a-6\leq 1$$

[obsidian monolith]

Alberto Z.

[proven night]

How to unify the solutionz

[willow bear]

what do you mean by "unify"?

@proven night

[proven night]

Nevermind

I already figured it out

I solved the inequality

What's next

[willow bear]

you mean inequalities, yes?

you solved both of them

[proven night]

Yes

[willow bear]

you want the set of all values of a which satisfy both inequalities at the same time.

how do you get that from each inequality's solution set?

[proven night]

As i said before, i already solve the inequality

[proven night]

But what should i do next in this question

[willow bear]

that's what im trying to have you reason out by yourself

you solved both inequalities. each one gave you a solution set, yes?

[proven night]

I solved it, thanks

I found the minmum a

[willow bear]

...

i mean

ok

[smoky panther]

Idk man i am struggling here i gotta get good at this shit in like 2-3 weeks and i am losing my mind

[willow bear]

ok but like

what exactly did you see that made you say that

[smoky panther]

I don’t remember but i think i saw a bunch of Ys in my text book

One sec ill check again

I think i am tweaking

[smoky panther]

They just added that shit and didnt explain anything

[willow bear]

this looks like implicit differentiation to me

[smoky panther]

Oh my god, i watch some voices aboyr it yesterday, I didn’t understand shit. I am not looking forward to retrying

[halcyon yew]

I did not know what on earth this is until like 10 seconds ago

but it seems like ur just assuming y is a function of x and differentiating both sides

then solving for y'

[willow bear]

that is in fact exactly what is done

[willow skiff]

just has a fancy name

[viscid thistle]

Why for the RHL it's (x-2) when x approaches 1?

[willow skiff]

The slope of that line is 1

And when x = 2, y = 0, so you have y = x - 2

0 = 2 - 2

you're looking at the limit from the right-hand side

[halcyon yew]

because for x>1 it's x-2 so RHL would be x-2?

[viscid thistle]

Yah got it .Thanks for the help

[willow skiff]

npnp

[winter comet]

the question itself was put here ;-; but yeh

[winter comet]

at that point why even pretend its not calc ._.

[daring tapir]

Lmao tell the tchrs xD

[winter comet]

😂

[viscid thistle]

What if we move calc down to precalc and move RA down to calc

[hidden lagoon]

what if we move calc down to 5th grade

[winter comet]

actually not a bad idea tho

[viscid thistle]

Intro to proof at 3rd grade

then introductory LA at 4th grade

[winter comet]

😢

[viscid thistle]

for 6th grade analysis ig

[hidden lagoon]

these 10 year olds need to understand Taylor series and integration techniques

[winter comet]

whats LA and RA?

[winter comet]

why not???

[viscid thistle]

#linear-algebra and real analysis

[winter comet]

oh

💀

[viscid thistle]

They must numerically compute integrals without closed forms by hand

[hidden lagoon]

if they know y=mx+b they’re ready for linear algebra

[winter comet]

ok sheesh i dont mean like being great at it but

being able to understand the concepts and how to do the problems

._.

[viscid thistle]

They should teach 1729-dimensional fourier transform to kids for them to multiply numbers efficiently

[hidden lagoon]

just give the riemann zeta function to them and tell them it’s graded on accuracy

[viscid thistle]

Just give the normal zeta function and let them make an analytical continuation

[final pier]

Guys I need some help I keep getting the wrong answer 😞

It’s on my pre calc final and highkey idk what I’m forgetting

[hot basin]

ok

what is the question

i got the answer

but whats the question so i can know which answer they need

[charred epoch]

Why can the coefficiant of a logarithm be moved into the exponent of the number in a logarithm?

Like 2log(x) is equal to log(x²) right?

[hot basin]

wait

yes

2 infront

goes back

to the power

x^2 yes

if its logx^2 it will be back to 2logx

[river drift]

yes, this corresponds to the exponent rule that [ (a^b)^c = a^{b \cdot c} ]

[obsidian monolith]

cloud

[charred epoch]

Hmm I figure maybe I don't understand why the all of the exponent rules work either

[hot basin]

right

you have to know the rules

https://youtu.be/LCmnKWXCIfk?si=z_LGjRxjV7d1idRz

[final pier]

How does it get ^3sqrt(4x

Idk where the 4 comes

[hot basin]

watch this video

[hot basin]

what is the question

i dont know what they asking u to find

[final pier]

The inverse

[hot basin]

oh

[final pier]

😭

It’s on my pre calc

And I kinda forgot

[hot basin]

simple

change x and y

x = y

and y = x

[final pier]

I know

But

[hot basin]

oh

[final pier]

When solving

There’s no 4

[hot basin]

let me solve it

and see

[final pier]

And a 4 comes into the equation

[hot basin]

is that cube on top

[final pier]

Yeah

[hot basin]

do we have to do (y+1)(y+1)(y+1) correct and then times 2?

or nah

wait

nope

i understand

x = 2(y+1)^3

divide 2

x/2 = (y+1)^3

now you cube both side

i got cude root x/2 = y +1

and i got the y by itself

so y = cuberootx/2 - 1

[final pier]

I figured it out 😭

But thank you mb I had to go to class

[smoky panther]

Alr thank you :)

[smoky panther]

I somehow did this on my own before watching the video :D i am so fuckin proud

[hot basin]

good stuff

proud of you

[hot basin]

✅

[smoky panther]

Thanks g

[viscid thistle]

precalculus chat really? I joined the wrong server. Even Missy can do that.

[hushed sphinx]

You shall be free to leave if you cannot abide the thought of someone being less advanced than you.

[surreal crescent]

Who else thinks this dude doesn’t even know what ln is

[daring tapir]

(I mean he was probably joking considering his name being sheldon as in the sheldon from young sheldon he had his bachelors at like 15 iirc tho he is fictional)

[winter comet]

you do realize there are more chats...

[hot basin]

.

[surreal crescent]

Everyone quick, go look up degenerate conics, this is interesting AF

[winter comet]

degenerate convicts are cool 👍

💀

[surreal crescent]

Also if two linear slopes intercept, are they tangent to each other?

[winter comet]

no, i would think the tangent line of a straight linear line is the linear line itself

linear line 💀

wait

yes

sorry i misread what you said

lol

[surreal crescent]

The only def of a tan is that you intercept once

[winter comet]

they the same line so ye they tangent to each other technically

[winter comet]

are you sure thats the official definition?

[surreal crescent]

It would be undefined then, because it has to both match the slope and only hit once

[winter comet]

if thats the definition

but its not undefined

so its probably not the correct definition

like

that definition works for curves, but not when you get to lines

i don't know what the definition of a tangent line is

but if you take the derivative you get the same slope, and it has to intersect the same point, so uh . . .

[surreal crescent]

I think I commited a syntax error

[neat nacelle]

I need a sanity check

[winter comet]

you are not sane

you're welcome

[neat nacelle]

$3\csc^2(x)-2\cot^2(x)-4=0 \
\implies 3\csc^2(x)-2\left(\csc^2(x)-1\right)=4 \
\implies 3\csc^2(x)-2\csc^2(x)=6 \
\implies csc^2(x)=6 \
\implies x=\asin(\frac{\sqrt{6}}{6}\
\text(vs) \
3\csc^2(x)-2\cot^2(x)-4=0 \
\implies \frac{3-2\cos^2(x)}{\sin^2(x)}=4\
\implies 3-2-2\sin^2(x)=4\sin^2(x)\
\implies 1=2\sin^2(x) \
\implies x=\pm\frac{\pi}{4}$

[winter comet]

told ya

jk 💀

[neat nacelle]

basically I get two different answer sets with $3\csc^2(x)-2\cot^2(x)-4=0$, where one is $x=\asin(\pm\frac{\sqrt{6}}{6})$ and the other is $x=\pm\frac{\pi}{4}$

[obsidian monolith]

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

GoldenPhoenix

[neat nacelle]

and I'm not sure where things are going wrong here

if anyone wants to try decoding all that latex then that'd be helpful, as it shows my work

[river drift]

work shown:

\begin{gather*}
3\csc^2(x)-2\cot^2(x)-4=0 \\
\implies 3\csc^2(x)-2\left(\csc^2(x)-1\right)=4 \\
\implies 3\csc^2(x)-2\csc^2(x)=6 \\
\implies csc^2(x)=6 \\
\implies x=\arcsin(\frac{\sqrt{6}}{6})\\
\text(vs) \\
3\csc^2(x)-2\cot^2(x)-4=0 \\
\implies \frac{3-2\cos^2(x)}{\sin^2(x)}=4\\
\implies 3-2-2\sin^2(x)=4\sin^2(x)\\
\implies 1=2\sin^2(x) \\
\implies x=\pm\frac{\pi}{4}
\end{gather*}

[obsidian monolith]

cloud

[river drift]

arithmetic mistake on 2nd last line, should be 6 sin²(x)

[neat nacelle]

actually it's arithmetic mistake on the third to last, it should be positive 2 because it's -2(1-sin^2)

which then explains the top as well

dangit

thanks

all units have order 2 dangit

[winter comet]

wouldn't that just get you back to where you were?

oh wait i see

wait naw i dont see

[neat nacelle]

yeah, and then in the first workings out I forgot that it's -2(csc^2-1) which then becomes positive 2

[winter comet]

oh so the bottom is correct

and the top has that mistake

yeh?

[winter comet]

i mean its accidentally correct

lol

or like the result is

[neat nacelle]

the bottom is correct in conclusion, I forgot a negative in the text but fixed it in the next line

[winter comet]

yea

[neat nacelle]

while I didn't notice it in the upper work

[winter comet]

whats syn?

like cos^2(x)?

or you mean discord syntax

yeah

cos^2(x) = [cos(x)]^2

is correct syntax

both are

[true sluice]

guys, what is the inverse of a function value absolute?

[willow skiff]

pls don't ask twice

in different channels

[willow bear]

and with the same incomprehensible wording...

[neat nacelle]

the preimage of f(x)=|x| is g(x)={x,-x} but it ain't a function.

[viscid thistle]

?

why is not a function

[neat nacelle]

because I'm slightly abusing notation, and it's not returning a single set, it's returning both a positive and negative value, i.e. each member of the domain is mapping to two members in the range

[viscid thistle]

wait you're responding to something?

I thought you just said that for no reason

[neat nacelle]

this thing

[viscid thistle]

how is each member of the domain mapping to two members in the range

|-x| = |x|

[neat nacelle]

f(x)=|x| goes from the reals to the non-negative reals, and maps two values to each real number

thus the preimage of f, f^-1(x) goes from the non-negative real numbers to the real numbers, and maps each number to its positive and negative values

pre-image means basically reading the function backwards(i.e. inverting it), and sometimes that makes it not a function anymore

|x|=3 implies that x∈{3,-3} (x is an element of {3,-3}, that is, x=3 or x=-3)

[viscid thistle]

okay sure i get what you mean

but in response to his question, you can construct a piecewise definition of one

yes it's not a function, but that wasn't really his question right?

[neat nacelle]

he asked for the inverse of a 'function value absolute' which I assumed (potentially poorly) was the absolute value function, thus I gave him the inverse relation, but I wanted to stress that it isn't a function in my response because at precalc and lower it's often assumed that the inverse of a function is a function.

[viscid thistle]

no inverse cuz |x| isn't a bijection

[final pier]

Can someone explain how to solve these for me

Since my teacher won’t teach me it

😭🙏

[viscid thistle]

,rotate 90

[viscid thistle]

do you know how to draw these graphs?

you can draw all these 3 graphs tbh

and "erase"

the regions that are not of concern to you

[neat nacelle]

every function has an inverse, they're only functions when the original is a bijection, otherwise they're relations.

[frosty marten]

can anybody help me out at help26

i have an exam in two days

[final pier]

Not really

It wasn’t something taught

[frosty marten]

anybody?

[final pier]

Just which parts do I graph

?

[neat nacelle]

Can you describe in words what the function is?

[thin thorn]

I'm not able to start this question
Can somebody help?

[wanton moat]

Hello everyone, i am a maths helper, good with algebra, calculus and statistics. Kindly feel free to DM me incase you need tutoring. Thank you guys and have a nice learning experience

[thin thorn]

Thanks for offering
Can you please help with the question above?

[winter comet]

i could be wrong, but if |x_0| < 1, then x_(n+1) is getting smaller, right?

wait

[thin thorn]

x0 might be positive too
It's not a guarantee

[winter comet]

nah actually its getting slightly bigger

ahh

yeh

hmmm

[thin thorn]

It might go both ways

[winter comet]

yeah

uh

[thin thorn]

Depends on the value of x0

[winter comet]

well the least we know is sqrt(1-(x_0)^2) is a positive number less than 1, right

nope

[thin thorn]

Someone suggested me to substitute x0 for Cos2∅ as it formed an identity
But I couldn't take it from there.

[winter comet]

nevermind

[winter comet]

yeah i thought of that too idk what it would do though lol

hmm

[thin thorn]

Yes that's true don't doubt yourself for that

[thin thorn]

I have the same reaction when I tried

I literally thanked the guy for giving me some insight and ended up reaching nowhere lol

[winter comet]

oh yeah its squared im dumb lol

[winter comet]

😂

[thin thorn]

I think I should ask the same person again

[winter comet]

XD

[thin thorn]

I'll message here when I get the approach

[winter comet]

alr

[winter comet]

wait but x_n has to be positive for n!= 0

[thin thorn]

I didn't get you?

[winter comet]

if you plug in x_0, x_1 will be positive and less than 1

and all the x_n after that will have to be positive and less than 1

[thin thorn]

Yeah that's true

I get it now

But does that take us anywhere?

[winter comet]

if you multiply a bunch of positive but less than 1 numbers

[thin thorn]

I'm really weird out by this now

[winter comet]

and its the limit to infinity

doesn't lim x - > infinity x_1 * x_2 * x_n.... = 0?

[thin thorn]

It approachs zero

[winter comet]

yeah

so if the bottom approaches 0

and the top is positive

[thin thorn]

Yeah
That's a cool insight

[winter comet]

then the limit has to approach 0, right?

too bad that aint an option

oh wait i meant infinity

the bottom approaches 0, then the limit approaches infinity

but the logic is flawed somewhere

lol

[thin thorn]

Not really it isn't necessary that a limit approaches infinity just like that

We have to check up the numerator a bit too

[winter comet]

why not?

[winter comet]

its a positive number less than 1, no?

yeah sum is wrong 💀

[thin thorn]

I think there's some flaw in our thinking

[winter comet]

yeh

[thin thorn]

I've solved many limits that seem to tend infinity just 'cause they are 1/0 but when I manipulate, they really don't

[winter comet]

hmm

this is precalc?

💀

something makes me think we have to use inverse cosine

[thin thorn]

Naah this is calc
But there isn't a channel for that here I guess

[winter comet]

calc 2?

[thin thorn]

I idk that

[winter comet]

oh there is

[winter comet]

series calc?

[thin thorn]

I'm actually from India
We don't have those systems here

We just have math that has calc

[winter comet]

does it use anything with sequences and series calc? like series expansions or smthn...

i don't really know what im asking cuz i don't fully understand that

lol

im just gonna keep assuming there is a way that we can get 💀

also there is a #calculus channel

[winter comet]

fair 💀

[thin thorn]

It isn't there in the tabs shown to me though

[thin thorn]

Those really aren't much used in calc here
Like you're asking convergence, divergence ?

[daring tapir]

Or hear me out since it is an mcq just let, x_0 = 0

[thin thorn]

What the hell man 💀

[daring tapir]

Nvm

Wont help

[thin thorn]

You really talk like the character on your pfp

I literally heard that guy's voice in the sentence

[daring tapir]

You kinda can do that even tho it doesnt help xD

[daring tapir]

Anyhow post that in #calculus it has a much higher probability of getting answered there

[winter comet]

i'm stil confused on why the answer is not infinity ;-;

[daring tapir]

Recursion's a bitch

[winter comet]

💀

is this something i would know better if i understood calc 2 more ;-;

maybe for some reason the denominator doesnt converge to 0

[daring tapir]

💀

[winter comet]

i dont see why it wouldnt tho 💀

[daring tapir]

Idk it shoud though

I mean wait

[winter comet]

i am stupid

possibly

no nevermind i dont get it

[daring tapir]

Guess we gotta ask so high

[winter comet]

fr

[daring tapir]

Cant ping him tho

He got his jee in 2 days

[winter comet]

bruh

that must be crazy

[daring tapir]

Hm it probably is

💀 neon can probably also solve this rather easily but for that op gotta post in #calculus

[steep nacelle]

what is calculus and precalculus

[daring tapir]

Troll spotted <@&268886789983436800>

[steep nacelle]

what

i fr dont know

i havent been taught it yet

[modern valve]

What? Doesn't seem like a troll to me

[winter comet]

difficult to say

precalculus is literally anything pretty much

[steep nacelle]

oh ok

[winter comet]

calculus includes limits, derivatives, and integrals

[daring tapir]

Uh okay then i guess

[winter comet]

and different kind of applications to them

[steep nacelle]

damn yeah idk what that is lol

[winter comet]

you learn it in calculus

[steep nacelle]

kk

[winter comet]

or sometimes precalculus

[final pier]

It’s literally just graph the function idk what else to do 😭

[winter comet]

a function takes an input value and returns an output value

you can use this information to plot points and try to draw the shape of the graph

or you can just know the shape originally, plot a few points and draw accordingingly

[final pier]

But the graphing looks really weird 😭

[winter comet]

wait do you know what a piecewise function is?

[final pier]

No

[winter comet]

oh, its a function that is composed of multiple functions on different intervals

[neat nacelle]

that's why I asked. If you can't read what it's trying to do, then you can't graph it.

[winter comet]

so on the interval -1 to 2, it could look like the graph of |x|, whereas on the interval of 3 to 5, it could look like the graph of 2x

[final pier]

Oooh

[winter comet]

so with ur piecewise function

[final pier]

Heres this 😭

[winter comet]

do you see the three intervals?

[neat nacelle]

notice how you have it cut into pieces?

[final pier]

Yeah

I mean I understand that there’s 3 different lines

[winter comet]

do you see like 2x + 3 for x < -2

[neat nacelle]

you need to find which graph puts the right shape piece on the right region

[final pier]

Yeah

Is that the one I start from?

[winter comet]

that means for all the values before -2, its the graph of 2x + 3

[final pier]

So I just use a graphing calculator or whatever

[final pier]

Ooooh okay

[neat nacelle]

you don't need one for this, really, it's asking about rough shapes, especially when they gave you this. Things to check would be the endpoints of each interval, and where the graph should be increasing or decreasing

[final pier]

Okay

[neat nacelle]

do you know the difference between the open and closed circles?

[final pier]

Yeah

It depends on the line under the < right 😭

If it’s open or closed

[neat nacelle]

yep

[final pier]

Okay I got this 😼

[neat nacelle]

just to check: those things are related, yes, but what do they mean

?

[final pier]

The teachers never taught us what they actually mean

Unfortunately

😭

Or they like vaguely went over it

Because we only get a semester of pre calc, algebra 2 and 1, and calc and trig and calc3

[neat nacelle]

when I say x<3, what does that mean?

[final pier]

X is less than 3

[neat nacelle]

ok, now what about x__<__3

[final pier]

X is equal too or less than 3

I think😭

It’s been a while

[neat nacelle]

so in the first one, can x be 3?

[final pier]

No

[neat nacelle]

so should a graph be able to touch 3 with that condition?

[final pier]

No

[neat nacelle]

and what about the second one?

[final pier]

OOOOOOH THAT MAKES SENSE

[final pier]

The second one it can be touching 3

[neat nacelle]

most math symbols and conventions are designed to show what they're trying to tell you. This is one of those cases

[final pier]

Cause it’s filled

[neat nacelle]

yep!

[final pier]

That makes sense 😼

[neat nacelle]

so I would look very closely at the conditions of the intervals and see how they compare to the endpoints on the graph

[final pier]

Okay

Thank you

[thin thorn]

Hey @winter comet
I got that one
I can explain if you want

[daring tapir]

Please do

[thin thorn]

See when you put xn=Cos2theta

x_n+1 becomes Costheta

[daring tapir]

Oh damn-

Aha its pretty much solved now then

[thin thorn]

Honestly it wasn't a moment ago
Even when I knew that substitution

[daring tapir]

Well the cosine GP formula to the rescue

[thin thorn]

Now after substituting
We note that Sin2theta is 2SinthetaCostheta

One of the Costheta below cancels

Then the half angle formula again

And all the CosTheta like terms cancel

Leaving us with 2^n times Sin(Theta/2^n)

And as n goes to infinity Sintheta can be written as theta

[daring tapir]

Yea no i kinda just remembered that as a formula in itself so uea xD

[thin thorn]

2^n cancels
The answer becomes Theta

Which is just arccosx

[daring tapir]

Hmmm damn

[thin thorn]

Aaah it was too long to type ngl

[thin thorn]

I'm kinda new to doing math full time so I don't really know many such patterns

[daring tapir]

Now you know one more!

[thin thorn]

Yeah that's growth

I have one more integration question

Maybe I should post that directly to the calculus section now

[viscid thistle]

Is the correct option d?

[viscid thistle]

any advice on to reduce mistakes on math tests(one solution im gonna try is to get more sleep)

[winter comet]

1. get more sleep
2. try not to make mistakes as you go, think through everything but don't go too slow cuz you gotta make sure you finish.
3. if you have time, check you work, you can mark any areas you were unsure about as you go.

[viscid thistle]

i do 2 and 3 but i still have so much mistakes

my grade is down atleast 10% just from mistakes

[winter comet]

theres a reason people say just keep practicing 💀

just make sure everything makes logical sense

like you actually know what you're doing, not just going through the motions

[viscid thistle]

i spend way to long practicing💀

[winter comet]

maybe don't over-practice and get burnt out, but 💀

what do you make mistakes on btw?

what kind of mistakes

[viscid thistle]

simple mistakes

[winter comet]

could you give me an example?

[viscid thistle]

arithmetic operations, swapping the symbols, and just simply doing dumb stuff for example when finding complex roots I forgot to divide the degree in the equations

its so dumb that even my teacher feels bad for me 💀

[winter comet]

ok i really really think like

knowing exactly what you're doing

it's hard to explain in a generic way lol

[viscid thistle]

knowing how and why what im doing works?

[winter comet]

as a basic example
x - 4 = 4
(x - 4) + 4 = (4) + 4
x = 8

very basic tho, i don't have an example because i dont know exactly what mistake you've made or smthn XD

but you get what i mean?

and also on each step

you can check if your previous step was correct

[viscid thistle]

ok

[winter comet]

just really think about it

[viscid thistle]

ill try that

[winter comet]

yeah

[viscid thistle]

ill give another example

yk systems of 3 linear equations

[winter comet]

yeh

[viscid thistle]

ok so normally they are easy

[winter comet]

yeh

[viscid thistle]

but i somehow kept messing them up on the test(i had to redo them 2-3 times) but i got the correct answer eventually

i dont know where or how i got it wrong

but that feeling was so weird

cuz it feels like u did it correctly but it just doesnt click

[winter comet]

if you spent more time thinking through each step, maybe it would have helped you not make mistake?

[winter comet]

yeah

[winter comet]

of course it depends on how well you're concentrating

we all have those days when you make bajillion mistakes but yeh XD

[viscid thistle]

thats everyday for me xD

[winter comet]

XD

[viscid thistle]

rn ima j try getting some sleep and just go w the flow ig

[winter comet]

alr

if ur taking an exam, gl

[viscid thistle]

ty lol

[winter comet]

xD

[viscid thistle]

finals this wed and im at border 90% (im cooked)

[winter comet]

💀

just pray to the math gods that you don't make a mistake

🗿

[viscid thistle]

i will

also what would u do if u ever got burned out

cuz mg burn outs are killing me

[winter comet]

you mean like tired of doing math, during a test?

or what do you mean

[winter comet]

and thus being more likely to make mistakes?

[viscid thistle]

tired from just grinding(practicing)

[winter comet]

XD

[winter comet]

ahh

take a break, do a different kind of problem, or try to wake up 💀

also

try to really think about what you're doing

don't just follow the method, understand the process and why it works and how you're doing it

[viscid thistle]

i picked up that habit after geometry xD

[unkempt urchin]

btw if you want to recheck your answer, try verifying it from a different approach, use a different method to recalculate the answer, or check for mistakes like wrong unit or wrong plus minus

[winter comet]

XD

oh yeah

[viscid thistle]

bet

[winter comet]

if you make a mistake on uh

systems of equations

you can always plug back in

[viscid thistle]

breaks dont work for me because i just spend half the day on break and the day is just done

[viscid thistle]

thats how i was able to verify it

[winter comet]

yeah

if you get an answer, you plug into all systems, and it works, theres like no way you made a mistake 💀

unless you've made four mistakes and they all happened to give the same answer 💀

[viscid thistle]

💀

[winter comet]

don't do anything during the break

[unkempt urchin]

sometimes you dont even realize you make a mistake because its not the calculation thats wrong, but the logic behing the working step (like tom give 2 apple each to his 5 friends but you instead 2 + 5 = 7)

[winter comet]

it makes you bored and then you WANT to do the math XD

[viscid thistle]

guess ill take a nap

[winter comet]

naw, then ur gone forever 🗿

[viscid thistle]

oh shi

[viscid thistle]

thats not a healthy way of putting itxDDD

[winter comet]

😂

[viscid thistle]

BRROOOOOO I LEGIT DO THAT TOO MUCH

its beyond sad how much i suffer from those exact types of mistakes

[winter comet]

thas what im saying the most general advice "think about what ur doing" mhm mhhm

💀

its so general that it aint even help 💀

[viscid thistle]

aight ima do what u j said

who knows ill have to experiment tho

[winter comet]

yeh

[viscid thistle]

1.Could anyone explain what is the greatest integer function?
2.And why is it different for integral and non integral values?
3.Why the values of c+ and c- are different?

[river drift]

the greatest integer function finds the greatest integer less than or equal to x

which is the same as "flooring" or "rounding down"

[viscid thistle]

Can you answer other 2 questions?

[river drift]

i'm not sure what you mean by different for integer vs non-integer values

and c+ is any number slightly larger than c, whereas c- is any number slighly smaller than c

[whole void]

the greatest integer less than or equal to 3 is 3

[viscid thistle]

Like decimals and integers