precalculus

[winter comet]

but

ln^2(x) implies exponent

non-universally 💀

semi-vaguely 💀

[river drift]

exponents over functions may be used to denote composition: [ f^2(x) = f(f(x)) ] which is more consistent with the meaning of the exponent in inverse function notation

[obsidian monolith]

cloud

[river drift]

so it's not universal or unambiguous to use it as an exponent

[plush zephyr]

hi

how to get started with pre-calc

trying to learn it prematurely for personal interests

[uncut mulch]

khan course for the basics

[plush zephyr]

too lazy tho

[uncut mulch]

you want to learn but don't want to do the work?

that's impossible

[plush zephyr]

i want to do the work, just not through watching youtube videos

sorry wrong wording

I'm more of a hands-on guy

[uncut mulch]

they have a combination of written documents too

[plush zephyr]

got links?

[uncut mulch]

go directly to the khan academy sit

[plush zephyr]

alr

[golden cairn]

Is this all correct?

Please tell me if I did anything wrong

I’m unsure about the domain and range I wrote for 2a

And also for 2b, what do they want me to write in behaviour of function near the npv?

[daring tapir]

hmm it looks correct

[daring tapir]

slope and stuff

[golden cairn]

Ok

And also for number 1, is that all they wanted me to write?

Like the equation of the horizontal asymptote

[daring tapir]

you just wrote a bunch of extra stuff below rest yea

[golden cairn]

Ok

[golden cairn]

Did I do these two questions correctly?

I’m not sure about the ph one

[agile sparrow]

We got chemistry question in precalculus before GTA VI

[golden cairn]

Yeah lol

[daring tapir]

lawl

[daring tapir]

lgtm

[golden cairn]

Okay

Is this correct? f(x)=x+3, and g(x)=1-x^2

I actually think it might be wrong

Is x^2+6x+9 supposed to be in brackets or no?

[timber saffron]

Yes, it should be

[golden cairn]

Okay so I have to multiply the - with everything I guess

[timber saffron]

yeah

[golden cairn]

I know ai is not a trustworthy math source but it keeps telling me the domain in this question is [-1,♾️)

I think it’s (-♾️,♾️) but I’m not sure

Oh wait

I think the restriction is x≠-1

[golden cairn]

How do I do this question?

[winter comet]

the domain of g(x) is [-1,infinity), but the domain of f(g(x)) is (-infinity,infinity)

[viscid thistle]

Hey, would you look at that? We got the same about me.

[viscid thistle]

Viper

I will d you smth

Dm*

[winter comet]

k

[golden cairn]

So what would I even write for the domain? This is my work

I am not sure if the graph is even correct

[winter comet]

no it was correct before, sorry i shoulda been more clear 💀

the domain of the new function is (-infinity,infinity)

the line is just y = x - 1 for all x

wait

im completely wrong

[winter comet]

this is correct lol

[golden cairn]

Ok

[flat shuttle]

How would you find the zeros for a quartic function like this

[chrome ether]

start by testing potential rational roots, rational root theorem

[flat shuttle]

ok

[young gust]

shouldn't the -1 be a bracket

it's a closed circle

should be [-1, infinity)

oh oops just realized i replied so late soz

[glass magnet]

do yall have access to a ti 84 in your class

[flat shuttle]

Yes

But I think I need to do it algebraicly

And show work

[shadow summit]

This is an annoying one. Ultimately I would use the remainder theorem, as it is by the easiest one to do. Start with substituting positive/negative factors of 10 into the function.

If that doesn’t work, there are other methods such as the Newton-Raphson method for finding roots, although that might be too advanced for you.

[hallow tide]

that's a rational root theorem problem

[whole void]

don’t test positive roots

[hallow tide]

i think he knows that

[whole void]

test some negative integers to start

rational root theorem

[flat shuttle]

Ok then

[plush zephyr]

just went into trigonometry topic in khan academy

its a but hard but not that bad

the trigonometric ratios of theta are also the coordinates of the terminal ray

[thin summit]

now study polar coordinates, and modulii of complex numbers

[spare badge]

I got stuck on PART B. Can someone explain why they removed the 2 before integrating, and then not multiplying it back in at the end. I’ve linked my working out, the mark scheme, and the question.

[summer ruin]

the integral of 4/(3+2u) is 2*ln(3+2u)

[golden cairn]

How do I graph 1/f(x) using this graph?

[chrome ether]

f is piecewise linear, find out what each “piece” is

[golden cairn]

I don’t know what that means

[chrome ether]

well then find out what it means

[golden cairn]

Okay

But what is the equation of the graph in the picture?

[chrome ether]

^^^^^^^

[golden cairn]

So there is no equation

I searched it

[chrome ether]

treat the horizontal line segment and the “diagonal” ones separately

[golden cairn]

Okay

But what is the meaning of 1/f(x)

I understand that it’s a reciprocal function

But what do I do

[chrome ether]

take the reciprocal of each piece and graph??

[golden cairn]

So there is no work to show other than the graph?

The teacher is usually asks to like show mapping

Or something

[chrome ether]

you want to find out what 1/f(x) is for each piece

probably write that down as well

[golden cairn]

Yeah but how do I figure out what each piece is?

[chrome ether]

this is basic stuff

finding the equation of a line through two points

surely you know how to do that?

[golden cairn]

Yeah

[chrome ether]

do that for each piece

[golden cairn]

Ok

So I just graph y=1/x-1

?

And y=1

[chrome ether]

on their specific domains yea

so 1/(x-1) on 0<x<2

1 on 2<=x<=4

[golden cairn]

Won’t the y=1 piece just stay the same?

Is this what I graph?

[chrome ether]

restrict the domains to match the domain of your original f(x)

so the 1/(x-1) part is only for 0<x<2

and the 1 part is for 2<=x<=4

[golden cairn]

Ok

[golden cairn]

What did you put into desmos to get that?

[chrome ether]

just put your domain restrictions in curly braces in the same line as the function

for example 1/(x-1) {0<=x<=2}

[golden cairn]

Okay

Thanks I get it now

[chrome ether]

np

[full galleon]

Ration functions are rlly fun tbh

[torn atlas]

I got my finals next week

Anyone got any study tips

[chrome ether]

come back with a more specific question

read over questions you've done; redo them to get the methods back in your head, etc

[spiral sonnet]

Can I get quick confirmation on a DNE question?

I have an ordered pair in cartesian form (0, -4)

asked to convert to polar form

r = 4
y = arctan(-4/0) = DNE

Or am I missing some law I don't understand

[chrome ether]

$\theta=\dfrac{3\pi}{2}$

[obsidian monolith]

elrichardo1337

[chrome ether]

using arctan directly doesn't really play nice with $\dfrac{\pi}{2}$ or $\dfrac{3\pi}{2}$

[obsidian monolith]

elrichardo1337

[winter comet]

is there any algebraic way to go about it or do you just have to visualize it on a graph lol

[chrome ether]

$x=r\cos\theta,$ $y=r\sin\theta$

[obsidian monolith]

elrichardo1337

[chrome ether]

plug directly into these

and it should be clear what's what

[winter comet]

der

i need to sleep 💀

[spiral sonnet]

At first I couldn't tell this was in response to me lol but thank you this helped a lot

[chrome ether]

np

[plush zephyr]

bruh

[obsidian monolith]

Tetronix

[plush zephyr]

and tangent of 0 is just the slope of the terminal ray

[golden cairn]

How do I do this

[viscid thistle]

What are a few algebra requirements that a person would need for doing calculus better

[viscid thistle]

index laws are priority number 1, I will say that

[viscid thistle]

I agree, any other things that come to mind?

Nothing particularly, im just in the middle of my final exams so not that qualified to talk about it. Other than knowing trig identities and partial fractions well will also save you in calc

What about calculations? Like there are some calculations that don't seem to make much sense at first but later when someone else points it out, you understand it fully.

Anyway to like analyze calculus better?

Do you have an example to hand? I know some questions can get very calculation heavy like when you have to integrate by parts multiple times if that's kinda what you mean

No no, not calculation heavy well, atleast at my level it doesn't that that excessively heavy on calculations but I meant as in something as simple as using substitution or eliminating two eqn after finding a similarity between them

Sometimes it's hard to catch if you don't have a sharp eye for small analytic details

Most people recommend practicing which is the number one priority but blindly practicing ultimately leaves you blind to new things.

Make sure you write down everything you do and write it neatly is also advice worth following. Cliche but it really does help since sometimes you might not spot things immediately but get a chance to review them again and (a higher*) chance of spotting them.

Well its tricky because maths is just practice, practice will train your pattern recognition and you'll eventually see the same things over and over again in terms of exam maths

What about the forgetting curve?

I had an exam on diff 1 today and I forgot some things from my product rule chapter

Which was comparatively alot easier

Then the things I was practicing one or two days ago

Anyway to effectively understand each topic?

[viscid thistle]

Yep

So I find that when I learn something, I will leave it a day or two then watch a youtube video on the subject and do practice questions

In the run up to my exams atm I have done 60 previous papers from my exam board and others to try and test not only what ive learned but having to recall each topic every now and then

Even just watching a youtube video about a topic will help with recall, at least what ive found.

That's kinda true for me aswell

It's better then other methods I tried

Alright dude thanks for the tips

[tired stone]

does anyone know how to do that? Im lost

[golden cairn]

I just finished that unit

[safe basin]

Linear functions have the same slope everywhere so the graph doesn’t bend

Meaning you only need to know 2 points on the line to draw the entire line

In this case the first and last point on the interval

[sand gate]

This shit hit hard

[daring tapir]

do you want a solution?

[sand gate]

Naah I just did it

I feel like aryabhatta now

[desert swallow]

in a dominance matrix do you achieve to aim transitivity

[tidal oxide]

is it useful for a senior hign school student to learn calculus?

[hushed sphinx]

Yes.

How that utility compares to other things the senior high school student could spend their time on instead, is a more difficult and context-dependent question.

[chrome ether]

just to get some perspective:

calculus is FAR from the "be all end all of math" that high schoolers often mistakenly believe it to be

[tidal oxide]

just for fun.but my teacher says I should spend time on "more important things".

[chrome ether]

most STEM majors stop at calc 2 or calc 3, if you decide to study math further in college you'll get to stuff that is not even hinted at in HS

if you're about to graduate: for now just enjoy the free time and start to get ready for college lol

but there are plenty of resources online if you want to learn some calculus on your own

[tidal oxide]

I'm 10th.I have bought The Calculus Lifesaver.Is that suitable for me?

[chrome ether]

idk what that textbook is

but

most standard calculus texts are essentially identical so i wouldn't worry too much about your specific choice

like i said, there are plenty of online resources you can use as well

ok just found a pdf of calculus lifesaver

looks pretty thorough to me

[tidal oxide]

There may not be that many on the Chinese Internet!

[chrome ether]

you could download whatever you need for offline use

are you currently in china?

[tidal oxide]

Sure.

[hushed sphinx]

The word "lifesaver" makes me worry a bit that it could be focused on procedures rather than understanding.

[chrome ether]

it's some 700 pages long

looks pretty good from what i've seen?

[hushed sphinx]

Okay.

[chrome ether]

at least it's not one of those crappy "AP calculus" books that those "test prep" companies feel the need to shit out a new edition of every year

[tidal oxide]

It's actually called the Princeton Calculus Reader in Chinese.

[chrome ether]

spivak would be my recommendation, but that is ideally read with some prior experience with proofs

[tidal oxide]

no Chinese version.😢

[chrome ether]

do you need it to be in chinese? or can you also read the english version?

[tidal oxide]

As normal Chinese students,we dont have the ability to read even simplist English.

[chrome ether]

ah i see 😔

unfortunately i don't know much about what chinese language texts there are

sorry

[tidal oxide]

Our toughest English exam questions are just articles from common English newspapers.

[tidal ginkgo]

damn

are you using proxy?

[tidal oxide]

emmm.... yes

[golden cairn]

Is this cos or sin function?

[river drift]

it can be either

any sine function can be rewritten as cosine and vice-versa

[hushed sphinx]

(Somewhere, I'm sure, a seismologist is crying).

[golden cairn]

Yeah but if I wanna graph it

Which do I graph it with?

[chrome ether]

doesn't matter

[flat yacht]

So ehh finally got to rewatch this video to make some notes on it, and I still do not get why they want to "choose" only one dx here. Like, I see just as many x's as dx's, so why aren't there just as many x's as dx's in the simplified concept?

I hope you at least fairly remember what that stuff was about lol

[river drift]

all the terms containing more than 1 "dx" will limit to 0

like choosing n = 3 for example:

\begin{align*}
(x^3)' &= \lim_{\Delta x \to 0} \frac{(x + \Delta x)^3 - x^3}{\Delta x} \\
&= \lim_{\Delta x \to 0} \frac{\cancel{x^3} + 3x^2 \Delta x + 3x \Delta x^2 + \Delta x^3 - \cancel{x^3}}{\Delta x} \\
&= \lim_{\Delta x \to 0} \frac{3x^2 \cancel{\Delta x} + 3x \Delta x^{\cancel 2} + \Delta x^{\cancel 3}}{\cancel{\Delta x}} \\
&= \lim_{\Delta x \to 0} 3x^2 + 3x \Delta x + \Delta x^2 \\
& = 3x^2 + 0 + 0
\end{align*}

[obsidian monolith]

cloud

[potent salmon]

what are the roots for sqrt -9

[golden cairn]

How do i find the equation of this?

[hushed sphinx]

It looks like it is a (vertically flipped) exponential function with asymptote y=3. Do you know the general shape of the expression for such a function?

[golden cairn]

I actually ended up figuring it out

The reason I couldn’t get the graph earlier was because I thought it was a square root function

I don’t know why

How do I do this one?

I have never seen a graph like this

Maybe it’s a sinusoidal

I’m not sure though

[hushed sphinx]

Definitely looks like a sine curve.

(Or cosine, perhaps -- as Elrichardo said you can use one or the other, but one will be slightly simpler).

[agile sparrow]

Looks like x-x^3/3!+x^5/5!.... to me

Could be
1-x^2/2!+x^4/4!.... too

[agile sparrow]

I think it's this one

[vapid plaza]

Yeah he mentioned in the video that we only “care about” the terms with no dx or only dx^1 because anything dx^2 and above are “negligible” (goes to zero after dividing through by dx and taking the limit as dx -> 0)

[flat yacht]

but then why take one dx? I'm confused, maybe I get your explanation tomorrow, it's kinda late rn too

Why does a day have such a small amount of timeeeee

[worn tendon]

how can i change a vectors direction without affecting its magnitude

[agile sparrow]

the new vector with diff direction

make it a unit vector

[river drift]

apply a rotation to it

[agile sparrow]

then multiply with magnitude or orginal vector

[agile sparrow]

isn't it

Too complicated

[worn tendon]

can you give an example

[agile sparrow]

devide any vector by its own magnitude it becomes a unit vector

[normal shadow]

sin, cos, and tan work with a triangle, where you divide one portion of the triangle by another

if you dont have a triangle

how can you use these concepts

for example the sin of x always goes between 1 and -1

if there is no triangle that contains the side x, then how can you do anything with sin or cos or tan

[hushed sphinx]

More fundamentally, (cos(t), sin(t)) are the coordinates of a vector of length 1 whose direction is an angle of t counterclockwise from the x-axis.

[normal shadow]

uhh

ima google translate that hold on

[lofty mountain]

Is it possible to learn all of Calc 2 in one month

Taking into account the person is studying 3-4 hrs a day and working a 9-5

[normal shadow]

thats what im doing with AP Precalc, im sure its possible

[lofty mountain]

Nice

[winter comet]

yes

its at least possible

[normal shadow]

yo i chat gpt that sentence and i still dont get what it means

you have a point on a circle which has a radius of one

i get that part

but what point, what do you do with that point, etc.

no clue

[hushed sphinx]

Sorry, I tried to word it in as elementary terms as I could.

[normal shadow]

dang

[winter comet]

😭

[normal shadow]

💔

[winter comet]

you're talking about parametrics tho?

[hushed sphinx]

Perhaps I'm just not understanding which kind of explanation you're looking for.

[winter comet]

i would say you can use sine waves and cosine waves in real life, such as with sound and circuits and ect.

[normal shadow]

no i mean

like you have sine

and sine is opposite/hypotenuse

on a triangle

but like when you have no triangle, and you plug a random number into the calculator in sine

then it gives you another number

what is the calculation it is doing

[winter comet]

ah

[normal shadow]

because its not like it makes up a triangle to get numbers from and solve

[river drift]

we extend the definitions of sin and cos based on the unit circle. When 0 < theta < 90, these definitions match, but the unit circle allows us to define trig functions for more angles

[winter comet]

you'd prob solve it with power series expansion but the calculator apparently does smthn else 😔

[hushed sphinx]

Are you asking what actually happens inside the calculator, or how the answer it's supposed to give is defined?

[normal shadow]

the second one

[normal shadow]

my definitions only work then when a triangle is present, and enough information is given while the theta being less than 90

right?

ok nvm i get what youre saying

but then the unit circle

??

[river drift]

well first, we can imagine a hypothetical triangle with hypotenuse of length 1

then since sin = opp/hyp, sin(theta) gives one of the side lengths of that triangle, and cos gives the other

[normal shadow]

alright

i get it so far

but what is the unit circle thing then

how does that come into play

[golden cairn]

How do I find the equation of this graph?

I know it is x^2+4

But there is a point of discontinuity at (1,3)

[viscid thistle]

The roots are -2 and 2

[golden cairn]

Yeah

[viscid thistle]

Put in bracket form

And expand

[viscid thistle]

Wdym by that bit

[golden cairn]

But what about the point

Like the hole

[river drift]

let's position our triangle in a particular way so that one edge is at the origin. then the other edge of the hypotenuse is a distance 1 away from the origin

then by virtue of giving the side lengths of the triangle, we get that the coordinates of the purple point are x = cos(theta), y = sin(theta)

[viscid thistle]

Whst do u wanna do with it

[golden cairn]

How do I include it in my equation?

How do I indicate it

[viscid thistle]

Yep I only do a levels bruh never heard of that

Srry

[golden cairn]

Ok

It’s undefined there I guess

[hushed sphinx]

You're probably supposed to multiply by (x-1)/(x-1).

[golden cairn]

I already got the equation

I just don’t know what to do with the point of discontinuity

[viscid thistle]

Why would u need to do anything with that??

[hushed sphinx]

[golden cairn]

Why?

[hushed sphinx]

Try it.

[golden cairn]

Multiply -x^2-4 with that?

[hushed sphinx]

(x-1)/(x-1) is undefined when x=1, and is 1 otherwise.

[golden cairn]

So is the point of discontinuity whatever cancels?

Or something

Idk

Like when something cancels out

[river drift]

then to define cos(theta) and sin(theta) for other angles, we can say "they're the coordinates of that point on the circle"

[golden cairn]

Oh so the non permissible value?

Because x can’t equal 1 in the denominator?

[hushed sphinx]

Yes, that's the point.

[golden cairn]

Ok

So the equation will still be the same as what I said?

[hushed sphinx]

No, $-x^2+4$ is not the same as $\frac{(-x^2+4)(x-1)}{x-1}$. The former is defined when $x=1$; the latter is not.

[winter comet]

isn't it -x^2 + 4?

[golden cairn]

No I said -x^2+4

[obsidian monolith]

Troposphere

[golden cairn]

Yeah

[hushed sphinx]

Yeah, typo, sorry.

[golden cairn]

Okay I get it

That’s way simpler than what I thought

[hushed sphinx]

It's just a guess that this is what you're supposed to do, though. It may also be that you're supposed to say

f: R\{1}->R, f(x)=-x²+4.

[golden cairn]

What’s the R supposed to be?

I can’t read that

I haven’t seen anything like that before

[hushed sphinx]

$$f: \bR\setminus{1}\to\bR: \quad f(x)=-x^2+4$$

[obsidian monolith]

Troposphere

[golden cairn]

Well the teacher hasn’t taught us that

So I guess the previous thing was what I am supposed to do

[hushed sphinx]

Probably. ¯_(ツ)_/¯

[winter comet]

or maybe they just didnt teach that notation

like maybe f(x) = -x^2 + 4 for x != 1 or smthn lol

[golden cairn]

How do I do this?

For the r value I get -0.3084

But I can’t get the correct answer for the second calculation

Never mind

I just typed the wrong number into my calculator

[winter comet]

A = 10^(5.1) * A_0
A is the new earthquake, A_0 is reference/"standard" earthquake

[golden cairn]

Oops

I should have deleted it from here

I already got the answer

Sorry

[viscid thistle]

Simple question, but if this figure was a regular tetrahedron with side lengths 1

Then does the red point have coordinate (1/2, 1/2, 0)

assuming also that the left most blue edge has a fixed coordinate of (0,0,0)

[chrome ether]

no

the faces of this tetrahedron are equilateral triangles

[viscid thistle]

oh yeah i totally forgot about that lol

[chrome ether]

lmao rip

[viscid thistle]

oh yeah forgot to say thanks

anyway the base would just trivially be a 30-60-90 traingle right?

so uh let's say (1/2, sqrt(3)/6, 0)

yeah okay that probably looks right

then pythag to get the coordinates of the apex

[chrome ether]

is the red point the center of the base?

[viscid thistle]

yes

[chrome ether]

yea that sounds right then

[viscid thistle]

it should be right underneath the apex right?

for a regular tetrahedron

my main goal is to find the coordinates of the apex

Which is a smaller problem for something else i'm doing

[chrome ether]

yeah

[lofty mountain]

Can smb help me pls??
I'm seeing D is the ans but why?

[hushed sphinx]

It would have to be the second of the two D's.

But a larger problem seems to be that (0, -3) is not on the curve at all.

[winter comet]

😂

[burnt yarrow]

I know the answer is -6, 6 but i’m a bit confused where the x+9 goes, like why does it dissapear?

[lofty mountain]

Can smb help me pls??
I'm seeing D is the ans but why?

[lofty mountain]

Thank you so much

[paper tartan]

could you specify what you mean a bit more?

as it stands, I can tell you that g(x) is defined at x = -9 because g(9) = 0

so x = -9 is in the domain of g

but I'm not sure if that's what you're asking here

[hushed sphinx]

The only thing that can go wrong there is that the denominator is 0 -- but as long as you make sure the denominator is nonzero, you can divide whatever the numerator turns out to be by it.

(Note that x+9 itself makes sense no matter what x is -- so it does not restrict the domain).

[burnt yarrow]

ohh okay thank you!

[normal shadow]

i dont understand how im supposed to know to divide everything in the equation by x^2

i get that in this equation, the highest degree is x^2, but I dont understand why I have to divide everything by it, and how it realtes to the rule of how if the degree is higher in the denominator, then the horizontal asymptote becomes 0

are you supposed to just divide everthing in the equation by the highest degree if you want to understand how the rule of [Degree of numerator < Degree of denominator] means that the end behaviors of a function approach y = 0

[winter comet]

you have to divide everything by the highest degree because then every term of lower degree will have a factor of x in the denominator of that term, and thus those terms will tend to 0, and you will be left with the constants that were the coefficients of the terms that had the highest power :l

and if the denominator has the highest power, then all the terms in the numerator will tend towards 0 while the denominator tends towards a constant, so the limit is just 0

[torn atlas]

Are these considered equal? Does the arrangement of the 2 matter

[normal shadow]

thank you, but why the sad face ):

[winter comet]

because if i did not already understand it then i would not understand what i just said, but if it helped you then gud :D

💀

[normal shadow]

lmao

can you then explain to me why im wrong here, the quotation marks in the explanation given don't really help

what does it mean "removable"

also am I allowed to @ anybody, i dont think I am but im not sure

[summer ruin]

the limit f(x) as x->3 exists

[river drift]

a function $f(x)$ has a "removable continuity" at $x = a$ if it is discontinuous there but we can define a function $g(x)$: [ g(x) = \begin{cases} f(x) & x \ne a \ b & x = a \end{cases} ] such that $g(x)$ is continuous (we removed the discontinuity). Note this is the same as saying [ \lim_{x \to a} f(x) = b ] for some finite number $b$

[obsidian monolith]

cloud

[austere blaze]

Yes but f(x) at a should also be b

[river drift]

that doesn't have to be the case

for example, we could have $f(x) = g(x) \frac{x - a}{x - a}$

[obsidian monolith]

cloud

[winter comet]

yes

[torn atlas]

Thanks 🙏

[limber forum]

Can someone help please
?

This question is driving me crazy

[thick herald]

is this pre calc?

[limber forum]

Im not sure

[thick herald]

cool

[limber forum]

??

[viscid thistle]

Here why would ds and dt become a single point rather than 2 different points on the graph?

[river drift]

he is saying that the point (t + dt, s + ds) is getting closer to the point (t, s) as dt approaches 0

[white rapids]

How do you determine between a cosine and sine function regarding it's extremum

[frank vessel]

is this a level

[white rapids]

@winter comet I need help pls

[winter comet]

example?

[white rapids]

Uh

I'll just send the question

[winter comet]

i mean technically a cosine wave could be represented as a sine wave and vice versa

if you add pi/2 :P

[white rapids]

I did another example like this but the same thing when x = 0 it's either a sine or cosine function

[white rapids]

@winter comet

[winter comet]

technically it could be either

but uh

[white rapids]

If x = always 0, how do you determine which to use???

[winter comet]

wdym?

the function only intersets (0,-7) once

[white rapids]

Ye

They only look at the midline

I dont get it

[winter comet]

does this check out

[white rapids]

So how do you determine if it's a cosine function

[winter comet]

it intersects the y axis at the middle of the graph

is that a cosine or a sine wave

[white rapids]

🤷‍♂️

[winter comet]

😭

[viscid thistle]

a base cosine wave intersects at y=1 when x= 0

[winter comet]

im assuming they just center it at 0 :l

[viscid thistle]

sin intersects y=0 at x=0

👍

[white rapids]

???

[viscid thistle]

thats without transformations

[viscid thistle]

what happen

pls elaborate your "???"

[white rapids]

I mean

If it's either a cosine or sine function,

[winter comet]

does this help visually

[white rapids]

Both functions intersect eitherway

[winter comet]

yeah but one intersects at a maximum, and one intersects at the midline

does that make sense

[white rapids]

Ye

Wait where is it suppose to intersect

The y or x axis

[winter comet]

the y axis

it intersects the x axis infinite many times

right

but y axis only once

[white rapids]

Ohhh right ye ye

Man this hurting my head

[winter comet]

your problem's graph intersects the y axis at the midline

which basic graph intersects the y axis at the midline?

[winter comet]

based on like this or smthn

lol

[white rapids]

Cosine graph ye

[winter comet]

nah the sine

cuz uh

[white rapids]

Oh crap

[winter comet]

y = 0 is the midline for both

so the graph intersects the y axis at the midline

if its a sine graph

[white rapids]

Uh...

[white rapids]

I mean looks like from here the y intersection point is the maximum

[winter comet]

for the cosine graph...

like theres 2 different graphs right

one is sine and one is cosine

[white rapids]

Yep

[tawny bramble]

After a bit of guessing I found those two functions. I can send full proof but it filled with unreasoned assumptions

[limber forum]

Yes please

[tawny bramble]

Basically I try to keep it as simple as possible. I assume that those functions will have the form of
y = kx+b
y = a*sin(bx+c)+d
You can tell right ahead that
y = 1.23x - 2
Then I take a derivative of sine
y' = ab*cos(bx+c)
I can form a set of equations for x = 2 and y = 0.46
1.23 = ab*cos(2b+c)
0.46 = a*sin(2b+c)+d
Under assumption cos(2b+c) = 1 it simplifies to
ab = 1.23
d = 0.46
From that assumption I take one of the solutions
cos(2b+c) = 1 => 2b+c = 0 => c = -2b
Final set of equations
b = 1.23/a
c = -2.46/a
d = 0.46
In my example a = 1.23

[limber forum]

I still dont get it

[limber forum]

Anyway thanks

[late timber]

have sense

[worldly cloud]

This only goes up to Pre-Calculus, not Calculus?

[uncut mulch]

get access to uni roles
in that survey you filled out when joining

if you have calculus questions, you can also claim a general help channel

[hushed sphinx]

Alternatively, check "show all channels" in the server pop-up menu. The early university channels such as #calculus don't require special roles to access.

[worldly cloud]

👍

Taking AP Calculus AB and BC in the fall/spring, so I'll definitely be needing some help.

[golden cairn]

For cos^-1(-1/3) I get 1.911 and the questions says over the interval [0, 2pi], so does that mean I just add pi into 1.911 to get the second solution?

[hushed sphinx]

That sounds like you're assuming cos(1.911) and cos(1.911+pi) are the same -- that's not how it goes.
It's cos(1.911) and cos(2pi - 1.911) that are the same.

[golden cairn]

Oh

Okay thanks

[golden cairn]

How do I graph this?

[shadow summit]

Find the asymptotes, the peaks, where it crosses the x-axis and y-axis, ect. Just how you graph a regular function.

*note it won’t cross x-axis

[golden cairn]

But it’s weird because it’s like a recipricol parabola or something

So I don’t get it

[shadow summit]

Try plugging values in for x, and see the general shape you get.

[golden cairn]

I have the asymptotes and the vertex for it

But I put it in desmos and I don’t understand how to get the points to draw the two curving lines at the top

[round current]

jesus chrsit if this in precal c im cooked

[winter comet]

why?

[golden cairn]

How do I graph that?

Please explain to me

I know how to graph a parabola but this is just different

I remember doing it in grade 11 precal but I forgot how to do it

And I can’t find my old notes

[winter comet]

plot points

or you can use calculus if you know it

i guess you could also know that there are asymptotes at x = -2 and x = 3 and then plot some points around that, but i don't know how you could know theres are asymptotes at y = 0 without limits

[golden cairn]

I don’t know calculus

[winter comet]

and then concavity and stuff :l

[winter comet]

i would say the best bet is to notice the asymptotes at x = -2 and x = 3 and then plot some points, idrk

[golden cairn]

Okay

My exams on June 11th and I have no idea how to do this😩

[winter comet]

do they actually tell you to graph it?

[round current]

im taking algebra 2 over the summ 💀 it is easy which is scarry

the kids at school would cry about algebra 2 being hard

💀

[winter comet]

💀

[round current]

i cant be smart

so im cooked

i dont think they teach the hard stuff too

on the online course

[winter comet]

i mean

[round current]

im taking precalc courses too and its scary

[winter comet]

it probably sounds harder than it is 💀

[round current]

maybe lol

[winter comet]

lol

how were you at algebra 1

[round current]

i was in geometry

[winter comet]

i dont think algebra 2 has too much geometry lol

[round current]

good

i forgot all of geometry

[winter comet]

💀

[golden cairn]

Yeah I took that question straight off the provincial exams

The ones from 2019

I have a whole pdf of it and it’s actually driving me crazy

[winter comet]

whats the exact question they gave you?

they told you to graph it?

[golden cairn]

Let me check exactly what it said

They actually straight up just tell you to graph it

And state the y intercept

[winter comet]

weird

i would just find the vertical asymptotes and plot points lol

maybe thats what they want you to do

[golden cairn]

Yeah hopefully because I don’t remember if the teacher even taught this to us

I even checked all my notes and I couldn’t find anything like this

[winter comet]

i'm guessing the exam is different from what ur teacher gonna do then

[golden cairn]

Yeah but the exam should have stuff that we were at least taught how to do

And this is a 2019 exam

So she could have gone over it or something

But I checked this years exam that happened in January and it looks much easier than the 2019 one

So I think I’ll be fine

[digital perch]

After finding the asymptotes, seek to solve the function by zones, as the asymptotes would be 3 and -2, then seek to solve the function or give it some form when x is greater than 3, it is between 3 and -2, it is less than -2

[winter comet]

wdym "give it some form"

[digital perch]

For example, with asymptotes you can analyze the value that the function will acquire, whether positive or negative. When you analyze the value of the function when it is negative, you will find that it results in a quadratic function where its vertex is at x = 1/2 which matches the computer graph.

[winter comet]

whats the function when its negative?

[digital perch]

(x-1/2)² - 25/4

[winter comet]

how did you get that?

also doesn't it go down?

[vagrant current]

does any1 have any good resources to study pre-calc?

[paper tartan]

Khan Academy is a good source:
https://www.khanacademy.org/math/precalculus

[undone sage]

this feels like a bad proof isn't it?

tthe proofs from my other book are much better although it's on a different topic, this feels like it's jst making shit up, I get it but I don't get how the proof came about

[viscid thistle]

This doesn’t belong in pre calculus

As for whether it’s a bad proof or not, what even constitutes that? If you can understand it

And it’s coherent

[vapid plaza]

If you’re asking for the motivation of the proof, the main idea is that to get from g(x)k(x) to g(x+h)k(x+h) you first replace k(x) with k(x+h), then replace g(x) with g(x+h) (or the other way around). The first part corresponds to g(x)k’(x) and the second corresponds to g’(x)k(x)

But I do see a typo in the proof

[eager narwhal]

same

[proven night]

Hello

I want to know whether this kind of solution is good and is acceptable by the majority of teachers?

I mean, we "see" that x is 81, it's obvious, but we did not simplify the equation step by step so that we finally have x on the left side and a number on the right side.

[terse rivet]

Help pls

It looks simple but am going crazy over this

[random yoke]

Notice 25=5^2

[severe pond]

Is this the full precalc curriculum? I’m about to take precalc a year after alg 2 / trig and I forgot everything. But now I’m relearning trig by myself, but still not very comfortable. I start July 5

[willow skiff]

probably a bit extra honestly, so contains more than you need to know to do calculus

matrices and PnC have to go somewhere but strictly speaking, you can leave those to do later

conic sections are nice but not 100% a must either

[formal moss]

ghost ping?

[hushed sphinx]

You're right to be skeptical there.
Your rewriting has shown the original equation is equivalent to f(x)=f(81) where f is defined by f(x)=(x/9)^x. This is helpful, and shows that x=81 is a solution, but doesn't end the investigation.
The problem is that this function f is not injective. For example, f(1) = f(6.3767297896022341657...), so we'll need additional reasoning before we can be sure 81 is the only solution to your original equation.

Exactly how that additional reasoning can go depends a bit of what you know already. For example, are you already aware how how the graph of x^x looks?

[paper tartan]

I don't know what your curriculum looks like, but Khan Academy's is very comprehensive

it'll probably meet most to all of your curriculum requirements

[proven night]

No

[proven night]

Already yes

[hushed sphinx]

Hmm, I'm getting mixed signals here :-)

It might be possible to just wing it, if you can show that f(x)<1 when 0<x<9, and that f(x)>=1 and is strictly increasing on [9, infinity).

[severe pond]

I picked up a precal course on Udemy that included trig, and when looking thru the precalc section it started off with polar coordinates and learning to graph weird shapes, but this differs from what I’m seeing in other precalc courses that are mostly intermediate algebra. Should I not pay much heed to this?

[winter comet]

yeah those are polar coordinates, those appear in calc 2 and probably after

good to know, but you won't use them in calc 1

the intermediate algebra precalc stuff mostly just prepares you for calc 1 prolly

[severe pond]

I need to grind through some algebra again then, I went through most of the trig section over the last couple of months and did some stuff I didn’t even do in my actual trig class

[winter comet]

like what stuff?

[severe pond]

Well it was an alg 2 / with some trig not an actual dedicated trig course

[winter comet]

did it have trig identities?

[severe pond]

I didn’t do any actual graphing of trig functions, but this Udemy course has me doing that with a b c d transformations

[winter comet]

oh

[eager narwhal]

wait is polar coordinates part of the second half of this course, if it has one

[severe pond]

Yeah. Part 1, full trig, part 2, full “precalc” but it starts with polar coordinates and weird shape graphs like roses and lemoids or whatever the hell

[eager narwhal]

ok I think the second part should start with polar coordinates and polar graphs

I took an SVHS course and the second part also started with polar coordinates

[severe pond]

What’s svhs

[eager narwhal]

but why is the first part just trigonometry?

[eager narwhal]

silicon valley high school

online courses

[severe pond]

Oh

No idea, it’s a trigonometry + precalculus package

[eager narwhal]

oh

ok

[indigo dew]

help!

[winter comet]

do you know the power reducing trig identity?

[daring tapir]

0

[winter comet]

o true

[daring tapir]

see the bounds

[winter comet]

ye

[indigo dew]

NO

[winter comet]

but like

knowing how to calculate the antiderivative of sin^2(x) is important

[daring tapir]

either the author made some mistake or its 0

[daring tapir]

sure but for this que you dont need to do shit xD

(also i prefer the IBP way anyways)

[winter comet]

you don't NEED to but if you don't know HOW to then it would be preferred to learn how to rather than looking at the bounds :l

[winter comet]

why bruh

[naive yarrow]

0

[winter comet]

the power reducing is much simpler :l

[indigo dew]

your ans is correct I am saying idk power reducing trig

[daring tapir]

idk i found the IBP way myself, while the power reducing thingy i read in a book

[winter comet]

oh
sin^2(x) = [1-cos(2x)]/2

[daring tapir]

soo im biased

[winter comet]

ahh XD

[indigo dew]

so do I need to know pwer reducing identity for this question?

[daring tapir]

there are multiple ways

[winter comet]

you don't NEED to for this question, no

thats what they were saying

but honestly i think its good to know :l

[daring tapir]

for this que specificaally since the bounds are equal it basically means it's asking you area of a line

which is 0

[winter comet]

its asking you for the area with height sin^2(x) and width 0 😂 for each dx

like not infinitely close to 0

i mean literally 0

im probably making it more confusing for nor eason

🗿

[daring tapir]

lol

[indigo dew]

can you tell me what to do after this

[winter comet]

if you look closely it says the limits of integration are actually pi/4 and pi/4

not pi/4 and -pi/4

[indigo dew]

lower limit is -π/4

[winter comet]

but anyways as far as finding the antiderivative of 1 - cos(2x) you should integrate each term seperately

[winter comet]

oh it is?

[indigo dew]

ye

[winter comet]

that just said pi/4

is that amistake or sum

XD

or are youd

doing a different integral

[viscid thistle]

(This is not precalc?)

[golden cairn]

How do I do this question?

Actually never mind

I think I just reciprocate the y values of the original graph

Or something

Is the vertical asymptote at -2?

[jagged vine]

can someone explain how to do this and fill ? first day of online calc

[chrome ether]

plug in values of x super close to zero

0.1, 0.01, 0.001, -0.1, -0.01, -0.001, etc

[jagged vine]

so like

in the middle is is 0 and DNE

or 0 and 0 top n bottom in the middle

[willow skiff]

yes, 0 and DNE

dividing by 0 is undefined

whereas if you have say $\frac{\pi (x + 1)}{x + 1}$, you can cancel out the $x + 1$s

[obsidian monolith]

שוק

[willow skiff]

so technically it's undefined at $x = -1$ but the limit still exists there

[obsidian monolith]

שוק

[willow skiff]

so your situation is the 3rd image from the left

the pi (x + 1)/(x + 1) example is the 2nd image

[eager narwhal]

yeah

[gloomy cliff]

Is there a formula for an equation like this?

[hallow tide]

that's not an equation but no not really

[gloomy cliff]

alright thanks I couldnt find anything on it

and I guess thats why

[vapid plaza]

Using change of base you can write it as (log a)^2/(log b log c) but apart from that not much meaningful cancellation you can really do

[covert parrot]

I am studying over the summer to get a better understanding of precalc before I go into precalc. One topic I am struggling with is summation notation. I am specificly struggling with converting arithmetic, geometric, and log equations into summation notation form. like how would you convert 2((1-3^14)/(1-3)) into sumation notation?

[summer ruin]

$\sum_{n=0}^{n-1} ar^n = \frac{a(1-r^n)}{1-r}$

[obsidian monolith]

Transparent Elemental

[covert parrot]

$\sum_{n=0}^{n-1} ar^n = \frac{a(1-r^n)}{1-r}$

[obsidian monolith]

McNoob

[covert parrot]

how would I convert $\sum_{n=0}^{n-1} ar^n = \frac{2(1-3^(14))}{1-3}$

[summer ruin]

don't you find that your expression looks suspiciously familiar to the one on the right?

[obsidian monolith]

McNoob

[covert parrot]

yes I do

sorry I was just practicing the laex

[summer ruin]

identify constants r,a,n

[covert parrot]

so is the sumation form

$\sum_{n=0}^{13} 6^14$

[obsidian monolith]

McNoob

[summer ruin]

it's not 6

[covert parrot]

oohhh I am stupid

it would just be $2(3)^{14}$

[obsidian monolith]

McNoob

[covert parrot]

right?

[summer ruin]

yes

[covert parrot]

some people use k in sommation notation is that basically the same thing as n?

[summer ruin]

sorry I made a typo, it should be $\sum_{k=0}^{n-1} ar^k = \sum_{k=0}^{13} 2 \cdot 3^k$

[obsidian monolith]

Transparent Elemental

[summer ruin]

instead of r^n

[summer ruin]

yes

except here k and n are supposed to be independent, k is the summation index, n is the upper bound of summation

[covert parrot]

how would you know if it is k=0 or k=1? Or to better phrase the queation how would you judge the summation index?

[summer ruin]

the geometric series is a series of the form a + ar + ar^2 + ar^3 + ...

[covert parrot]

yes

[summer ruin]

so it starts with the term r^0

[covert parrot]

oh, so basically the summation index is where you start the series

I have a question so the series is defined by $S_n=$\sum_{n=0}^{n-1} k^2

[obsidian monolith]

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

[covert parrot]

are the partial sums 1,4,9,16,25 and so on?

[summer ruin]

these are elements of the sum, not sums themselves

[covert parrot]

what is an element of the sum?

[summer ruin]

the numbers that make up the sum

[covert parrot]

are sums like 1+4+9+16+25?

[summer ruin]

you didn't actually do any summation here, you just listed the elements

that's what's meant by partial sum

[covert parrot]

ok

so when they ask the question the third partial sum is (blank). is the answer 14?

[summer ruin]

yes

[covert parrot]

how do you figure our the recursive and explicit formulas?

I think the explicit is just the equation its self

[summer ruin]

it is done on a case by case basis

[covert parrot]

so the recursive formulas are different in geometric and arithmetic series.

[tiny moss]

I have watched a video about tips before taking calc, and i question my self with this: Does taking the first, second, third and so on of a derivative of a function and plugging it to another one works?

i was really confused when i ask this question to myself, what ive think is it would work out

[river drift]

could you show an example of what you mean by that?

[tiny moss]

alr hold on ill think a simple one

[winter comet]

i was confused when you asked this question to us

💀

[tiny moss]

lmao

[tiny moss]

like this

[river drift]

i mean you can do that, sure, although i'm not sure why you need to calculate g(f'(x))

[tiny moss]

sooo

it works

[winter comet]

wdym "works"

yes it mathematically "works"

XD

[tiny moss]

i was blackedout for second bruh

alr lets say it mathematically works

[winter comet]

💀

yeah

[hushed sphinx]

I don't get how f'(x)=4x is supposed to imply g(x)=x².
Do you know something about these functions that you're not telling?

[winter comet]

are you getting technical about the arrow 🤔

😂

[torpid socket]

Hi there, new here. Didn't find any introduction channel. I'm quite good at precalculus and enjoy it dearly. Hit me for any help, would be glad to do so. I'm excited to help and learn.

[winter comet]

hi

[elder owl]

do any calc questions appear on the SAT? if so, what do they look like?

[river drift]

the SAT only covers algebra

[safe basin]

Yeah no calc on sat

[river drift]

too many students would not have covered calculus by the time they took it

[safe basin]

Altho maybe it’ll provide a shortcut on some questions

If ur familiar with it

[elder owl]

could one potentially do calc on it as extra credit or something

because im doing dual enrollment

so i mightve already taken calc by that time

[winter comet]

💀

[river drift]

there are no extra credit questions on the sat

[winter comet]

imagine if there was extra credit on the SAT

😂

[river drift]

if you want to complete a test that shows your calculus knowledge that's what the ap exam is for

[winter comet]

i mean with dual enrollment you'd be taking the class so you'd get the credit anyway

and the colleges can see that

so... 💀

thats the bonus you get

besides knowing it before college

[winter comet]

if you pass the class as a dual enrollment in college you don't need to pass the ap exam-

[elder owl]

oh

nice

[torpid socket]

Hello

[torpid socket]

Never attempted SAT so cant help. I have taught precalculus to students for their high school.

[chrome ether]

SAT does not go far beyond basic algebra

they way dumbed down the test these last few years from what i can tell

[shadow ether]

pretty sure theres sometimes a bit of easy trig there

there is def geometry tho

[winter comet]

really?

[chrome ether]

i took a "research study" SAT right when i was starting college

it was all algebra 1 stuff 😭

[winter comet]

i thought there were pretty hard questions on there

maybe just hard for me 💀

[chrome ether]

there's only ever like one (1) slightly nontrivial grid-in question

[safe basin]

when i took it there was some basic trig and complex numbers

nothing past whats covered in algebra II tho

[shadow ether]

yeah thats what i thought too

[chrome ether]

i remember there was like

one unit circle question 😱

when i actually took the SAT for college apps

[winter comet]

i swear i thought the sat was hard 😭

now im looking at problems 💀

i really thought it was harder than this ☠️

[viscid thistle]

wait the SAT is just algebra 1?

[winter comet]

i guess

i really don't know what algebra 2 is

but yeah...

[viscid thistle]

neither do i

and im taking it next year

[winter comet]

i meant like i don't know what algebra 2 consists of vs algebra 1

[chrome ether]

it's predominantly algebra 1

just bc the average level of math proficiency continues to drop

[winter comet]

💀

they should like

teach math

ngl

more in school

[chrome ether]

it's not that simple 😭

you've got everything from covid fucking over younger students to politicians meddling in the educational process to internet induced brainrot to

[winter comet]

._.

[winter comet]

btw, do you think linear algebra or differential eqs first

unrelated to what im replying to 💀

[chrome ether]

order is somewhat interchangeable, some parts of diffeq use ideas from linalg

[winter comet]

alr

basic ideas or like

like a basis to linear algebra is good and then either one?

[river drift]

linear algebra is a prerequisite to differential equations at my university

[winter comet]

it isn't here

idk if that means i'd be learning the linear algebra concepts, or if it would be dummied down

[river drift]

the theory of linear differential equations (which are the ones you'll study a lot in an intro course) uses the concepts of linear transformations a fair amount

[viscid thistle]

how do diffeqs and linalg connect

[winter comet]

are linear transforms...like...learnable on khan academy or smthn 😂

[chrome ether]

3blue1brown is a good place to look to get an introduction

[winter comet]

im leaning towards diff eqs because its in person class and linear alg is online :l

[viscid thistle]

oh is that how their connected?

[winter comet]

alr

[river drift]

khan academy does have a linear algebra course, although like most of their college-level courses it's a bit minimal

[chrome ether]

i haven't been on that site in years 😭 how bad is it now

they're snapping up all that "AI" bullshit so i've lost faith in their quality

[winter comet]

yea but would it be enough of an...introduction...

[chrome ether]

that's a bit threadbare hmm

[winter comet]

💀

[fathom kindle]

Yooooo

I love math, anyone willing to sit down and teach me on some math? LMAO

I'll pay n schtuff

[winter comet]

dont pay

[winter comet]

what math 💀

[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/

[winter comet]

if you have questions you can post here and sum can answer

[chrome ether]

if you're looking for tutors this is not the place

[winter comet]

or problems and people can help

[severe night]

how does the teacher get these 2 numbers

[chrome ether]

completing the square

[severe night]

how to do that

[chrome ether]

the way i usually do it

move all the constants to one side

so you have $4y^2-2x^2-4y-8x=15$

[obsidian monolith]

elrichardo1337

[chrome ether]

now factor the LHS partially to get $4(y^2-y)-2(x^2-4x)=15$

we want to turn the left hand side into 4(y-[something])^2-2(x-[something else])^2

that's where completing the square comes in

[obsidian monolith]

elrichardo1337

[chrome ether]

add $4\left(-\frac{1}{2}\right)^2-2\left(-\frac{4}{2}\right)^2=1-8=-7$ to both sides (so, really, subtract 7) to get what's in the picture above

[obsidian monolith]

elrichardo1337

[safe basin]

Idk how much math you’ve done but the connection comes from differentiation being a linear map

with certain conditions I’m lazy to type out

[viscid thistle]

oh huh

wait how

do you make a vector space from a function

im stupid, the vectors are the argument lists

and output values

wait nvm

[safe basin]

so linear basically means it satisfies additivity and scaling, ie if

f(x+y) = f(x) + f(y) and f(ax) = af(x)

So think about how derivative of a sum is the sum of derivatives and how u can pull constants out

These properties make it amenable to linear algebra

[viscid thistle]

ohh

[safe basin]

There are certain details I’m leaving out but that’s the essence

[safe basin]

Do you know about function spaces

[viscid thistle]

no...

so differentiation is a matrix that works on function spaces

what's the actual matrix

[safe basin]

Like C[0,1] for instance is the space of continuous functions on the interval [0,1]

[viscid thistle]

ok

how are they represented as a vector space

[safe basin]

the rzn I’m saying that is to understand when we can recognize it as a matrix

So if you have R[x], this is the space of polynomials with real coefficients for instance

[viscid thistle]

ok

[safe basin]

If you truncate it to a certain degree n, you might see this as P_n, you can represent it as a matrix

[viscid thistle]

but how do function spaces work

[safe basin]

Taking what I used as an example, P_2 would have a basis of {1, x, x^2}

[viscid thistle]

but how are they vectors

[obsidian monolith]

chipotle

[safe basin]

Ignore the commas

[viscid thistle]

wait but wait

what are 1, x, and x^2 as vectors

[safe basin]

U familiar with these axioms?

[viscid thistle]

wait your talking about a vector space representation of GF(2)

[viscid thistle]

yeah

[safe basin]

Ok so polynomials in the space I described would be described thru a linear combination of that basis

Where the numbers in a matrix represent the coefficients

[viscid thistle]

i get that

ohh

so 1 is [0, 0, 1], x is [0, 1, 0] and x^2 is [1, 0, 0]

ooh function spaces seem like exactly what i need to solve a problem

[safe basin]

I think u reversed it but yea

[viscid thistle]

whats the most general function space?

not taylor polynomials

[safe basin]

Hold on there’s something I’m trying to find

I don’t know how much that’ll mean to you but yeah

If you’re looking into this I think any standard linear algebra book would have an exposition on function spaces

Whenever I did it we started with smth like C[0,1]

Where the inner product on it (generalization of dot product) is now $\int_{0}^{1} f(x)g(x) dx$

[obsidian monolith]

chipotle

[safe basin]

Typing latex on phone is so annoying man

[viscid thistle]

yeah

actually i get some of that

what does cauchy sense mean

[safe basin]

Schwartz?

[viscid thistle]

just limits that converge, they converge to values in the space?

[safe basin]

Oh a cauchy sequence

[viscid thistle]

yeah

is that what that means in the chart

but is there a function space

that can capture most normal functions

used in real and complex analysis

[river drift]

a function space is a vector space where the vectors are functions

[viscid thistle]

?

[safe basin]

It means all cauchy sequences converge, idk where to start with explaining tbh

It’s like sequences where points get arbitrarily close to each other

To a particular point (usually)

[river drift]

well the set of functions defined on the real numbers forms a vector space, and so does the set of continuous functions

[viscid thistle]

i know what that means

[viscid thistle]

wait what is this vector space and how does it work

and is there one for any group/group-like object ig/ring/field/whatever absalg thing where you can do functions

and are these differentiation matrices invertible?

[safe basin]

Ah then yeah that’s what Cauchy completion refers to

“a metric space M is called complete (or a Cauchy space) if every Cauchy sequence of points in M has a limit that is also in M.”

Formally

[viscid thistle]

ok

[river drift]

a vector space consists of a set of objects we call vectors, and a definition of addition and scalar multiplication, subject to certain axioms. for functions the usual definition is pointwise, i.e. (f+g)(x) = f(x) + g(x), (k*f)(x) = k*f(x)

[viscid thistle]

i know what vector spaces are

and how that works

[safe basin]

What’s the question?

[viscid thistle]

whats the vector space of all functions c to c

[river drift]

it's just that, the vector space of functions $\bC \mapsto \bC$. i don't think there's necessarily a standard notation for it, although you could call it something like $F(\bC)$ or something (or just say, ``let $V$ be the space of...'')

[obsidian monolith]

cloud

[paper tartan]

why am I seeing functional analysis in the precalc channel

maybe precalc finally updated its curriculum

[white rapids]

Can someone help please?