precalculus

[hard hornet]

ya

[viscid thistle]

Before I go further should I memorize these elementary graphs

[hard hornet]

which elementary graphs

[viscid thistle]

Inverse graphs of cos and sin

[hard hornet]

mmmmm

its okay to memorize em

but u dont need to

[viscid thistle]

Lol

[hard hornet]

if u know how to solve inverse cos and sin

which scares me

cuz i havent answered ur question yet

inverse cos of root(3) / 2

[viscid thistle]

Yeah so I see that on the unit circle, cos of root(3) / 2 gives us an output of 1/2

but

[hard hornet]

mhm

[viscid thistle]

It somehow is the angle

[hard hornet]

no

[viscid thistle]

which is pi/6

[hard hornet]

whats

oh

yes

remember

cos/sin ofan angle gives you a ratio

thus

arc cos / arc sin of a ratio gives you an angle

its the opposite

trig functions, an input is an angle, u get a value
inverse trig functions, an input is a value, and output is angle

[viscid thistle]

Are you able to show a picture of it or in TeXit form

[hard hornet]

mmmmm show wuat

[viscid thistle]

What you just wrote

[hard hornet]

uhhhh

what dont u understand about it

for example

cos (pi) = -1

sin (pi) = 0

anything u input in a trig function is an angle

0 to 2pi

and your ouput, well, thats the output (the ratio)

inverse trig

is just the opposite of that

you input a value or a ratio, and it spits out an angle

[viscid thistle]

This is what I wrote

[hard hornet]

yup thats right

[viscid thistle]

I dont understand why he made us write "ratio of right triangles"

[hard hornet]

because trigonometry functions measures the ratio of right triangles

sin = opposite / hypotenuse
cos = adjacent / hypotenuse
tan = opposite / adjacent

[viscid thistle]

Fock

This is gonna take a while for me

to get

[hard hornet]

its hard to grasp at first

but once u nail it

it all clicks

lmk if u got anymore questions

[viscid thistle]

I completely dont understand why its the ratio

Do you have a video

link

[hard hornet]

cuz we defined it to be that way

[willow bear]

can i get a recap on what the problem is here

[hard hornet]

we defined a sin function

originally

it was like

what is arccos (root(3) / 2)

and idk where im going now

we defined a sin function

to be

the length of the opposite side of the angle, divided by the length of the hypotenuse side of the angle (triangle

[viscid thistle]

Okay so In regular trig functions our Domain contain our angles, and the range contains our Trig Ratios?

[hard hornet]

mhm

inverse trig is just the opposite

[viscid thistle]

Alright let me retry

the problem

[hard hornet]

ok

one thing that might help

is draw out the triangle

[viscid thistle]

Alright

I dont even get how to draw it lmao

[hard hornet]

draw any right triangle

[viscid thistle]

What is the angle root 3 over 2

[hard hornet]

ok

so arc cos root3 over 2

draw a triangle

in which

the adjacent side is root 3

and the hypotenuse side is 2

remember

[viscid thistle]

Cos = adj/hyp

[hard hornet]

the adjacent side of the angle

yup

remember that inverse trig is the opposite of regular trig functions

if you want to solve for the angle

you label the sides

and then figure out angle

[viscid thistle]

oh sheet

Is this drawing correct?

@hard hornet

[hard hornet]

yup

its beautiful

[viscid thistle]

lol

[hard hornet]

u good for now?

[viscid thistle]

hold on

How

I dont ge ti

it

How is pi/6 the answer

[hard hornet]

when you draw out the triangle

what is the angle

remember that arc cos, or inverse trig functions

u input a ratio, it outputs an angle

it outputs the angle of the triangle

in which u built with the input (the ratio)

u built a triangle

with ratio root3 / 2

root3 being adjacent to the angle u want to know

and 2 being hte hypotnuse to the angle u want ot know of the triangle

[hard hornet]

so what is the angle

noOOOOo

Like i said

regular trig functions, INPUT IS ANGLE, OUTPUT IS RATIO
inverse trig function, INPUT IS RATIO, OUTPUT IS ANGLE

[viscid thistle]

okay so in cos-1(x) the input is not an angle

[hard hornet]

its a ratio

ur inputting the ratio

to a triangle

and its spitting out

the angle of the triangle

[viscid thistle]

Thats my problem

I was thinking that the input was always an angle in both functions

[hard hornet]

o nooooooooo

[viscid thistle]

Wait

so what I wrote in my example

Ik its incorrect

but I wrote

[hard hornet]

wuts ur question

[viscid thistle]

Is what I wrote now correct?

[hard hornet]

basically

but its weird to have a root3/2 angle

lol

[viscid thistle]

I know

But it helps me understand

the differences

[torn oriole]

radians> degrees

[viscid thistle]

OOOHH

Now I think I understand why he made use write that the range is our angles also

So in the question

The output should be an angle

And our angle for the trig ratio is pi/6

@hard hornet

[hard hornet]

yes

[viscid thistle]

@hard hornet

I attempted this one

But now Im stuck

[hard hornet]

remember

arc cos spits out an angle

u can think of arc cos(1/4) as an angle

arc cos (1/4) = theta

what is sin(theta)

hint || u dont need to know the angle ||

[viscid thistle]

root of 15?

[hard hornet]

no

try again

[viscid thistle]

fock

[hard hornet]

remember

what is sin

[viscid thistle]

It means I dont get it

still

[hard hornet]

root 15 is part of the answer ill tell u that

remember

sin is opposite over hypotenuse

[viscid thistle]

In a regular function sin is the output

[hard hornet]

ok

let theta be equal to arc cos of whatever

what is sin(theta)

then

remember

u dont have to solve for the angle

ill brb

[willow bear]

let θ = arccos(1/4). then cos(θ) = 1/4. what is sin(θ)?

[viscid thistle]

LOL

It makes sense to me

[willow bear]

oh brilliant

yes

good

sqrt(15)/4 is correct

[viscid thistle]

No way

I'm screwed

I think

Alright 1 sec let me try this

[hard hornet]

wat is the viewing angl

theta + alpha?

im confused lol]

[viscid thistle]

its asking for x

distance

[hard hornet]

what is the viewing angle though

[viscid thistle]

So I have to use ratios to find angle I think

??

<@&286206848099549185>

https://cdn.discordapp.com/attachments/363224154469826562/578463577808044061/unknown.png

In this section we are looking at inverse functions if that helps

Inverse Trig Functions*

[torn swift]

I like how memes please gave up lol

[willow bear]

$\theta(x) = \arctan\left(\frac{30}{x}\right) - \arctan\left(\frac{10}{x}\right)$

[obsidian monolith]

Ann:

[willow bear]

not sure how to maximize that w/o calculus though.

[viscid thistle]

No idea

[hard hornet]

lmao i gave up cuz i have no fkin clue what its asking

idk wat tilt angle means

[compact tendon]

@willow bear you want to find the extrema?

What do you aim?

[willow bear]

the goal is to maximize θ(x), but given that this is in #precalculus i am not entirely sure if taking the derivative and setting it to 0 would be appropriate

[hard hornet]

my thought too

so there has to be some weird trick or the question is weird

[viscid thistle]

lol that ez

a^2+(10)^2 = 20^2

= 17.321

[wet scaffold]

For problem 3 what’s the connection between 2 series with the same remainder and their difference being divisible by n?

[spring thunder]

Well let's just think about two numbers (a and b) which have the same rest when divided by n

[wet scaffold]

The same rest?

[spring thunder]

$$a = nq_a + r$$ $$b=nq_b + r$$ $$(0\leq r < n)$$

I'm just used to saying rest lel

Remainder

[wet scaffold]

Ok

[obsidian monolith]

emeric75:

[spring thunder]

So now if you take a-b

[wet scaffold]

Is q the number of times the number can be divided?

And r is the remainder?

[spring thunder]

Yeah q means quotient in here

And r remainder indeed

[wet scaffold]

Ok

I see so when u subtract those numbers the r cancels out

Correct?

[spring thunder]

$$a-b = (nq_a + r) - (nq_b + r)$$ $$a-b = nq_a - nq_b + r - r$$ $$a-b = n(q_a-q_b)$$

Yah

[obsidian monolith]

emeric75:

[spring thunder]

So the difference is divisible by n

[wet scaffold]

Ok I see now

That’s cool thanks

[spring thunder]

👌

[graceful egret]

<@&286206848099549185>

[hoary yoke]

Well first off @graceful egret

Ravenue = Selling Price of n clothing - Producing cost of n clothing

Lets see if you can derive ravenue formula

[graceful egret]

(150-0.5x)-(3500+.75x^2)

[hoary yoke]

No but close

You see it says that price of 1 is 150-0.5x

But production says for X amouts of clothing

[graceful egret]

?

i dont understand

[hoary yoke]

Basically 150-0.5x is price per suit

[willow bear]

total price isn't 150 - 0.5x

it's x times that

[hoary yoke]

You need x suits

Ye ^ as ann said

So price for x suits is x(150-0.5x)

[graceful egret]

x(150-0.5x)-(3500+.75x^2)

[hoary yoke]

Yeppie

Now expand and simplify ig

[idle dust]

hey solarkoid

i have a dum question in #calculus and im confused

can you help?

[hoary yoke]

Ill try

[idle dust]

ok

[graceful egret]

(150x-0.5x^2)-(3500+.75x^2)

then do i foil

[willow bear]

no, because there is nothing to "foil" and you should stop using that mnemonic anyway

[graceful egret]

oh ok

do i simplify and take derivative?

[idle dust]

hey ann you know @frigid vale

he told me today that you hate him

[willow bear]

huh what?

@graceful egret yes

[graceful egret]

-1.25x^2-150x-3500

derivative is -2.5x-150

so with that how do i find total revenue and profit

<@&286206848099549185> i have 4 problems i need help with and if someone can help i will pay you in lots of love and fortnite skins

[idle dust]

bruh sound f(x)

[graceful egret]

bruh

[serene heath]

u serious about those fortnite skins?

[graceful egret]

if you help me with these 4 problems

sure

[tawny nacelle]

hey i want the tomato skin thing with the skinny D

[graceful egret]

how much is it

[tawny nacelle]

like 50 bucks or so

¯_(ツ)_/¯

[graceful egret]

lmao

nahhhh

[tawny nacelle]

jk i dont play fortnite

its shit

[graceful egret]

neiter do i

[tawny nacelle]

wow

what a scam

[graceful egret]

what

ur a scam

[tawny nacelle]

post the goddamn question bruh

[graceful egret]

i did

[tawny nacelle]

rEEEEE

[graceful egret]

and the guy stoped half way through

[hoary yoke]

I help u

[tawny nacelle]

thats what u get for scamming me

[hoary yoke]

And u give me hug

[graceful egret]

ok

i will give hug

[tawny nacelle]

it'll tkae more than a hug to get me involved

[hoary yoke]

Ok can you post simplified profit formula?

We will need it

[graceful egret]

i really wish i could

-1.25x^2-150x+3500

is that it?

[hoary yoke]

P(x)=x(150-0.5x)-(3500+0.75x^2)=-1.25x^2+150x-3500

[graceful egret]

ok

[hoary yoke]

Wait what does revenue and profit have in difference

Owo

[graceful egret]

revenue is profit -cost

[hoary yoke]

Oh

[graceful egret]

wait

profit= revenue -cost

[hoary yoke]

So profit is what you get?

Not counting what you will lose?

[graceful egret]

im confused i dont even know what im solving at this point

[idle dust]

🅱

[graceful egret]

any other day i wouldnt care about your random post mr.b guy

but i gotta final tomorrow

no time to mess around

[hoary yoke]

Tomato i dont know englosh well

Idk difference between profit and ravenue

,w revenue

[idle dust]

revenue is the money u get

profit is the money you get at the end

ok say you have a company

you sell a shirt

[hoary yoke]

ok thank you Bno

[idle dust]

u had to buy teh shirt

[hoary yoke]

Now i understand

[idle dust]

bruhv

[hoary yoke]

@graceful egret ok total ravenue is x(150-0.5x)

Since thats what we get for selling x clothes

[graceful egret]

ok

[hoary yoke]

So on total revenue is 150x-0.5x^2

And on total profit we alrdy derived

[graceful egret]

-1.25x^2+150x+3500

this right?

[hoary yoke]

Ye

[graceful egret]

ok so how many suits to maximize profit

find derivative and set equal to 1?

[hoary yoke]

Ok now lets make an observation

Wait you know derivative?

[graceful egret]

yes

[hoary yoke]

This is precalc dude

[graceful egret]

wups

im in survey of caluclus

[hoary yoke]

-1.25x^2+150x+3500

Ok nvm lets solve it

Its plus150x not minus

[graceful egret]

-2.5x+150

[hoary yoke]

Yep

Now we set that to 0

[graceful egret]

ok

[hoary yoke]

Since slope of maximum of a parabola is equal to 0

There was also another way

[graceful egret]

60!!!

[hoary yoke]

Since the parabola has negative a value we can find the vertex x value (or how many clothes) in -b/2a

We plug in values and get

[graceful egret]

oh ok interesting

[hoary yoke]

,calc 1500/25

[obsidian monolith]

Result:

60

[hoary yoke]

Ye

So 60 clothes and they will maximize their profit

[graceful egret]

okok

[hoary yoke]

Now to find the maximum profit

[graceful egret]

so now plug 60 into x

for the derivative?

[hoary yoke]

Plug in x=60 in the original equation

[graceful egret]

okok

[hoary yoke]

We need P(60)

[graceful egret]

ok

[hoary yoke]

To see the maximum

Since 60 clothes gets us maximum profit

[graceful egret]

8000

holy shit i think im learning

[hoary yoke]

Ok

[graceful egret]

this feels weird

[hoary yoke]

Naisuuu

[graceful egret]

so how do i find price per unit now

[hoary yoke]

I dont get it well but i think

Remember p=150-0.5x

[graceful egret]

ah

[hoary yoke]

I think you should plug x there

And 60

Since that gives us maximum profit

So we need maximum revenue

[graceful egret]

so 120

[hoary yoke]

Ig

Check now

I wanna know

[graceful egret]

i cant check untill there all dont 😦

done

[hoary yoke]

All homework?

Or everyone who does homework?

[graceful egret]

what

no not on homework

[hoary yoke]

So what i dont understand

What are u doing

From where is this question

[graceful egret]

mymathlab

[hoary yoke]

Ok when you check for answers

Please tell me if we did good on that one

[graceful egret]

will do

[hoary yoke]

Dm me

Thank u

[graceful egret]

i got 3 more to do

[hoary yoke]

Ok if you need help post it here

[graceful egret]

AND UR GONNA HELP ME ON ALL OF THEM 😄

[hoary yoke]

Okay

[hoary yoke]

U wont have me on exams doee

[graceful egret]

haha i know

[hoary yoke]

A is easy

[graceful egret]

i learn best from getting the answers and reverse engineering them

[hoary yoke]

You will get it

Do a urself

[graceful egret]

0.002x^3+0.07x^2+15x+700

[hoary yoke]

Ye thats monthly cost where x is amount of chairs

[graceful egret]

so plug in 70?

[hoary yoke]

You want current monthly cost and currently 70 chairs are made

So ye

Plug in 70

[graceful egret]

2779

[hoary yoke]

If you did it correctly idk but ill take it

So thats monthly cost now

Ok try b as well

Its also easy

[graceful egret]

ah ok im also going to double check a 1 sec

[hoary yoke]

Okee

[graceful egret]

54.692

rounded to nearest cent 54.69

[hoary yoke]

Wat did u do

To get that answer

[graceful egret]

plug in 71 to x

and subtract 2779

[hoary yoke]

Ye correct

And you get that?

[graceful egret]

yes

[hoary yoke]

So ye thats b

[graceful egret]

awesome

[hoary yoke]

[graceful egret]

c i have no idea

[hoary yoke]

,w marginal cost

[obsidian monolith]

Display results online and refine query

Requested by Solarkoid#3482

[hoary yoke]

What

[graceful egret]

lol

ahhaah

[hoary yoke]

Ye i also have no idea

[graceful egret]

ok

we can just move to another problem

and see if someone can help with this one later

[hoary yoke]

Next also needs marginal cost

[hoary yoke]

Ok

WHAT IS THIS

WHAT ARE THE TERMS

[graceful egret]

ikr 😦

you wont understand from definition i dont think

[hoary yoke]

Ok wait lemme find some exampkes on internet

[graceful egret]

ok

[hoary yoke]

Ok lets back to marginal cost part

[graceful egret]

ok

[hoary yoke]

okay so

Marginal cost is derivative of total cost function

So take derivative of C(x)=0.002x^3+0.07x^2+15x+700

[graceful egret]

0.006x^2+0.14x+15

[hoary yoke]

Ok now it asks you to find marginal cost at x=70

[graceful egret]

54.2

[hoary yoke]

Okay

I will trusr tou with that

So now find the marginal cost between 72 and 70 chairs

[graceful egret]

1.98

[hoary yoke]

Round to the nearesr cent

[graceful egret]

now what is C(72)

[hoary yoke]

And you can also do the d part

Lets move onto the elasticity

[graceful egret]

wait

how do i do e

[hoary yoke]

Ok apparently

$E(f(x))=\frac{f'(x)\cdot x}{f(x)}$

[obsidian monolith]

Solarkoid:

[hoary yoke]

Also are you majoring in economics? What is this

[graceful egret]

business so kinda haha

[hoary yoke]

Ye basicallt thats the elasticity formula

You can do this by yourself

And also find the elasticity at x=2

[graceful egret]

wait but i still need e on the last problem

[frozen needle]

does that notation really work?

[hoary yoke]

I have no idea

I found on stack exchange and wikipedia

So it must

[frozen needle]

if physicists decided on that, then it can't really be trusted

I would've expected something more like
E(f)(x)

[spring thunder]

thats' an econ shit not physics

[hoary yoke]

It was also in this form

$E_{x}(f(x_{0}))$

[obsidian monolith]

Solarkoid:

[hoary yoke]

Which just oddly looks like partial derivative in terms of x

Sooo

Ye no i think i did it correctly

See if As x increases from 2 derivative gets smaller or bigger

If it gets bigger the profit should also increase

Im not too sure

[graceful egret]

so whats the elasticity

[hoary yoke]

I have no idea

Im not majoring in economics. Yet

[graceful egret]

😦

[hoary yoke]

Just google

[graceful egret]

what about e on the last problem tho, ill come back to this one because i think i have some notes on elactisity

[hoary yoke]

Just plug in 72 for x in C(x)

And round to nearest cent

[graceful egret]

but it says to use the answer from part d

use the

omg spelling lol

[hoary yoke]

O

Uhhhh

I honestly donno

Economics is bs

[graceful egret]

2889.38

im just gonna go with thta

[hoary yoke]

I would think C(70)+1.98 (marginal cost) but not too sure

Hey thats pretty close

Ye use C(70)+1.98 i think

Since it says to predict

[graceful egret]

why would i predict when i have the funciton right there

??

[hoary yoke]

Okay go with your way ig

[graceful egret]

stupid teacher askin stupid questions smh

[hoary yoke]

Its better

Okay ill go now

Dm me the answers ok?

[graceful egret]

ok

[hoary yoke]

Dont forget

[graceful egret]

thank you

sure

[hoary yoke]

Ill see it tomorrow

Also send a hug

U promised

[graceful egret]

okok

[hoary yoke]

[graceful egret]

<@&286206848099549185>

[hard hornet]

i dont know my econ

but take derivative and set to 0?

[graceful egret]

so

[hard hornet]

i have no clue what its asking

but im guessing it has to do with derivative

[graceful egret]

1/2x^-1/2

=0

[hard hornet]

i think sooo????

wait no

uhhhh

[graceful egret]

lol

[hard hornet]

thats probably wrong

i canth elp u here

[graceful egret]

fuck

ive asked so many people

[hard hornet]

same

[graceful egret]

im triggered

do you know elasticity?

[hard hornet]

nop

[viscid thistle]

@willow bear Thanks for the help yesterday, In class he revealed that it was only solvable with calculus...

and @hard hornet thx

He said that we were supposed to use a graph lol

[hard hornet]

Draw it first

And see what you can ded e

Deduce*

[viscid thistle]

Domain is all real numbers except for 2. Range is all real numbers except for 1.

Not really sure what it means by concave up or down.

[viscid thistle]

Hello

I kinda forgot how to go about solving lims like this

[frozen needle]

you can first factor by 2x

[viscid thistle]

Wait what

[frozen needle]

$\forall x< 0,~\sqrt{4x^2-9x+3}+2x=2x\prn{-\sqrt{1-\frac 94\frac 1x+\frac 34\frac 1{x^2}}+1}$

[obsidian monolith]

Tuong:

[viscid thistle]

O_o

Interesting

[frozen needle]

so now the question is
how fast does that square root thing approaches 1

[viscid thistle]

Tried this btw but then it would be inf*0

I don't really get what you did and how it helps

[royal gull]

Multiplying by conjugate might help?

[frozen needle]

,align \forall x< 0,~\sqrt{4x^2-9x+3}+2x&=-2x\prn{\sqrt{1-\frac 94\cdot\frac 1x+\frac 34\cdot\frac 1{x^2}}-1}\
&=-2x\prn{-\frac 94\cdot\frac 1x+\frac 34\cdot\frac 1{x^2}}\frac{\sqrt{1-\frac 94\cdot\frac 1x+\frac 34\cdot\frac 1{x^2}}-1}{-\frac 94\cdot\frac 1x+\frac 34\cdot\frac 1{x^2}}

[obsidian monolith]

Tuong:

[frozen needle]

the last factor, the big fraction one, is special

you recognise a $\frac{\sqrt{1+h}-\sqrt 1}h$ thing as $h$ tends to $0$

[obsidian monolith]

Tuong:

[viscid thistle]

Isn't that a derivative thing

[echo plaza]

eh

[frozen needle]

yes

[echo plaza]

I'd prefer conjugate tbh

[royal gull]

Same

[echo plaza]

$\sqrt{4x^2-9x+3}+2x \equiv \frac{3-9x}{\sqrt{4x^2-9x+3}-2x}$

[obsidian monolith]

CaptainLightning:

[viscid thistle]

What does the triple line thing mean?

equivalent

it means equivalent

Why not use = then?

Looks possibly right?

[frozen needle]

Well that's the correct answer

[viscid thistle]

Thank god

Also thank you

[frozen needle]

You're welcome

[viscid thistle]

can soomeone help me w precal interest rates

[torn swift]

Because interest rates change as well?

[viscid thistle]

@torn swift idek what nominal rate is

[grizzled orchid]

@viscid thistle u know what nominal means

[viscid thistle]

no, the textbooks defines it but its weird

[grizzled orchid]

basically "in name"

[viscid thistle]

oh

[grizzled orchid]

nominal interest rate is the interest rate before adjusting for inflation

[viscid thistle]

ohh

[grizzled orchid]

why do math classes love to shove so much bullshit econ in there lmfao

[viscid thistle]

ikr

bruh i dont get the question asfdf

like it says argument

what are we arduing against

[grizzled orchid]

well

nominal basically means "in name" right

so nominal interest rate can be thought of as "the interest rate in name only"

nominal interest is not what you actually play

*pay

it's what you nominally pay

but your exact value is different

[viscid thistle]

ohh ic

Thx

WHY is this in my precalculus textbook

[grizzled orchid]

your nominal rate is uncorrected to account for interest, and therefore it is "nominal," not real or effective (both other types of interest mostly for real estate and investment primarily )

[viscid thistle]

like its not even a math problem

ohh

[grizzled orchid]

because textbooks think that tying their math to shitty economics will convince kids that its cool

[viscid thistle]

🤦

[grizzled orchid]

"see? math is used in real life too!!!"

smol time math like alg Ii and precalc is too simple to be applied to real life situations in useful ways

like u can calc or lin alg

so this is what these textbooks are left with lmfao

[hybrid charm]

Every single thing you mentioned is difficult

[grizzled orchid]

wut

what did i mention

[hybrid charm]

algebra 2,precalc,real life situations,calc, linear alg, textbooks

[grizzled orchid]

when did i say they weren't???

wot

[hybrid charm]

you said precalc is too simple

[grizzled orchid]

no...?

or like, kinda

but i meant that it is too simple to model very complex problems irl

so basic financial stuff is a go to "example" or "model" in textbooks

[fossil bronze]

Can anyone help me understand perfect square trinomials?

[torn swift]

[fossil bronze]

What makes something a perfect square trinomail?

And what are imaginary numbers?

[hard hornet]

imaginary numbers in a nutshell

we want to find a solution for

$x^2 + 1 = 0$

[obsidian monolith]

MemesPlease:

[hard hornet]

how to do that?

welp there is a way

we introduce imaginary numbers

how do you do squareroot again on texit

[fossil bronze]

Im not sure

[echo plaza]

\sqrt

[frozen needle]

\sqrt{aaaa}

[hard hornet]

ah thank you

so

we call

$x = \sqrt{-1}$

a solution

[echo plaza]

\sqrt{-1}

[hard hornet]

o

thanks

[obsidian monolith]

MemesPlease:

[hard hornet]

such that

when we square this number

we get a negative number

because normally, you cant squareroot a negative number

thus we call this

I M A G I N A R Y

thats imaginary numbers in a nutshell

applications, endless

you're probably looking at one rn

[fossil bronze]

Wdym

How can i look at one

[hard hornet]

ur phone

that stuff uses imaginary numbers

[fossil bronze]

Oh

[hard hornet]

i think

[echo plaza]

j 🤢

[sturdy gyro]

How are imaginary numbers used irl

[timber plinth]

If it's 2dimensional, they're pretty much a gimme

Esp if rotations are involved in said 2 dimensional thing

They're also often used (with some extending, of course) in physics once you get into the hell that is QM iirc

[random quartz]

you can do QM in real

[timber plinth]

can and should are very different

[serene heath]

the possibilities are endless, use ur imagination

[fossil bronze]

how do you write (x-5)^3 in standered form?

[torn swift]

you mean expand it all out?

[fossil bronze]

yes

[viscid thistle]

(x-5)^3 is just (x-5)(x-5)(x-5)

can you multiply polynomials?

[fossil bronze]

foiling?

cuz thats the only thing i can think to do with it

[torn swift]

you can either memorize the form with of a cubed monomial or just good ol' (x-5)(x-5)(x-5)

[viscid thistle]

ye its something like

ugh

nvm i dont memorize ^3

[torn swift]

a^3 +3a^2b +3ab^2 +b^3 I believe?

[fossil bronze]

well 5 to the 5th is 125

so would it bne

[torn swift]

I mean the easiest way for me to expand it is to expand (x-5)^2 then multiply that with x-5

[viscid thistle]

yes

[torn swift]

I can only remember the perfect square one

[fossil bronze]

x^3+3x^10+15x+125

[viscid thistle]

a^2 + 2ab + b^2

[torn swift]

but we can derive the perfect cube really quickly then

[fossil bronze]

i think that might be it

[torn swift]

just do (a+b)^2 (a+b)

so the one I wrote was correct lol

[fossil bronze]

thats not one of the answers in my lesson

heck

[sturdy gyro]

Use pascals or just multiply

[torn swift]

don't think they will get to pascals until they do the "large" powers

probably anything past 3 qualifies as large lol

[primal badger]

Or just use the binomial theorem

$nCr*x(^n-r)*y^r$

Where

[obsidian monolith]

MacMinty:

[primal badger]

Then do the sum for r= 0, 1, .... n

[obsidian monolith]

MacMinty:

$nCr*x^n-r*y^r$

[primal badger]

Whatever, the notation is a bit weird

x^(n-r)

[willow bear]

$nCr \cdot x^{n-r} \cdot y^r$

[obsidian monolith]

Ann:

[willow bear]

@primal badger is it this you were going for

[primal badger]

Yes there it is

$n \choose r$

[obsidian monolith]

MacMinty:

$n \choose r$

[viscid thistle]

$\binom{n}{r}$

[ruby otter]

Help

[willow bear]

what's giving you trouble here

@ruby otter

[ruby otter]

is it possible to differentiate here

coz the limits in the integral and and the function is itself both are function of x

[willow bear]

honestly, this is bad notation and shouldn't be used

[ruby otter]

ya instead of x it must be some other variable right??

[willow bear]

i \textit{think} what was meant here is $\frac{\left[ \int_0^x e^{t^2} \dd{t} \right]^2}{\int_0^x e^{2t^2} \dd{t}}$

[obsidian monolith]

Ann:

[rigid beacon]

How do people type LaTeX so fast lol

[willow bear]

like that's what i would take it to be

@rigid beacon practice

[ruby otter]

is it possible to solve without using l hospital

[willow bear]

i don't think so

this is actually one of the few cases where LH is reasonable

[ruby otter]

😂

[limber bone]

e^2x^2/e^2x^2 = 1

do lh

assume thats inf / inf ofc XDXD

[rigid beacon]

L'Hopital is always reasonable

[willow bear]

no

wrong

[limber bone]

lhopital is always reasonable

ye

lol

[willow bear]

no

[rare zephyr]

Whats the derivative and integral for a logarithmic function

[ruby otter]

a function

[rare zephyr]

?

[willow bear]

$\lim_{x \to +\infty} \frac{\sum_{k=1}^{54} \frac{(-2)^k}{(2k)!} x^k}{\sum_{k=1}^{54} x^k (1+e^{-kx})}$ try doing \textit{this} with LH. \ have fun applying LH 54 times lmfao

@limber bone

[obsidian monolith]

Ann:

[ruby otter]

$$\lim_{x \to 0} \left( \frac{2^x +3^x +4^x}{3}\right) ^{1/x}$$

[obsidian monolith]

Radical Ninja:

[willow bear]

the derivative of $\ln(x)$ is $\frac{1}{x}$, the integral of $\ln(x)$ is $x \ln(x) - x + c$, and you really should've asked this in #calculus @rare zephyr

[obsidian monolith]

Ann:
Compile Error! Click the reaction for details. (You may edit your message)

[rare zephyr]

@willow bear So derivative and integral of complex functions like trigs and logs are calculus or derivative and integral in general are calculus?

[willow bear]

derivatives and integrals are already #calculus material.

[ruby otter]

but limit of integral is #precalculus 😂

[rigid beacon]

What no it isn't

Integrals are calc bois

I've seen limit definition of the derivative in my precalc class

But that's it

[rare zephyr]

Im too scared to go and ask questions on #calculus, everything posted there is too complex. And the questions I'll likely ask that's calc-related is likely some lvl1 stuff

[willow bear]

don't sweat it

seriously everyone starts out somewhere

[rigid beacon]

Yea

[ruby otter]

whats happening here??

[rigid beacon]

Dear God why would someone ask you to solve that

[ruby otter]

fine how did the (ln2 + ln3 + ln4 )/3 = that?

nvm i got it

i didn't notice the e^ there

[dapper geyser]

what the fuck am i lookinga t

[grizzled orchid]

lol what the fuck is that

[dapper geyser]

wait its not actually hard though

[grizzled orchid]

ugh yes i guess

[dapper geyser]

just plug in the 0s and yeah

[grizzled orchid]

its ugly as fucking shit though

yeah

ugh yeah

[dapper geyser]

okay that question must've been written by somebody trying to be cool

[grizzled orchid]

ye

simplifies to

[dapper geyser]

e^ln24

or wait nvm

im dumb

e^((ln24)/3)

[grizzled orchid]

$e^{\lim_{x \to 0} \frac{1}{3} (\ln(2) + \ln(3) + \ln(4)} = 2\sqrt[3]{3}$

REEE

[dapper geyser]

oof

almost

[grizzled orchid]

hmm

[dapper geyser]

why is discord so slow wtf

[obsidian monolith]

hegel:

[grizzled orchid]

ere we go

[dapper geyser]

nice

[grizzled orchid]

ugh this is so fucking stupid tho

wait

@dapper geyser doesn't that simplify to 8

or

no oop

[dapper geyser]

no

[royal gull]

[dapper geyser]

(ln24)/3

is ln(cbrt(24))

[grizzled orchid]

ye

.> i know my brain crashed for a moment

[dapper geyser]

i tried to type it earlier but the comment wouldn't send and i had to reload discord

oof

same tho

[grizzled orchid]

lol

wow that's a dumb problem tho

[dapper geyser]

yeah

[grizzled orchid]

i hate problems like that where they're just there to try and throw enough shit at you that you give up before trying

honestly one of the reasons i don't like math comps is that many problems are overly esoteric in an mostly useless way

[viscid thistle]

lmaoo

anyone get this

[short sorrel]

how do we get log_11(5) from log_5(6) and log_5(11)?

[viscid thistle]

ikr ??

[dapper geyser]

um

well

log_a(c) = [log_b(c)]/[log_a(b)]

@viscid thistle

[viscid thistle]

bruh what

idt thisss

*get

[vestal peak]

helppp

[hard hornet]

find the volume of both shapes

and set them equal to each other

do you need help finding the volume of both shapes?

@vestal peak

[vestal peak]

Yeah

I do

[hard hornet]

ok

lets start with the cylinder first (the ice cream cone one is a it hard)

so, what is the volume for a cylinder?

a good way to think about it

is the area of a circle multiplied by it sheight

so what would the formula look like?

[vestal peak]

tr2H

?

[hard hornet]

whats t

[vestal peak]

pie

pi

lol

mb

[hard hornet]

$pi * r^2 * h$ right?

[obsidian monolith]

MemesPlease:

[hard hornet]

where

[vestal peak]

yeah

[hard hornet]

pi r^2 is the area of the circle

multiplied by height gives you volume

now, we know that the radius of the circle is 2x, and the height is h

so what is the volueme of the cylinder in terms of radius and height

or in terms of x and h

[vestal peak]

radius is 2x

height is

idk

[hard hornet]

height is h

just h

so what would the formula for volume be?

[vestal peak]

pir-squredheight

[hard hornet]

and your radius is 2x

[vestal peak]

Yeah

[hard hornet]

so what is the volume of the cylinder in terms of x and h

not in terms of r and h

in terms of x and h

[vestal peak]

pi(2x)Squared(h)

[hard hornet]

correct

$V = pi * (2x)^2 * h$

[obsidian monolith]

MemesPlease:

[hard hornet]

when you expand it out

you get

$V = pi * 4x^2 * h$ right?

[obsidian monolith]

MemesPlease:

[vestal peak]

yup

[hard hornet]

ok good

now lets move to the ice cream cone

[vestal peak]

wait

[hard hornet]

what do you notice about the shape?

yea?

[vestal peak]

i have the answer

i want to show you

it

[hard hornet]

mmm go for it

[vestal peak]

Cause i cant understand it at all

number 11

[hard hornet]

$4/3 * pi * x^3 + 1/3 * pi * x^2 * 4x = pi * (2x)^2 * h$

[obsidian monolith]

MemesPlease:

[hard hornet]

oh the work is a bit weird

but the writing is right

find the volume of hte hemisphere

find the volume of the cone

add those together

and then set that equal to the volume of the cylinder

[vestal peak]

why does he do

4/3*2 pi x squared

what the hell is that

and why did he times the 3 by 2

[hard hornet]

its so cluttered, the work is kinda sloppy

[vestal peak]

Pearson maths answers

[hard hornet]

pearson o god

i'd say we run through this together, the answer they have isn't too clear

[vestal peak]

ok

quick tho

i g2g soon

[hard hornet]

ya

ok

[vestal peak]

Cool ty

[hard hornet]

$(1/2)(4/3) * pi * x^3 + 1/3 * pi * x^2 * 4x = pi * (2x)^2 * h$

first one

is the volume of a sphere

but then we have to multiply it by 1/2

lemme edit it

cuz its half a sphere

[torn swift]

\pi

[vestal peak]

ok

[obsidian monolith]

MemesPlease:

[hard hornet]

if you were wondering, thats how they got the 4/3*2

and then the volume of the cone in terms of x

and then all you have to do

is simplify, and find a relationship between x and h

and thats about it

[vestal peak]

why does

it add 4/3 pi x cubed

[hard hornet]

mmm hwere

[vestal peak]

After working out the hemisphere

[hard hornet]

oh thats the volume of the cone

in my formula

[vestal peak]

what the hell

[hard hornet]

its a bit hidden

but your height

is 4x

the cone's height is 4x

you know the height of the whole thing is 5x

and the hemisphere's height is x

[vestal peak]

yeah

[hard hornet]

so 5x - x = 4x