Hey I'm tryna learn maths, please check the description it's not going to take a lot of your time

[vale stone]

" Name a topic each or a few at a time here and just mind your business go somewhere else if u wish. then I do my research on that and type it in here with explaining how or why and logics with the concept.
you aren't responsible for focusing on what I replied. but whenever you come here to dump your concepts per day or so, you can choose to reply on my replies to someone else and say if I'm correct or wrong or question something.
Like that no one will lose their valuable time in tutoring me. and I would still be able to do things in the way I asked "

and by what topics or concepts u name here, it isn't limited to high school. the bigger picture just should contain all of high school maths, but we could go overboard too

and please don't type paras, just a few or one topic each for a day.
My goal is to complete all of high school maths before may 31st

hello

I am open to answering questions on why I need this

but anyways

to anyone who's here reading this and to everyone who's not

I wish you a merry life:)

[hardy nest]

Well what do you already know

[vale stone]

little

[carmine beacon]

do you at least know algebra 1

[vale stone]

well

yes

but what I know doesn't matter

[hardy nest]

I would say linear equations 😭😭😭😭

[carmine beacon]

I mean it kinda does but ok

[vale stone]

it does but

not based on the topics you give

it would be better if they're in high school year 1 to 4 but it's fine if it's any topic really

I'd love to go deep.

and u don't even have to put a lot of work

[carmine beacon]

if you're gonna go deep you have to have the basics of hs math

[vale stone]

yep

[carmine beacon]

so start doing algebra 1, geometry, algebra 2, then precalc

[vale stone]

I will read all that requires in me learning yor concept

but

[carmine beacon]

and make sure you understand the concepts

[vale stone]

I can't just read books in order I can but it isn't what I feel like to do

[hardy nest]

On a scale of 1 to 10 how proficient are you with lines

[carmine beacon]

tbh for hs math there's no order (except for precalc which you want to do last)

so just start with geometry or algebra 2

[vale stone]

probably a 2 or a 3

[carmine beacon]

I can find a decent textbook for you

[vale stone]

no I don't want textbooks

you just name topics

and I study all that requires to go in depth

even if it requires a textbook

[carmine beacon]

ok I'm confused

[vale stone]

what I wanna do is "I" Going into it myself

[carmine beacon]

why are you doing this

do you want to get really deep into mathematics

[vale stone]

yes

[carmine beacon]

do you wanna learn it for like school

[carmine beacon]

ok

[vale stone]

no

well physics but

physics requires math

and both require deep thinking

so I like them both in a way

[carmine beacon]

ok so your final, overall goal is to excel in physics?

[vale stone]

uh

yeah

but while I'm on discord it's gonna be maths

alright please stop focusing on the clutter

[carmine beacon]

alright this is def gonna require textbooks

search online for an algebra 1 textbook

and work through it

repeat this for geometry and algebra 2

and precalc

[vale stone]

just say a few or a topic you wish to (if u can please) and I will send my responses after researching it with textbooks, web pages, and courses

[carmine beacon]

for calculus

use ocw.mit.edu and look for the calc courses

[vale stone]

alright thanks for the resources

[carmine beacon]

I'm really free rn so I can find some good textbooks for yo

u

https://bim.easyaccessmaterials.com/index.php?location_user=cchs

use khan academy as practice

and for precalc

[hardy nest]

Well then maybe the first topic I recommend is linear spaces. So essentially all of the following

• linear equations (y = mx + b)
• linear functions & properties
• system of simultaneous equations
• basic intro: vectors & matrices
• "linearity"; the general concept
• linear transformations
• vector spaces

You don’t have to dive deep into the last one, but this would give you an exceptional knowledge on "lines” and their properties. Vectors are important for physics

[carmine beacon]

https://iblog.dearbornschools.org/mrkcalc/wp-content/uploads/sites/1385/2017/09/Precalculus-by-Demana-Waits-TEXTBOOK-1.pdf

if you're looking for the traditional method of learning math I'd recommend my way

but if you want a more abstract method

you should go with @hardy nest 's method

[vale stone]

I don't really want resources, I like the feeling of going into it on my own. I just want concepts for explaining each day to make it all feel human and fun for me

[vale stone]

alright thanks

I will reply to you after I understand those and when I can defend those

[carmine beacon]

alright then I'd recommend rel's method

best of luck in your mathematical endeavors

[vale stone]

thanks

[hardy nest]

If you really want to btw you can ask for help here for the things you’re having trouble understanding, but good luck to you

You should be able to determine if some function f : A → B is considered "linear" or not based on how it behaves… you'll study this when you look at linearity & linear transformations

[vale stone]

ah I see alright

thanks

I was looking for something like this

where u give a concept for me to research.

[cunning quail]

I would suggest the topic of algebraic understanding of power of point like a function on the plane

[fallow quarry]

I have some fun topics for you to research!

Give these a try:

• Cartesian Products
• Set Theory Math Symbols
• Derivation of e
• The Fundamental Theorem of Arithmetic
• Decimal Binary Conversion

[heavy furnace]

okay thanks

[hardy nest]

@vale stone btw hows it gong

[heavy furnace]

hey

so

here's what I found out till now

after watching like 20 minutes of videos

so

a linear equation

ax + by

= c

where a b and c are constants

they'd make a line in a graph!

okay

now

if I have another equation

a2 x + b2 y

= c2

where a2 and b2 and c2 are constants as well

I have another line in a graph!

so

the thing where you want a solution

mainly comes based on intersecting points

for example

you got a line stretching from y axis 7 to x axis 8

that's the first equation

or thingy

and u have another line of the equation

which extends from 0 to somewhere

if they intersect

at ONE point

then

that's what is called

a UNIQUE solution!

BUT

if they are lines which are on TOP of each other in the graph

then they have infinite solutions, because u can just add or multiply each thingy thingy

that's called a

infinite solution?

anyways

so these 2 are where you actually get solutions of inter sections

BUT

what if

The lines are parralel

then you get something called

no solution-

that means there's no solution at all

it doesn't mean anything

so mathematically tho u dont have to draw graphs all the time to DETERMINE if theyd give solutions and which type or not.

there's a simple thing like a trick or something like a law.

what it means is

If a1 by a2 is not equal to b1 by b2

then it only has one solution

and it intersects

what they really mean here is

[hardy nest]

@heavy furnace are u usap on a different account

[heavy furnace]

the a part in the first line

[heavy furnace]

totally not

[heavy furnace]

by it's part in the second line (what I mean by a1 and a2 is, the constants which x is multiplied to)

if it's not equal to b part (b1 times y in the first line and b2 times y in second line) aka b1/b2

them you'd only get one solution

because logically you cannot change the variables to get many answers

so only one

alright moving on

to know if you'd get infinite solutions

you have to check if a1/a2 = b1/b2 BUT also = c1/c2 (the answer in the first line divided by the answer in second line)

if that's true and they all equal

then you have infinite solutions

why?

because you can change things and add or multiply and they'd differ in constant answers but remain the same pretty much

uh that's not grammatically correct but somewhat like it

but

how do u know if u dont have a solution?

you check if a1/a2 = b1/b2 BUT is not equal to c1/c2

because

logically

if the thingies that multiply x in the first line and the second line are divided aka a1/a2, and they EQUAL to the thingies that multiply y in the second line aka b1/ b2

are NOT equal to the thingies that answer the whole thing divided as c1/c2

then there's no sense

because x and y are same in both lines

so if a's and b's are samely proportional or equal in fractions

the c's should be too

but if they're NOT

you won't get a solution

this is what I learnt in the 20 minutes of YouTube videos

I also learnt that

you have to convert the right answer to a constant or something

for example

2x + 3y +2 = 0

and 4x + 6y -4 = 0

you can't do it before logicalizing it

so

take the +2 and -4 and take it to the right side

and you'd get

2x + 3y = -2

and 4x - 6y = 4

now 2/4 = 3/6 but is NOT equal to -2/4

so you have a parrelel line formation that doesn't inter sect to give a solution

and boom you're gone

there's also many more things of this chapter

that are covered by a 2 hour video

and I learnt this in the first 20 minutes of it

also

I think I can learn by these one shot completion videos of chapters on YouTube. upto an extent

they seem pretty human and easy to me.

[hardy nest]

Yes essentially, a linear equation is any equation that can be written in the form

y = mx + b
or a(x) + b(y) = c, where a, b and c are coefficients

If two lines have the same rate of change, or slope, they would never intersect, meaning they are parallel. The only time they would intersect if they are the exact same line

However if they have different slopes, they will always intersect somewhere

Unique solution → a solution is unique if it satisfies a given problem

Infinite solutions (dependent) → lines are dependent if they are the same line everywhere - if one if a scalar multiple of another. More specifically something is considered dependent if it can be written as a linear combination of something else but you'd learn this more when you learn vectors
y = 2x + 6
2y = 4x + 12
These are the same lines, and they intersect everywhere so when trying to find the intersection points you'll find out they are the same lines

Inconsistent → not just for lines, but if a system of equations is inconsistent it means there are no solutions

Parallel lines are two different lines with the same slope, they never intersect
Perpendicular lines are two lines that intersect at exactly a 90° angle. Two lines are perpendicular if their slope are negative reciprocals
y = 2x
⇒ y = -(1/2)x
These two lines are perpendicular

My explanation could definitely be more rigorous but that’s the general concept at least

[heavy furnace]

hey just a question

wdym by y = mx + b

I can just say 2 = 0.5 times 1 + 1.5

would that be a linear equation?

doesn't linear mean the power of it is 1 ?

[hardy nest]

y = mx + b
⇒ m is the slope
⇒ b is the y-intercept, it's where x = 0 on the graph, it is literally where the line touches the y-axis

y and x are variables, inputting an x gives you a 'y', assuming you know what 'm' and 'b' are

[hardy nest]

A linear equation is a equation where the highest exponent is 1

[heavy furnace]

What exactly is the slope?

[hardy nest]

The highest exponent is 0 and I don’t even think that equation is true. I don’t think the right hand side equals the left hand side when combined

[heavy furnace]

ah I see

[hardy nest]

The slope of a linear function is the steepness of the line, or the rate of change. Suppose you know two points on the line (x1, y1) and (x2, y2), you can find the slope, m, by finding the change in y over the change in x. The "change in" something is notated with the delta (Δ) symbol,
m = (Δy)/(Δx)
⇒ m = (y2 - y1) ÷ (x2 - x1)

A function with a slope of 'm' has a rate of change of m y-units per 1 x-unit. It moves vertically 'm' units and horizontally 1 unit

[heavy furnace]

what is the slope over here

I can't exactly understand how to get the change of y or x

I know what it means

but what exactly is it

[hardy nest]

Notice how when y = 2, x = 1. The slope is 2 because for 1 change in the x-value, there is a change of 2 in the y-value

y = mx + b
y = 2x + 0
⇒ intersects the y-axis when x = 0
y = 2x

The slope here is 2

[heavy furnace]

OHHHH

so lowkey like proportions

if it's a 5 on the x axis and 15 on the y axis

then it's a 3 slope

because for every chunk of x, there's 3 chunks of y

ah I see

I see

wait

doesn't that mean the highest integer in x,y is the slope

?!?

oh no

similar but no nvm

[hardy nest]

Consider you know two points on a line, (1, 2) and (2, 4) and the function intersects the y-axis when x = 6. To find the slope, we subtract the change in y over the change in x

Δy = (4 - 2) = 2
Δx = (2 - 1) = 1
m = Δy/Δx = 2/1 = 2
The slope is 2, but what about the y-intercept? If the graph intercepts the y-axis when x = 6, the y-intercept (b) is simply 6.

y = mx + b

y = 2x + 6
↑ also because, when x = 0, you get y = 6 as a result, and this is exactly what the problem said

The change in something is simply the difference between its constituent values, it doesn’t matter what way you subtract/find the difference, because eventually you will end up with the right slope

It’s just recommended to do

higher - lower

So you can avoid negatives

[heavy furnace]

so, I have to points one here. and one. . here
the first point is 1 of x axis and 2 of y axis, and the second is 2 of x axis and 4 of y axis.
when x = 6 , the line intersects y axis, so it takes a turn somewhere. to find the turn, we subtracy the change in y over the change in x. how
you mean the difference in the first points' y axis and the second points' y axis to the vice versa for x axis?

so bigger to smaller aka.
2-1 for x and 4-2 for y.
so I get 1 and 2
and the m aka the change of slope rate is the change in y over the change in x basically triangly y divided by triangly x

and the triangly y is 2 because 4-2 bigger to smaller of the 2 points

and triangly x is 1

[heavy furnace]

okay but what if I don't have two points, like here?

I just do y/x or x/y and that's the slope?

but hows that the slope I don't know for sure if the lines bending or turning somewhere

[hardy nest]

It doesn’t “take a turn” it just intersects the y-axis.

When x = 0, y = 6. This means when the line is at the y-axis (x = 0), the y-value is 6

[hardy nest]

Lines don’t bend

[heavy furnace]

okay how do I know for sure it isn't a straight line

[hardy nest]

If the line “bends” or somehow curves it’s not a line

[hardy nest]

If you see an exponent higher than 1, the graph would not be a straight line

[heavy furnace]

I see

[hardy nest]

For example
y = 2 + x^2
y = x^3 - 2x^2 - 3x
These aren’t straight lines anymore, so the concept of “slope” is a bit more abstract

Slope is specified towards straight lines that change at a constant rate

[heavy furnace]

oh but u told me it intersects at y axis if x is 6?

wait

uh

[heavy furnace]

oh I see

so something that takes turns sharply or changes sharply without bending

precisely* not sharply

[hardy nest]

Algebraically though, to determine if something is a “line” you simply look at the exponent. If you see any exponent higher than 1 it’s not a line

[heavy furnace]

why is that

for example if I have an equation x square + y square is a point

it can be 3 square and 2 square giving 9 and 4

right?

why's it not gonna be a line

[hardy nest]

If you see an exponent that is 0 or 1, it is a line, but equations with an exponent of 0 are horizontal lines meaning they have don’t change
If the exponent is 1, it’s still a line except it’s a diagonal line and it does change according to its slope

[hardy nest]

Sorry I should’ve specified
Something is considered a line in algebra if the highest exponent on a variable is either 0 or 1

What truly matters is the exponent of the variable(s). If at least one variable has an exponent of 2, it’s not a line

We call this "highest exponent" as the degree

[heavy furnace]

what's the basic logic of it tho, aren't equations to take points and draw lines between them? how do you curve something between 2 points there?

[hardy nest]

The degree of a (non constant) linear function is 1

[heavy furnace]

hmm

so

a constant linear function can be any degree

like 3 square, etc..

but the x and y parts

should be just 1

but why?

you're anyways gonna multiply those constants to variables

hmmmmm

uhm

I think

I'm cooked somewhere

[hardy nest]

Oh this is a line, because

1. There is no variable, which implies the degree is 0. Polynomial functions with a degree of 0 or 1 are lins

y = 3^2
⇒ y = 9
This simply creates a horizontal line at y = 9 that does not change

[heavy furnace]

hmm

wait

uh

ax + by = c

where do I put these in graphs

the ax part as x axis and by part as y axis? right?

[hardy nest]

With a function, the general format of it is
x → f(x) → y
For every x you input into the function f(x), you get a y. If the highest degree is 0 or 1, the y's will change linearly, they are just scaled by any other coefficients or shifts

y = 2x + 2
This is a linear function, because consider the chart…
y(0) = 2(0) + 2 = 2
y(1) = 2(1) + 2 = 4
y(2) = 2(2) + 2 = 6
y(3) = 2(3) + 2 = 8
y(4) = 2(4) + 2 = 10
For every x you input, it returns a y that is 2 times the value of the x plus 2. We inputted x = 3 and got y = 8….
8 = 2(3) + 2
↑ two times three plus two

The fact that there are other factors that change the relationship between x and y does not change the fact that it is a linear relationship

[heavy furnace]

I see

I see

I see

[hardy nest]

ax + by = c
Typically, it would be best to isolate y before you try to graph it, this means get y alone and by doing this, you see exactly what y equals,

2x - 3y = 2
-3y = 2 - 2x
y = (2 - 2x) ÷ -3
y = -(2/3) + (2/3)x

This means for every x we input, we will get a value (y) such that y is (2/3) times x plus a negative -2/3

[heavy furnace]

f of x gets a y, and f of x is 2 x + 2 then y is 2 x +2 . it starts at y(0) = 2(0) +3 = 2 . For every x you input it returns a y that is 2 times x plus 2 .

OHHH so it doesn't really change and it stays same so u can input any x u want so if I say x axis has a 3 . then you get a y axis of 8 because f of x is 2x +2 which gives y.

and if u get 3 chunks right side and 8 chunks up side. that's a function. so is a function just a point? and when does the non linearity come. Can u give example of that?

wait

I think I got it

can I keep f (x) = y where y = 2x -3y = 2?

wait but

huh

never mind I didn't get it

ok wait

let me read what u said above

properly

it's 4 am I'm just a bit clumsy rn

[hardy nest]

In math, a function is a special type of relationship between the range (y) and the domain (x). Essentially, for every (x) inputted into the function, there cannot be different (y)'s for the same x.

Consider this

f(2) = 3
f(3) = ±4

This is not a function because when you input x = 3, you get two separate y values, y = -4 and y = 4

[hardy nest]

A function is not just a point, it’s a relationship between two variables, the independent variable (x) and the dependent variable (y, which depends on what x you input)

[hardy nest]

You want the definition of linearity?

It might seem confusing right now because you’re missing some key topics

The first one is a function because for every x, there is exactly one y.

The second one is not a function because both 'b' is mapped to '2' and '3' indicating that for every x, there is not exactly one y (where b is the x)

[heavy furnace]

I see

which type of equation would give 2 answers tho

OHH

if the degree is 2 or more

u can write -2 times -2 too

OMG I UNDERSTAND IT NOW

so it must be linear to be considered a function

ah I see

[hardy nest]

Wait wait wait

No 😭

[heavy furnace]

huh

no?

so it can be illeniar?

but how? If it's 2 or more degrees u get multiple answers

or how do u actually get the 2 and 3 too

[hardy nest]

It does not necessarily have to be linear, but every for x that you input, it must map to ONLY one output. Let’s say this analogy

Consider a toy machine where you input a toy and it gives you another random toy. Let’s say you input a red toy and get a green toy, this is fine. But let’s say you input a blue toy and also get a green toy. This is not a function because both the red toy and blue toy correspond to the same toy (green toy).

[heavy furnace]

I see

but a question

if y is x square

then there are multiple answers no?

wait

ok yea yes

but then how can

illeniar things be included?

[hardy nest]

x^2 + y^2 = 2

This is an example of an equation, but it is not a function, because try isolating one of the variables.

y^2 = 2 - x^2
y = ±sqrt(2 - x^2)

where sqrt() is the square root function, √

This is not a function because when you input an x, you get two outputs (a positive y and a negative y)

P.S, x^2 + y^2 = 2 is the equation of a circle.

[heavy furnace]

okay

I think

I'm getting the gist of it

a bit a bit

whenever I see an equation it must contain x and y and both after the plus sign and must have a constant = answer.

[hardy nest]

Notice that for any x you go to in this graph of a circle, you get two y's

y = 0.7 and y = -0.7

(Approximately)

For when x = 1.2

[hardy nest]

Do you mean something like

y = x^2

[heavy furnace]

yeah

-4 times -4 is same as 4 times 4

so why is it being a function?

oh

I think

I get it a bit more

the answer is the same

but like the circle picture it has a different y factor

the answer always leads to 16

which doesn't differ

[hardy nest]

This is a function. It’s a very popular type of function that you explore in high school math after linear functions, called a quadratic function — a polynomial of degree two.

This is a function because for any x you input, you get exactly one y. Consider x = 2

y = (2)^2
y = 4

When you input x = 2, you will always get y = 4. But what about x = -2? You also get 4,

y = (-2)^2
y = 4

But this does not break the rule of the function. x = 2 and x = -2 produce the same output, but they are not the same input.

I just realized a flaw with my toy analogy. Let’s say you input a green toy in the machine and get a red toy, let’s say later you try to input the green toy in the machine again and get a blue toy. The green toy corresponds to both red and blue? It corresponds to two different toys, which means one input (green toy) corresponds to more than one output (red toy and blue toy), this toy machine is not a function.

[heavy furnace]

ah I see

[hardy nest]

If something isn’t a function, it means that when you input a singular x-value, you get more than one corresponding y values. Although, different x-values can correspond to the same y-value.. an example is this constant linear function

y = 5

This is secretly the same as

y = 5x^0

Where the degree is 1, because anything raised to the zeroth power is 1. If you input x = 2, you get 5, if you input x = 7, you also get 5. It does not matter what value of x you input, you will always get 2. The key thing here, is that, a singular x-value does not correspond to more than one y-value. It always corresponds to y = 5

[heavy furnace]

Hmmm

I see

I see

so

you mainly focus on the x value

that's why you give it priority during the equations too

and the question giver gives what f(x) = y is, so they can give y = 2x +2 or anything

so once I get x

I check that

they don't give more than one y value

alright

but

hmm

but

a function is a point then?

it's just a point... right?

a special type of point that isn't on the lines

you get a line only when u have 2 functions?

or do u get one with just one?

[hardy nest]

A function is a relationship between inputs and outputs

But on a graph, a function is a shape because of the behavior between x and y as you input different x values that corresponds to different y values. it draws a shape and connects it.

[heavy furnace]

yea but

but

if u take x as 3

you'd only get a point

on the paper

which is a x input's y output

[hardy nest]

It just tells you what is y when x = 3, so I guess you can think of a function as a connection of points

[heavy furnace]

so

u can change x

and get many points that are

sequancially similar

and then draw a line between them

And then calculate the slope

right?

but what do u do after that?

[hardy nest]

You can input an x into some linear function then get a y, you can do this with a few points and notice the general shape when you connect the points. According to intuition, you would fill the rest of the graph which a straight line (for linear functions)

[heavy furnace]

I see

if the function is linewr

it is a straight line

BECAUSE

it gives the same type Ofvalues

but if it's illeniar

it can change

only then the slope comes in

[hardy nest]

As you graph more and more, the shape of functions would be very intuitive to you; for example you would already know the general shape for functions like these

y = 3x - 2
y = 6x^2 - 12x
y = 4sin(x) - 3
y = 5(2^x)
y = 2log(x + 2)

etc….

As you progress more through the study of functions in high school, you can easily visualize the function on a graph as you graph and understand functions more

[heavy furnace]

WHAT

well uhmmmmm

[hardy nest]

?

[heavy furnace]

I may wanna say you

I don't know what sin or log is

haven't touched trigonometry yet

but

[hardy nest]

Well a function is linear because it has a constant rate of change. As x changes, y changes at a constant rate — that is the slope, even if there is a vertical shift or stretch of the function because of the coefficients and other values

[hardy nest]

Oh sorry

Log isn’t trigonometry

Do you know what an exponent is? Exponential function?

[heavy furnace]

yes

exponent I know but

I'm not really well affirmed with it

For example

[hardy nest]

You don’t have to know everything about exponential functions but

“Log” is the inverse of the exponential functions. The full word is the logarithmic function

[heavy furnace]

yea

I tried it

in calculation

it gives a small integer and a really complicated decimal

I think it's making big numbers small

[hardy nest]

It’s the function that undoes the exponent

2^x = 8
To solve for x, you would log both sides
log(2^x) = log(8)
x • log(2) = log(8)
x = log(2^3) ÷ log(2)
x = 3log(2)/log(2)
the log's cancel
x = 3

[heavy furnace]

by taking it in between of integers

[hardy nest]

Using the properties of the logarithm

I don’t wanna make this confusing for you tho sorry

[heavy furnace]

No no

it's not confusing at all

it's very fun actually

[hardy nest]

sin(x) isn't as scary as you think but it is different than the functions you are typically used to

The sine function is a periodic function so it repeats after a certain interval

[heavy furnace]

I see

[heavy furnace]

wait 3log(2) /log(2)

[hardy nest]

The shape of it is like this

[heavy furnace]

oh

log (2) / (log(2)

cancel

okay

[heavy furnace]

I see

think I've seen it somewhere in a video

cool shape

that's sin but

I thought sin was for triangles

[hardy nest]

log(8) = log(2^3)

According to logarithmic rules, there is a rule that states

log(a^b) = b • log(a)

When raising something in the logarithm by a power, you can "factor" the power out of the logarithm

[hardy nest]

Its more complicated than that

[heavy furnace]

I see

[hardy nest]

It is for triangles but it’s more connected to circles

You learn it when you’re in algebra 2

[heavy furnace]

uh people keep saying me about, algbera 2 , calculus 1 , calculus 4 etc..

are these uh

levels of it or something?

[hardy nest]

Well it depends on where you are in the world

[heavy furnace]

I see

[hardy nest]

In the US, there are courses such as "Algebra 1", "Geometry", "Algebra 2" and "Precalculus"

Algebra 2 is typically taken as a 10th grader

[heavy furnace]

ah I see

around 15

[hardy nest]

Yes

[heavy furnace]

same here too

[hardy nest]

Calculus 4 isn't as standard

But I see your point

[heavy furnace]

hmm

[hardy nest]

But for linear functions and stuff do you have questions, it could be related or unrelated

[heavy furnace]

do u know

when I could calculate the rpm of my fan or the weight in tons of my home's cement / concrete?

or the durability

[hardy nest]

Rpm ? Rate per minute

?

[heavy furnace]

no rotations per minute

if I turn on a fan

[hardy nest]

Oh

[heavy furnace]

is that included in this somewhere?

[hardy nest]

The rotations per minute, I’d have to look into that

But the weight in tons of your homes cement, yes that’s in the domain of calculus

[heavy furnace]

I see

[hardy nest]

Calculus deals with rates of changes and accumulations (area, volume, etc). It allows us to find the area or volume of irregular regions, and calculus is actually where those simple algebraic / geometric formulas for the area or volume of shapes come from

[heavy furnace]

SO I'D BE ABLE TO take a paper, and a pen, and write how much years a shop will last or the weight of it in tons and then impress the shopkeeper to get me a free bag of chips?

[hardy nest]

It generalizes the concept of "slope" to nonlinear functions

[heavy furnace]

hmm I see

[hardy nest]

Such as y = x^2 or y = x^3 - 2x^2

[heavy furnace]

I have a question though

how do u

apply it to irl objects

I mean

how do u take the shape in graphs

from irl objects

[hardy nest]

Wdym how many years the "shop will last"

[hardy nest]

This is not really covered in calculus 1 or 2, but it is in calculus 3 (multivariable calculus)

[heavy furnace]

when the concrete will fall or break although?

[heavy furnace]

I see

I see

calculus seems fun

[hardy nest]

If you let the region R be a 3-dimensional irregular region in space, then to calculate the volume of it, you would need to attempt to model this shape as a function and consider the bounds you are finding the volume inbetween of

[heavy furnace]

I see

[hardy nest]

It doesn’t have to be a function, but an equation

[heavy furnace]

there's gonna be 3d graphs tooo omg this is gonna be soo fun, I feel like doctor Strange

[heavy furnace]

I see

what are you doing right now?

in the sense which course or which calculus or what-

[hardy nest]

The lines and linear functions?

[heavy furnace]

uh nnno-?

I mean

what do u mean

lines and linear functions?

where

[hardy nest]

Like are you asking what class does this linear function stuff fall in?

What we did earlier

[heavy furnace]

oh yea sure

[hardy nest]

Alg 1, geometry, alg 2, calculus, etc?

[heavy furnace]

although I was asking what you are u doing-

like irl

[hardy nest]

Typing right now 😭

[heavy furnace]

I meant something like quantum mechanics phd Or uhmm- a COURSE

like

a studying course

[hardy nest]

Right now it’s in the scope of algebra 1, but that’s only because we’re just getting started. Matrices and vectors are typically first introduced in precalculus (after algebra 2, which is usually 11th or 12th grade). Everything above that like linear transformations and vector spaces are in a college level course called linear algebra, it's usually taken after calculus 1 and 2, and before calculus 3

Don’t be fooled by the name "linear algebra" because it’s way more complex than the usual y = mx + b stuff

[hardy nest]

What I do irl like specialization or field?

[heavy furnace]

yeah

[hardy nest]

I’m still in high school 😭

I just happen to know almost all of calculus 1, calculus 2, some of multivariable calc and a good chunk of linear algebra

[heavy furnace]

ah I see, that's what I was thinking. when I ordered my quantum mechanics book, the guy talked about linear algebra and calculus. and this seems so easy so I thought why I struggled a bit there.

[heavy furnace]

oh I see

[heavy furnace]

huh

you're 14!?!

or 15

and you know all this

[hardy nest]

I won’t disclose my age but im still a minor

[heavy furnace]

ah I see

[hardy nest]

I self studied it myself

[heavy furnace]

no wonder you seemed smart

alright I think I'll go learn algebra 1 and algebra 2 and geometry now

I found yt vids that explain them in hours completely

tysm for helping me

[hardy nest]

An example of an average linear algebra question can be something like

Consider the matrix A,
A = [2 5 1]
[3 1 2]
[1 0 7]
1a) Find the eigenvalues and eigenvectors of this matrix A, which are the vectors v(→) that satisfy Av(→) = λv(→).
1b) Write the eigenspace for this matrix which is the set of eigenvectors and interpret what it means in terms of a vector field.
1c) Find the set of eigenvectors that form a basis in A. In other words, find the eigenbasis and determine what it means in the context of matrix A.

It gets more difficult but I couldn’t think of anything else

Not to mean to overwhelm you but linear algebra isn’t just lines everywhere

[hardy nest]

I can link some videos here too, wait

https://youtu.be/Ft2_QtXAnh8?si=C3bQ1b5prQKz4gPZ

This video is probably perfect for everything we just did. It might expand on some topics more than I did as well

[heavy furnace]

oh thanks

thank you

[heavy furnace]

yea I can guess that

[hardy nest]

Desmos has a 2D graphing calculator, you can experiment with it and just input random functions, if you have any questions about it you can ask here

[hardy nest]

Thank you, I only self studied it because I accidentally stumbled across the first topic in calculus and looked more into it.. now I’m here

They have a 3D graphing calculator website as well. For example,

[heavy furnace]

I see

okay I'll check it out

[heavy furnace]

oh hmm

[hardy nest]

To find the volume of this 3D shape, you'd use calculus, but more specifically you would use something called an integral. The process / mathematical operation of using an integral on something is called integration. You would integrate the function f(x, y) in respect to both the x, y and z axis.

The symbol for an integral is an elongated S symbol,

∫

and it is always followed with a differential at the end, something like 'dx' , 'dy' , etc. This differential is the width of the infinitesimal rectangles enclosed between the function and something else

Volume = ∫∫∫ (x^2 + y^2) dV
R
where dV is the differential representing volume and R is enclosed region within the function. R is a set containing the "bounds" or inequalities for the shape, these inequalities are the bounds you input in the integrals.

I hoped this make sense conceptually, but you don’t need to worry about the math right now

[heavy furnace]

oh

yea

there were something called integrals and deriviatives I heard

[hardy nest]

∫∫∫ f(x, y) dV
This is a triple integral, and it's first learned in calculus 3 as well as double integrals. In calculus 1 and 2 you focus on single integrals (∫)

[heavy furnace]

and limits I'm pretty aware of

[hardy nest]

Perfect actually

The derivative

[heavy furnace]

o I see

[hardy nest]

The derivative is the generalization to slope, as I said earlier

[heavy furnace]

oh

y/x is a derivative?

I mean the triangle y by triangle x

[hardy nest]

To find the derivative of a function, or the "slope" of a function at a specific point x = α (the rate of change at the function when x is α), the mathematical operation is called differentiation

The Fundamental Theorem of Calculus (part 1) essentially states that differentiation and integration are inverse operations.

[hardy nest]

y = mx + b

The "derivative" of a linear function is always the slope

[heavy furnace]

Oh

[hardy nest]

A limit looks at a functions behavior as x gets closer and closer to some value, α, without actually reaching it. We say that "x approaches α"

[heavy furnace]

I have a doubt

[hardy nest]

The limit returns the y-value that the function approaches as x approaches α (x → α)

This states, the limit as x → c on the function f(x) is equal to L… the value of L is the value that y approaches as x approaches c.

[heavy furnace]

so an x and y being 1, 2 and another of the next of that function being 2, 4
The slope would be 4-2/2-1 aka the slope would be 2.
and supposedly this is the derivative. how or what is integration then? u add and multiply?

[hardy nest]

It’s gonna be hard to explain integration in baby steps, can we briefly talk about differentiation first (if you would like to continue), then integration?

The definition of the derivative is actually simple if you know what the average rate of change formula is

[heavy furnace]

Oh I see

yeah sure then

[hardy nest]

What the integral does is it accumulates values enclosed between the function and the x-axis, effectively finding the area enclosed within anything

Conceptually

[heavy furnace]

Oh

Wait

I think

I understand it a bit

there's a shape

that has lines with

[hardy nest]

We don’t have to understand the math behind it right now, just the concepts

[heavy furnace]

yea

the concept

[hardy nest]

Are you familiar with this formula

?

It’s… literally what we did earlier actually 😭

[heavy furnace]

what's the

f(b) - f(a)

[hardy nest]

It’s the function f evaluated at x = b and x = a respectively.

Consider the function,
f(x) = x
when x = 1 or x = 2, we say
f(1) = 1
f(2) = 2

f(b) just means the function evaluated at x = b. It is the output retrieved when you input b.

b is a variable and can represent an actual value… but for right now it’s a variable

oh cool I got the active role 😭

[heavy furnace]

Lol

uh

one sec

Let me try to process it

[hardy nest]

It might be unfamiliar if you’re not familiar with function notation such as f(x), g(x) or h(x)

[heavy furnace]

"it might be unfamiliar if you're not familiar"

lol

alright so

[heavy furnace]

how does that relate to y

I mean

how is it y2-y1/ x2-x1

or

[heavy furnace]

oh I see

so f of x is x

x is just itself

[hardy nest]

y = f(x)

[heavy furnace]

okay but f(x) = x

so

y = x

[hardy nest]

y = f(x)
f(x) = x

Which just implies y = x

[heavy furnace]

what-

[hardy nest]

But in higher level math we don’t use "y" much

[heavy furnace]

oh

they don't like it?

[hardy nest]

f(x) just means a function that inputs x

[heavy furnace]

I see

[hardy nest]

f(2) means the output of the function when x = 2. In other words, it’s the y-value when x = 2

[heavy furnace]

Okay but

to understand the slope

don't we need the y value OVER the x value?

why have they said f(b) - f(a) then?

[hardy nest]

Yes it’s the change in y over the change in x

[heavy furnace]

oh

[hardy nest]

f(b) just means the output of the function when x = b. It’s the y-value produced after inputting x = b

f(b) - f(a) ⇒ y2 - y1

They’re kind of interchangeable

[heavy furnace]

I see

I see

OH I SEE

that's why u said b = y

or itself

y = itself then

b = b

so y = y

so y2-y1 is same

but

u can calculate it for x too

and by that by by that

so

It is y2-y1 / x2 - x1

but I don't see any

divided by sign

where am I going wrong here then

what does it really say at the end

[hardy nest]

'/' is the division symbol

It indicates a fraction

Which means you are doing the numerator ÷ denominator.

6/3 ⇒ 6 ÷ 3 = 2

That’s why the fraction 6/3 simplifies to 1

If you consider the graph of the function y = x^2 - 2, and wanted to find the average rate of change between the point (0, -2) and (sqrt(2), 0), you would use the slope, or average rate of change formula

The line between these two points is also called a secant line which will be important later on

[heavy furnace]

Oh

[hardy nest]

A secant line is simply a line that passes through two (or more) points

Key word: line

[heavy furnace]

Oh

wait

it's still tryna

unclear

why is there

f(b) - f(a)

and b - a on the down of it or near x

if f(b) - f(a) really is y2 - y1

then

b - a how can it be equal to x2 - x1

[hardy nest]

Does this make it clearer

f(x) = 2x

let’s say a = 1
f(a) = f(1) = 2(1) = 2

So when a = 1, f(a) = 2 because the y-value returned when x is 1 is equal to 2.

[heavy furnace]

Oh

Wait

I think you're onto something

so

f(a) is y2 but it's of 'a'

are u tryna tell me

that a is x2 Or x1?

[hardy nest]

Yes basically, f(a) can be read as "f of a"

You know how for functions like f(x) we say "f of x"

Suppose a was some numerical value like a = 2.

We would be saying "f of 2" instead, which simplify means the function's output value (y) when the input is 2.

[heavy furnace]

I see

I see

but

we are calculating slope

this includes the y factor

but

where's the x factor then

WAIT

the x factor IS

OMG

OMG

OMG

OMG

there's no SPECIFIC X

whatever u take

IS x axis

so

2,1

is basically

huh

one sec

One sec

one sec

shit

alr

2,1

f of a

what is f of a

it's 2x

okay

it's 2x

so f of a or y is 2

okay

f of b

4,2

shit

but

wait no

I'm not cooked

I'm not cooked

f of a or y is 4

f of b or y2 8

so f of b - f of a

is

y2 - y1

and b - a

IS X2 - X1

OH MY GOD

OH MY GOD

OH MY GOD

I'M ONTO SOMETHING

[hardy nest]

😭

f(b) - f(a)
= (y2 - y1)

b - a
= (x2 - x1)

y2 is simply the value returned when you plug in 'b' for x into the function f
(where b is a numerical value like b = 2, 3, 10, etc… depends on context)

[heavy furnace]

bro

I just said that

but it took me sooo long

but

how

HOW IS THIS

how was this

created

From stones and trees

[heavy furnace]

Alright

Let's move on

okay

now I understand

the slope formula basic

[hardy nest]

Ok do you understand the average rate of change and secant lines

The average rate of change between two lines is given as the "slope".. so consider two points
(a, f(a)) and (b, f(b)), the average rate of change between them is

(f(b) - f(a))/(b - a)
⇒ (y2 - y1)/(x2 - x1)
where (x1, y1) = (a, f(a)) and (x2, y2) = (b, f(b))

[heavy furnace]

yes

okay

I don't like the multiple brackets but yes

I think I feel it

[hardy nest]

Ok well

[heavy furnace]

alright

[hardy nest]

What if we say
b = a + h
where h is some value, the difference between a and b? For example,

a = 7
b = 10

b = a + h
⇒ b = 7 + h, where h = 3 right?

[heavy furnace]

uh

uh

one sec

okay

b is 10

then h is 3

yea

ok yea

[hardy nest]

We can change the average rate of formula to be

(f(b) - f(a)) ÷ (b - a)
(f(a + h) - f(a)) ÷ (a + h - a)

⇒ (f(a + h) - f(a)) ÷ (h)

[heavy furnace]

uh

f of b

Minus

f of a

total bracket

Divided by

b-a

yea that's the formula

f of a + h

minus

f of a

[hardy nest]

Do you underused that , the algebra behind it

[heavy furnace]

total total bracket

[hardy nest]

We’re extremely close to getting to the definition of the derivative

[heavy furnace]

Yes as I'm talking I'm tryna understand

[hardy nest]

Ok hold on

[heavy furnace]

divided by

a+h-a

hmmmmmmmm

one sec

I'm cooked

lemme try to deep think it

okay

okay

OHHH

instead of b- a

it's a + h

no

lemme deep think it again

[hardy nest]

When we let b = a + h, in the denominator, the a's cancel out giving us only h in the denominator

[heavy furnace]

Okay

I understand it

I understand it

yes

although

I feel we are going in a rabbit hole

a very deep rabbit hole

soon I won't be able to remember that it's y2 - y1 / x2 - x1

if it goes like this

[hardy nest]

Youre generalizing the concept of slope to all functions

All functions will have a “slope”, that’s called the derivative

Anyways

[heavy furnace]

alright

[hardy nest]

Do you know what a limit is? Right?

[heavy furnace]

next

yes

[hardy nest]

The basic concept of it

[heavy furnace]

yes

what does it have to do with derivatives tho?

[hardy nest]

The derivative is mathematically defined as a limit

Same with the integral

Limits are the first topic in calculus 1 actually

[heavy furnace]

wait what

[hardy nest]

But, do you know how when you zoom in on a function graph

[heavy furnace]

the slope is a limit?

[hardy nest]

It looks like a line?

[hardy nest]

We’ll get there wait

[heavy furnace]

u mean

zoom in?

like

ultra zoom in

so it is just a point or a line

?

[hardy nest]

Look at this clip

As we zoom in on this function, it ends up looking like a line, a linear function

[heavy furnace]

yeah

where does it get the slope from

[hardy nest]

This implies that all functions are just made up of linear functions

[heavy furnace]

wait

it can't change

[heavy furnace]

yeah

But

it can't

physically

[hardy nest]

Consider this graph with a and a + h

[heavy furnace]

move

differently

[hardy nest]

Well, what if we make it so that

a and a + h are extremely close together, the difference in h becomes very small, as if it’s approaching zero

When we let a get closer and closer to a + h, we are essentially drawing a line segment that touches the function at that specific point

This type of line is called the tangent line

A tangent line touches the function at exactly one point

[heavy furnace]

Oh

so

when a hits a+h

it's a tangent line?

[hardy nest]

Yes, or we can say, that as h → 0

This is the definition of the derivative

f'(a) = lim(h → 0) : (f(a + h) - f(a)) ÷ (h)

[heavy furnace]

uhhhhhh

ug

[hardy nest]

The slope of the function evaluates at x = a is given as the limit as h approaches 0 for the equation (f(a + h) - f(a)) ÷ h

[heavy furnace]

uh

[hardy nest]

I’m sorry it gets rlly complicated here but let’s go back

f'(x) is the notation for the derivative function of f, it's notated with an apostrophe (')

[heavy furnace]

I see

just a question

aren't slopes

decide with 2 points?

[hardy nest]

Yes, the average rate of change (slope) between two points

[heavy furnace]

okay but

what does tangent lines and getting closer

affect

that concept

how does*

[hardy nest]

Consider this though,

x → f(x) → y
For normal functions, as we input 'x' into the function f(x), we output a 'y'

x → f'(x) → slope of f at x
However, for the derivative function, as we input 'x' into the derivative function, we get the slope of the normal function f(x) evaluated at a specific x. In other words, the derivative function returns how fast the function f(x) is changing at any point x

[hardy nest]

The slope is just the average rate of change

But as we make one of the variables approach the other… in this case, letting
b = a + h,
⇒ h = b - a
Then letting the difference between b and an approach zero (the limit as h → 0), we get the instantaneous rate of change of the function. We get how fast the function is changing at any x, doesn't matter if it's linear or curved.

This image is also good

The derivative is equal to the slope, Δy/Δx, as the change in x approach 0

Δy = f(x + h) - f(x)
⇒ Δy = y2 - y1
⇒ y2 = y1 + h, where h is the difference between the y-values

This is the same thing in the image, just worded differently
f(x0 + Δx) - f(x0)
So the change in y is equal to the function evaluated at some x (notated as x0), with Δx being the change in x, which is the difference between y2 and y1 — the y2 being f(x0 + Δx) and the y1 being the f(x0)

https://youtu.be/EzePYR390dw?si=n7mONBXxOYISLjos

You might still be confused, please watch this video closely to see what he’s doing and saying

[heavy furnace]

I understand it but

[hardy nest]

Hearing an audio and a visual would be better

[heavy furnace]

I understand it but I still don't comprehend it

I might need a break

it's 7 am for me rn

I haven't slept for a lot of hours

for like

24+ hours

so

it's basically

it's not basically

uh

the change as x approaches 0

but

that means

you calculate the slope between two things, and you calculate it all the way till those two things join but don't join

uh

uh

right?

similar to that?

or lemme just watch the video

[hardy nest]

f'(x) = lim(Δx → 0) : Δy/Δx

You should get rest

I hope the video makes it make a bit more sense conceptually

(You should know that division by zero is undefined… so we say the limit as the denominator (change in x) approaches 0)

[heavy furnace]

for derivative of f of x = lim as x2-x1 approaches 0 OH MY GOD THIS WHAT YOU DID TOO I JUST REALIZED, y2-y1/x2-x1. The process is repeated or done till n approaches 0

[hardy nest]

I get what you’re saying and that’s basically it, when one point gets infinitely closer to the other point, but not quite reach it since the denominator would be equal to 0 when trying to calculator slope. This is why we use the limit

[hardy nest]

Yes but since Δx → 0, we can replace it with a variable, 'h', and say that h → 0

[heavy furnace]

I see, and u just repeat the process almost infinite times as n approaches 0

but

how is this practically possible

[hardy nest]

As for the numerator,
Δy = y2 - y1 = f(x + h) - f(x)
where h is the difference between y2 and y1. it's the value you need to add to y1 in order to get y2.

But the difference between y2 and y1 shrinks as the difference between x2 and x1 approach zero..

This means that the difference as a whole approaches 0, so the limit as h → 0

[hardy nest]

By “infinite” I mean we get closer and closer to a value without reaching it. We infinitely get closer to it but we are never quite at the value

[heavy furnace]

I see

[hardy nest]

Because if we do say h = 0, the denominator (Δx = x2 - x1 = h) is equal to 0, and division by zero is undefined

Did you watch the video?

Its not quiet long but you can watch it later

[heavy furnace]

I think

I understood it a bit

I will watch the video

but I got the gist of it

u have to do it to a very close value to 0

'precise enough'

I mean

of the value

of the slope

what you'd get when you reach very close.

[hardy nest]

You’ll learn the mathematical definition of a limit and what it means to "approach" because it is a bit ambiguous

Because if we say x approaches α, then x also approaches α + 1… it also approaches α + 5, so what does "approaching" specifically α even mean

You don’t learn this in calculus 1 or 2 at least

You learn it in a class called real analysis which is essentially the rigorous study of functions

Proving definitions from scratch and such

[heavy furnace]

oh

oh

[hardy nest]

Calculus is apart of analysis

But it doesn’t rigorously define limits and such

[hardy nest]

slope = (y2 - y1) ÷ (x2 - x1)
If we do it in function notation
slope = (f(x + Δx) - f(x)) ÷ (Δx)

The definition of the derivative is literally this… but as Δx → 0

You get the rate of change of the function at any point on the graph

But I think it’s time for you to go tbh

[heavy furnace]

alright

it was so fun

what I've managed to understand with you

[hardy nest]

https://youtu.be/-aTLjoDT1GQ?si=EB0CUb2ozKcj0Orl

This is a longer video but I believe it does explain more

[hardy nest]

Thank you, I don’t think im a very good teacher but I try to explain stuff in the best way I could to make ppl understand 😭

[heavy furnace]

in this past 4 hours, you've thought me the best I've ever been thought before in my life

you really are one of the greatest person for me

nobody has made me understood something this complex before

they just told me to read books

thank you so much

[hardy nest]

f(x) = x^2
f'(x) = 2x
This means that anywhere on the graph of f(x), the graph is changing at a rate of 2x. To find the rate of change (slope) of the function at x = 6, we plug in x = 6 into the derivative function
f'(6) = 2(6) = 12
So when x = 6 on the normal function, the function is changing at a rate of 12 units of y per 1 unit of x

We’ll get to the point where we understand why the derivative of x^2 is 2x but that’s that for right now

[heavy furnace]

I see

okie thanks

[hardy nest]

Last video

If you try finding the "slope" of this quadratic function given the two points (6, 36) and (7, 49) you end up getting 13. However the actual rate of change of the function at x = 6 is 12. This is because the quadratic function is not actually a straight line, it is curving a bit, and as a result the slope between the two points is not fully accurate

So while using the average rate of change formula for non linear functions gives you the secant line between two points, it doesn’t give you the accurate rate of change of the function, because it doesn’t take into consideration the fact that the function may curve or bend

⇒ At x = 6, the function is changing at a rate of 12 units per y, for 1 unit of x. This is what f'(6) tells us

⇒ However, when trying to find the slope between the two points using
m = (y2 - y1) ÷ (x2 - x1)
We get 13, this means the average rate of change of the function between x = 6 and x = 7 is 13 units of y per 1 unit of x, although how fast the function is truly changing at x = 6 is 12 units of y per 1 unit of x, not 13.

The true speed at which the function is changing is always found with f'(x). You can't simply use the average rate of change (slope) formula because you are assuming that the points on that interval are separated with a straight, constant line; when in reality they could be separated with a curve that looks like line, but slightly curves (this scenario). The only reason why our values were so close (12 and 13) is because of the steepness of the quadratic, it was so steep that the curve looked like a straight line, however it actually isn't.

@heavy furnace I hope this helps

bye!!!!

[carmine beacon]

Oh if you're learning about lines rn you should also try systems of linear equations and matrices

If you'd like to balance it on top of whay you're currently learning, of course

[heavy furnace]

helpo

hello*

I woke up

[heavy furnace]

sure I'd like the concept

[heavy furnace]

I see

[vale stone]

@hardy nest

[hardy nest]

@vale stone

[vale stone]

yo

so

it's may 22nd

I have like

8 days

to study something truly rigorous

[hardy nest]

What topic are you on

[vale stone]

and prove to my father

[vale stone]

uh not much recently

but uh

trigonometry

ish

started

[hardy nest]

Are you still going on the linear algebra route?

Linear algebra is typically after calculus 2 and before calculus 3, but I don’t know if your father considers it “rigorous”.

[vale stone]

no no

rigorous in hs terms

not that far

precalc aka trigonometry geometry etc.. derivatives etc.. basic calculus and calculus 1

so I need to learn these things

aka all of high school's math topics mainly

and some probability and statistics

wait

actually

I can

you're right I don't need to be rigorous in small terms

alr

linear algebra is the goal

[hardy nest]

If you’re trying to just become proficient in high school math, for the linear algebra route you can stop at linear transformations

Vectors and matrices are a very deep topic because there so much to them

Then after, I can help with trigonometry, but there is a lot of algebra you should know too

Calculus 1 is literally just differentiation.

Calculus 2 is literally just integration

[vale stone]

I see

[hardy nest]

We’ve already learned the definition of the derivative, which is what calc 1 is all about, it doesn’t dive into integration too deeply unlike calc 2

[vale stone]

I see

[hardy nest]

I hope you don’t get confused on what I mean when I say this btw

They are obviously not JUST differentiation or integration, but these span majority of the courses

Calculus 2 also includes vectors, sequences and series, as well as parametric equations and polar functions