precalculus

[willow bear]

loosely, linear algebra is the study of vector spaces and linear transformations between them

if you ask me to elaborate i will tell you to study it 🗿

[compact spade]

this doesn't intuitively seem possible

is it

[river drift]

what is the size of each matrix? What has to match for the two matrices to be multiplied?

[compact spade]

sizes don't match, eg. impossible?

[winter comet]

um

actually

i mean its not possible

but its not because the sizes aren't exactly the same

uhh

[compact spade]

what

[winter comet]

basically you gotta make sure the number of columns in the first matrix has the same number of rows in the second matrix i think

idk how to explain

wait i swapped them lol

💀

[river drift]

you had it right the first time...

[winter comet]

did I

bro im confusing myslef

💀

what am i doin 😔

[river drift]

you can think of the entry in the i'th row and j'th column of the product matrix as the dot product of the i'th row vector of the first matrix, and the j'th column vector of the second matrix

for such a product to be defined, such vectors must be the same length

[burnt wolf]

That’s only if you’re adding or subtracting

[winter comet]

but you can't multiply them, right?

[winter comet]

no its if your multiplying

[burnt wolf]

You can multiply them!

[winter comet]

if you're adding or subtracting they have to have the exact same dimensions...

[burnt wolf]

Yeah

[winter comet]

right

ok

[river drift]

you can find BA but not AB

[burnt wolf]

Why is that?

[kind schooner]

AB is 3x2 * 3x3, BA is 2x3 * 3x3

[compact spade]

h

[winter comet]

do you know how to multiply matrices?

[compact spade]

no

[winter comet]

hmm

[compact spade]

if i did i wouldn't be here

[winter comet]

true

[compact spade]

:(

[river drift]

let's put it this way: A is 3x2, B is 3x3. you need the "inside dimensions" of AB: 3x2 3x3 to match

[kind schooner]

wait is this the same book im using what book is this

[compact spade]

idk man

the bad one

[burnt wolf]

Ohhh, the last thing I’ve learned was composing matrices 💀

I have no room to speak here

[kind schooner]

it looks like precalc w/limits by ron larson

[winter comet]

idk if you want to but this is gud video channel so its prob good video https://www.youtube.com/watch?app=desktop&v=2spTnAiQg4M&ab_channel=TheOrganicChemistryTutor

[compact spade]

yeah probably

sounds like it

[river drift]

composing linear transformations represented by matrices is the same as multiplying matrices

[burnt wolf]

Wait really? I’d no idea, better go practice now

[winter comet]

i haven't actually done matrices in a while

so i keep tripping up and being unsure 💀

i need to review them lol

[surreal crescent]

Just hear me out:

1/3=0.333…
3(1/3)=1
3(0.333…)=0.999…
0.999…=1

[kind schooner]

thas crazy

[winter comet]

technically...

[surreal crescent]

And that’s calc boys and girls

[winter comet]

literally lol

like with respect to limits, yes that does check out

[river drift]

the calculus involved is making the number "0.333..." have any particular meaning in the first place

[surreal crescent]

Is limits that fancy f

[winter comet]

no

you're probably thinking of integrals

[surreal crescent]

I just want to use the fancy f

[winter comet]

limits

[river drift]

if you mean $\int$ that's an "s" (an old fashioned variant called the "long s")

[obsidian monolith]

pnoןɔ

[winter comet]

integrals

yea

[surreal crescent]

It’s like if an asymtote was roided up

[burnt wolf]

I’ve gotten through everything up to limits without finishing Algebra II or Pre-calculus 💀

[winter comet]

limits are used for asymptotes... if thats wut ur saying

[burnt wolf]

Now I’m drilling pre-calc so I can take it over summer with less difficulty

[winter comet]

thats good

[surreal crescent]

Yes but what does it mean

[burnt wolf]

Yeah!

[winter comet]

the integral means the area between the curve and the x axis

[river drift]

as the "s" suggests, it's a sort of sum (being calculus, it's really the limit of a sum we call a "riemann sum")

[surreal crescent]

Of a particular point, or of like the point of derivative or smth

[river drift]

everything in calculus is really a limit in disguise

[burnt wolf]

Indeed-y

[winter comet]

i mean there are integrals with bounds

so like

f(x) = x

from -2 to 2

the shaded region is the integral

except the negative area cancels out with the positive area

so i don't know why i used a confusing example

💀

[surreal crescent]

So it’s

$S(-2,2)$

[obsidian monolith]

Root

[surreal crescent]

Why did I think that would work

[winter comet]

lol

i think its like

backslash int_lowerbound^higherbound

or something

not sure

[river drift]

the integral $\int_a^b f(x) \odif{x}$ is the (signed) area between the function $f(x)$ and the $x$-axis between the points $x=a$ and $x=b$. because of the fundamental theorem of calculus'', it turns out that the integral is an operation related to the anti-derivative'', so we use $\int f(x) \odif{x}$ to mean the ``anti-derivative'' of $f(x)$, i.e., the function whose derivative is $f(x)$

[obsidian monolith]

pnoןɔ

[winter comet]

we're making this look way more complicated than it is

just take a funciton. its a line right? take the area between that line and the x axis, and thats the integral

lol

[surreal crescent]

$int_-2^2$

[obsidian monolith]

Root

[winter comet]

hmm

didnt work 💀

[obsidian monolith]

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

[winter comet]

oh lets go kinda worked not really

😭

maybe its .

not _

i'm not sure

don't take my word for it i'm really not sure

lol

[river drift]

the _ worked as intended, but it only includes one character by default

[surreal crescent]

$\int_-2^2f(x)$

[river drift]

to include more characters, enclose it in braces: _{-2}

[winter comet]

oh

[surreal crescent]

$\int_{-2}^2f(x) $

[winter comet]

well that kind of worked........?

good enough i guess 💀

[surreal crescent]

It’s like learning to code exept it’s only /print

[burnt wolf]

For real 😭…

[winter comet]

💀

[river drift]

you can click the reaction to get the error message

the problem is that you backslashed the dollar sign

[winter comet]

i thought you sighed lol

[surreal crescent]

$\int_{-2}^2f(x) \ $

Well now that just didn’t work

[river drift]

no spaces before the ending $

delete the backslash at the end

[surreal crescent]

$\int_{-2}^2f(x)$

[obsidian monolith]

Root

[surreal crescent]

Yay

wtf does the / do

[river drift]

\ is a prefix for latex commands

for example, \$ is a command that prints a dollar sign (the dollar sign otherwise is a signal to start/end math mode), \int prints an integral sign, \frac prints a fraction, etc

[surreal crescent]

Huh, anyways ima go to bed, I don’t think math is a good eneugh exuse to get 5 hours of sleep

[river drift]

do you know how a system of linear equations corresponds to a matrix equation?

[compact spade]

yes

[river drift]

so then you are trying to find if, in part 49a, [ \begin{bmatrix} x \ y \end{bmatrix} = \begin{bmatrix} 2 \ 1 \end{bmatrix} ] is a solution to the system of linear equations

[obsidian monolith]

pnoןɔ

[agile sparrow]

a for 49 and 50

[compact spade]

i dont feel like you need matrix multiplication for that...

hmm

thank you though

[agile sparrow]

Sorry not a for 50

[river drift]

matrix multiplication comes out to the same operation as any other method, just in a different format

[agile sparrow]

Oh yeah a for 50

[compact spade]

very much not a for 49

you are a liar how dare you

[agile sparrow]

Oh yeah it's B mb

My mind calculation is fucked

And I doubted the wrong one

[compact spade]

This is word salad

[river drift]

ok let's break it into steps

first, write it as a matrix equation

[compact spade]

how would i go about doing that

[river drift]

ok so for example
[ \begin{cases} 3x_1 + 6x_2 = 7 \ 2x_1 + 3x_2 = 9 \end{cases} ] then by the way we define matrix multiplication, that's equivalent to: [ \begin{bmatrix} 3 & 6 \ 2 & 3 \end{bmatrix} \begin{bmatrix} x_1 \ x_2 \end{bmatrix} = \begin{bmatrix} 7 \ 9 \end{bmatrix} ]

[obsidian monolith]

pnoןɔ

[river drift]

(you can multiply out the left side to see how that's true)

[compact spade]

ok makes sense for now

next thing?

[river drift]

the next thing is to construct an augmented matrix

[river drift]

if we call the matrix equation in this example [ AX = B ] then we have: [A = \begin{bmatrix} 3 & 6 \ 2 & 3 \end{bmatrix}, \quad X = \begin{bmatrix} x_1 \ x_2 \end{bmatrix}, \quad B = \begin{bmatrix} 7 \ 9 \end{bmatrix} ]

[obsidian monolith]

pnoןɔ

[river drift]

an augmented matrix is just a matrix where we've combined the entries of two different matrices into one for calculation purposes

so in this example we have: [ \begin{amatrix}{1} A & B \end{amatrix} = \begin{amatrix}{2} 3 & 6 & 7 \ 2 & 3 & 9 \end{amatrix} ]

[obsidian monolith]

pnoןɔ

[agile sparrow]

Find Inverse or A and multiply it with B

You're done

[river drift]

the instructions say to use gauss-jordan elimination

[compact spade]

yes and also i gotta understand the damn thing not just solve it

[agile sparrow]

adj would be
3 -6
-2 3

[river drift]

so in linear algebra, we have 3 basic row operations

[agile sparrow]

And determinatant -3

[compact spade]

please do not mention linear algebra those words scare me 💀

[agile sparrow]

There u go

Linear algebra

Linear algebra

Or maybe

Higher algebra

[compact spade]

ahhh terrified ahhh

[agile sparrow]

Coordinate geometry

[compact spade]

ok sorry cloud you can continue

[agile sparrow]

Who's cloud

[compact spade]

the helpful one

[river drift]

the idea is that performing such operations does not change the set of solutions to our equation

[agile sparrow]

So I am not helpful

[compact spade]

you're trying and that's helpful in terms of morale

[river drift]

the aim is to make a matrix equation with the same solutions, but which is easier to read. we call this "reduced row echelon form"

[compact spade]

but im finding cloud's explanations more helpful personally rn

[compact spade]

ok, makes sense

[proven geyser]

is it possible to skip pre calc over the summer

[compact spade]

yes

[proven geyser]

without getting a B or below in ap calc

ok

is it like light light or light

cause i lowkey already know half the shit they have on khan academy

[compact spade]

possible? sure. feasible? idk, idk you or your cirriculum

[proven geyser]

besides matrices and the advanced trig

i am relativley good at math

curriculum follows calc ab

doesnt matter for pre calc cause its not in school

[compact spade]

sorry, the "makes sense" was an acknowledgement that i was getting it so far - i'm still not 100% of the way there

[proven geyser]

also

when we use the word "mod" in pre calc is it reffering to abs value or actual remainder from division

cause i was reading this master math trigonometry book and the topic was polar equations

and it said mod was abs value

but mod in cs and combinatorics means remainderf rom div

[river drift]

most preferably, we would prefer our system of equations to look like:
[ \begin{cases} 1x_1 + 0x_2 = a \ 0x_1 + 1x_2 = b \end{cases} ] which in the augmented matrix format looks like: [ \begin{amatrix}{2} 1 & 0 & a \ 0 & 1 & b \end{amatrix} ]

[obsidian monolith]

pnoןɔ

[compact spade]

so rref?

[river drift]

yes

[compact spade]

ok

but how do we get from our current thing to that

[river drift]

Gauss-jordan elimination is a procedure to apply row operations to make our matrix into reduced row echelon form (rref)

our 3 row operations are:

1. multiply a row by a scalar
2. swap two rows
3. add a multiple of a row to another row

in principle there are many different sequences of row operations that can get you to RREF, but gauss-jordan elimination gives you a consistent procedure

[compact spade]

still on this

[river drift]

unfortunately the amatrix environment is custom

[compact spade]

how get from that to

[compact spade]

that

[river drift]

I in my infinite wisdom picked an equation with no solutions. So choosing a different augmented matrix (we will follow the same procedure): [ \begin{amatrix}{2} 1 & 3 & 5 \ 2 & 7 & 14 \end{amatrix} ]

[obsidian monolith]

pnoןɔ

[river drift]

The first step in Gauss-jordan elimination is to find the first pivot: this is the top left-most number

[strong epoch]

Could someone help me with a at least probability question?

I hated matrices bro

[river drift]

if the pivot is 1, we will leave it alone. If not, we will multiply the whole row by whatever we need to in order to make the pivot 1

[strong epoch]

So much work lol

[compact spade]

I hate them too

[compact spade]

Ok, makes sense

So usually it will be a fraction?

[strong epoch]

Could I get help with question 9?

[river drift]

yes, usually it's multiplying by 1/(something)

[compact spade]

And you multiply the row?

[river drift]

yes, so for example if we started with: [ \begin{amatrix}{2} 3 & 9 & 15 \ 2 & 8 & 14 \end{amatrix} ] then we would multiply the first row by $\frac{1}{3}$ to make the pivot 1: [ \begin{amatrix}{2} 1 & 3 & 5 \ 2 & 8 & 14 \end{amatrix} \begin{array}{l} R_1 \to \frac{1}{3} R_1 \ \mbox{} \end{array} ]

[obsidian monolith]

pnoןɔ

[river drift]

once we've done that, we "eliminate" every entry below the pivot by adding a multiple of the current row

so in this example, we'll add (-2) * the first row to the second row

in order to make the 2 below the pivot into a 0

[ \begin{amatrix}{2} 1 & 3 & 5 \ 0 & 2 & 4 \end{amatrix} \begin{array}{l} \mbox{} \ R_2 \to R_2 - 2 R_1 \end{array} ]

[obsidian monolith]

pnoןɔ

[river drift]

next, we take a new pivot down and to the right of our first pivot, in this case the 2

we again multiply the row by whatever's necessary to make the pivot 1:

[ \begin{amatrix}{2} 1 & 3 & 5 \ 0 & 1 & 2 \end{amatrix} \begin{array}{l} \mbox{} \ R_2 \to \frac{1}{2}R_2 \end{array} ]

[obsidian monolith]

pnoןɔ

[river drift]

since we've hit the bottom row we'll now eliminate above the pivot, to make whatever's above the pivot a 0 by adding a multiple of the second row to the first row

[ \begin{amatrix}{2} 1 & 0 & -1 \ 0 & 1 & 2 \end{amatrix} \begin{array}{l} R_1 \to R_1 - 3 R_2 \ \mbox{} \end{array} ]

[obsidian monolith]

pnoןɔ

[river drift]

now that we've achieved rref, the solution is made obvious. this seems like a lot of steps, but once you know the pattern it's mostly just rote work

if you search "gaussian elimination" or "gauss-jordan elimination" on youtube or google i'm sure there will be plenty of other explanations as well

[compact spade]

ok, thank you so so much

[strong epoch]

Do you know about combinations and stuff ?

[compact spade]

Me?

[strong epoch]

Anyone

I’m so confused

On 9

[compact spade]

Ah ok

I don't, good luck :(

[strong epoch]

Thanks

If anyone knows how to do it reading this isn’t it 1 - (probability of it not happening )

So isn’t it 83 choose 7

1- that

I got it let’s gooooo

[strong epoch]

1 - 83C7/108C7

[arctic sundial]

Mock exam today

Wish me luck

[willow skiff]

Good luck!

[candid mortar]

having trouble with these sort of questions. But the sheet full of them that i have dont come with solutions. On this one i got
(25-a^2) / a(5+a)

[surreal crescent]

The best way I’ve found is the use and actual division symbol-➗ (in case you forgot, lord knows I have) between the fractions, then put any non fraction terms above 1 and flip and multiply it out

[willow bear]

that's correct but you can simplify it further still

25-a^2 factorizes

(also the goal is to simplify, right?)

[candid mortar]

yea id gotten caught up in my frustration toward this complex fraction business and didnt realize until i wrote it down here

[candid mortar]

ill give this a go

[ripe siren]

hi can anyone help me with 3 precalc math problems?

so for the 3rd pic its the same option choice for both of the last question

[willow bear]

@ripe siren do you still need help with these?

[ripe siren]

yes pls

[willow bear]

ok, let's start with the first pic, which i'll repost standalone for my own convenience

any progress on this one so far?

[ripe siren]

so i got someone to help me and for a) i got 27017.7 b) idk yet

[willow bear]

can you tell me how you got 27017.7?

[ripe siren]

so since 1 revolution is 2pi radians and there are 4300 revolutions per minute, it should be 8600pi rad/min

[willow bear]

aight cool

ok

[ripe siren]

ye then i simlified it

[willow bear]

you put it in a calculator.

i wouldn't call that simplification.

you were asked to round to 2dp and you did.

... might wanna go to a help channel or wait

@ripe siren ok so now you want to find the linear speed of a point at the edge of your saw

do you know in general the relationship between angular speed and linear speed for circular motion?

yes/no

(no numbers yet!)

[ripe siren]

yes as the angular speed increase linear speed increase, no?

[willow bear]

that's too vague

do you know how to calculate linear speed given angular speed and radius?

[ripe siren]

no i do not

[willow bear]

right ok in that case

[ripe siren]

srry i dont kow much math

[willow bear]

let's step back a bit

since there's something we need to go over

instead of speeds let's talk about lengths first

so you know what a radian is, yes?

(yes i'm going somewhere with this, i just want to explain things fully before we get back to your problem)

[ripe siren]

thank you for trying so hard to try to explain it to me but i got the answer right 🙂

sorry i cut u off like that

:((

i didnt mean to waste ur time

[willow bear]

bruh

[ripe siren]

im soo sorrryy

[willow bear]

i mean

ok??

do you still need anything from me?

[ripe siren]

genuinely

[willow bear]

cause now you've shown me that you will throw away any attempts of mine at giving you a coherent explanation (and not just feeding you a formula you could plug&chug into)

[ripe siren]

no, sorry that it :((

[harsh basalt]

yo do u think u could help me now possibly

[willow bear]

no guarantees

but deezniks and i appear to be done, so you can post your problem (in full) now

[harsh basalt]

i need to use the function in the rectangular coordinate system

which is pretty easy to get the mins/maxes

[harsh basalt]

would i just use the rectangular coordinate system again?

[willow bear]

if you want to grab my attention, DON'T do it by reply-pinging on a message that's got nothing to do with your question.

anyway, furthest point from the origin just means r is maximized. so you need to take the max point for that you found in part A, and your point will have polar coordinates (f(θ), θ) with that value of θ.

[harsh basalt]

sorry for doing that

so would the x value of the max from part A be theta?

[willow bear]

that's what i said, yes

[harsh basalt]

alright thank you so much i appreciate the help

[naive dock]

Anyone here good with angle of rotation?

[willow bear]

!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/

[viscid thistle]

Is that Yamato?

[naive dock]

Yeah

Find the angle of rotation for 13x^2+6sqrt(3)xy+7y^2-32=0

[viscid thistle]

Bro

[naive dock]

I got cot(2theta) = 1/sqrt3

[viscid thistle]

Is that even precalculus?

[naive dock]

It is

[viscid thistle]

Sqrt????

[naive dock]

Is it possible for me to transfer 2 to the other side?

[naive dock]

Square root

[viscid thistle]

Ahh

What grade are you in? Just checking to know what's coming soon.

[naive dock]

Im in college

[viscid thistle]

Thank God.

[naive dock]

2nd year 1st sem

[viscid thistle]

Alright so I am safe fore a few years

Angle of rotation as in those idk transformations?

[naive dock]

Wdym

[mental fable]

a few prob means 2 buddy

[naive dock]

You use it to remove xy term I think

It has a formula of cot(2theta) = (A-C)/B

[viscid thistle]

Nah, 5

[mental fable]

how old are u

[viscid thistle]

11

[mental fable]

oh hell nah

[viscid thistle]

Lol

15

[mental fable]

oh nice im 16

[viscid thistle]

Am in grade 9

[viscid thistle]

Nice

10?

[mental fable]

oh really?

[naive dock]

☠️

[viscid thistle]

You in grade 10?

[mental fable]

11

[viscid thistle]

Nice

[mental fable]

im not too sure about these grade things

[viscid thistle]

Ohh shit, imma be 16 when I finish 10

[mental fable]

cus i learnt the stuff youre learning when i was 14

[viscid thistle]

Lucky lol

Well, smart actually

[queen shoal]

how do you find derivatives of functions

[willow bear]

https://youtu.be/5yfh5cf4-0w

https://youtu.be/-aTLjoDT1GQ

https://youtu.be/FLAm7Hqm-58

https://youtu.be/rAof9Ld5sOg

@queen shoal here, you might wanna give these a watch to start with.

[fiery creek]

do you have a specific function you want to differentiate

if so I can walk you through it

[arctic sundial]

I got a 5! 38/40 for mcq and idk about frq yet but I def got more than 4 points out of 24 to meet the minimum for a 5 bc we went over one FRQ in class today

[quiet merlin]

Does anyone know how to do these number 6,7,9,11,12

[viscid thistle]

Yes

Wait what

No

[viscid thistle]

Definitely not

Someone else???

[crisp hinge]

I'm in algebra 1, i just am learning this stuff

but what are integrals? and what are derivatives?

and what are hyperbolic sine/cosine/tangent

[summer ruin]

you shouldn't worry about this in algebra 1

[crisp hinge]

no i should because half of my class is foreshadowing for this

and its like

what you're learning is baby math

compared to

[insert trig or other stuff like calculus]

[river drift]

almost all math is baby math compared to something. but math builds on itself, so you have to have a solid foundation

[crisp hinge]

true

but I stil dont know what that stuff is

can u explain?

[summer ruin]

well the simple explanation isn't going to be useful and proper explanation isn't going to be short

[crisp hinge]

i dont care

im fine with the long explanation

[summer ruin]

then you can just open a book/search the web/watch a lecture on it

[crisp hinge]

i did

but

I need somebody to personify it

because i dont understand what they say

[summer ruin]

this is why you learn basic stuff first

you're not going to find a lot of people that are willing to explain material for hours

[crisp hinge]

true

[summer ruin]

so it would be better if you have specific questions about stuff you don't understand

[crisp hinge]

alright

is altitude easier to explain?

because i need to know all of this for math olympiad

and im in 6th

[summer ruin]

altitude?

[crisp hinge]

so idk any of this

[crisp hinge]

yes

something about triangle

s

[summer ruin]

who knows

you need to specify everything

[crisp hinge]

i need to understand this

could you draw an altitude from B?

[summer ruin]

sure

[crisp hinge]

so how would i do it?

[summer ruin]

draw a straight line from vertex B to the side AC

[crisp hinge]

does the angle have to be a right though?

because then

if it isn't it can be anything

[summer ruin]

what angle exactly

[crisp hinge]

if the point we are drawing from B to is named E

[summer ruin]

the vertex can have any angle, the altitude always creates a right angle with the side it touches

[crisp hinge]

then BEC

[crisp hinge]

ok

makes sense

none of the resources i found said that

[summer ruin]

my first search result is wikipedia's page on altitude https://en.wikipedia.org/wiki/Altitude_(triangle)

and it says it right there

[crisp hinge]

my browser is broken

[compact spade]

57 is impossible right

plugged it into rref and it didn't work

[river drift]

,w rref {{5,-3,2,2},{2,2,-3,3},{-1,7,-8,4}}

[obsidian monolith]

Results provided by WolframAlpha

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

Requested by cloudedhorizons

[river drift]

it's not inconsistent, but it does have infinitely many solutions

[compact spade]

Oh shit

Ok how do I prove that

[river drift]

rewrite this back into a system of equations

[compact spade]

i did

x - 5z/16 = 13/16
y - 19z/16 = 11/16

[river drift]

then you can solve for x and y

[compact spade]

can i??

[river drift]

just move the z term to the other side on each side

then you have a formula for x and y in terms of z

[compact spade]

but thats in terms of z then

[river drift]

yes, that's why there are infinitely many solutions

[compact spade]

so

[river drift]

each value of z corresponds to a different solution

[compact spade]

ok got it

ty

:)

[compact spade]

not sure how to properly turn this into a system of equations

[summer ruin]

find the number of beef required to produce some amount of muffins, bones and cookies

[compact spade]

ohhh wow that's really really helpful thanks

oh shit wait

sorry

i thought that was a sarcastic message until i reread it and you were unironically being helpful sorry 💀

...i still do need a little more help though given that im stupid

[summer ruin]

since you want to find the number of muffins, bones and cookies it makes sense to give each quantity a variable to represent each number

[summer ruin]

then do this

[compact spade]

so, what would each variable be representing here

[summer ruin]

the number of muffins, bones and cookies produced 👆

[queen shoal]

Thanks

[queen shoal]

No

[deft adder]

The function f is given by f(x) = (sin(x) + cos(x)) (sin (x) - cos (x)).
A student is trying to solve f(x) = 1. Which of the following equations
could be used to find the values of x that solve the equation f(x) = 1?
A) cos(2x) = 1
B) sin(2x) = 1
C) cos(2x) =-1
D) sin(2x) =-1

How do i solve this? I feel like i get most of it but then I get stuck

[chrome ether]

what is “most of it”? show us what work you have

[rugged harness]

let's say i can calculate the volume of a rotating block by calculating the area of several smaller shapes that making up that block, can i calculate the area of that block by calculating the circumference of those shapes

well particularly the torus

[queen shoal]

Wait I do actually

[viscid thistle]

Greatness can be attained through rigorous exercise.

Though this is quite a difficult math

Is it related to functions?

[river drift]

it's introductory linear algebra

[viscid thistle]

College level?

[river drift]

linear algebra is usually taught at the college level, although some high schools teach a small amount as part of an algebra/precalculus curriculum

[night elbow]

the derivative of the function: ln(f(x)) is generally f'(x)/f(x)

[willow bear]

... which is just a particular case of the chain rule.

[fiery creek]

this requires a bit of something called "chain rule"

are you familiar with derivative rules @queen shoal

[queen shoal]

no

i dont know derivative

[fiery creek]

ok well that's a pretty tough derivative for someone who's never done derivatives before

[queen shoal]

i dont know chain rule either

[fiery creek]

i'd recommend learning the basics first

[queen shoal]

ok

what are the basics

[winter comet]

it's either a "complicated" algebra 2 problem or a "basic" linear algebra problem

[winter comet]

the derivative of ax^(n) = anx^(n-1), where a and n are constants

example:
lets say you want to find the derivative of 5x^2
here, a = 5 and n = 2.
thus, plug it into the "formula".
the derivative of 5x^2 = (5)*(2)x^(2-1) = 10x^1 = 10x

[crisp timber]

why is the derivative of x^n = x^(n-1)/n?
It is kind of noticeable that it is x^(n-1), but why is it divided by n?

[summer ruin]

why is the derivative of x^n = x^(n-1)/n?
it's not

,w d/dx x^n

[obsidian monolith]

Results provided by WolframAlpha

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

Requested by transparent_elemental

[winter comet]

it's multiplied by n

make sure you aren't getting it confused with the antiderivative

[crisp timber]

True lol. But then, why is it multiplied by n?

[winter comet]

to prove the generalized chain rule you can always fall back on the definition of a derivative

i forgot exactly how they prove the chain rule

you can look it up if you want

but that's just what you do

you multiply it by n

[limpid wasp]

use the limit definition of derivative, and use binomial theorem for (x + h)^n

[loud valve]

does anyone know how to solve these

or these

[barren gorge]

hi

[viscid thistle]

Someone tagged me?

[winter comet]

here mebbe?

[viscid thistle]

Nah nah someone else

[queen shoal]

Thx

[willow skiff]

The hint is very useful about negative numbers

Is your problem with how to get the domain or writing it in the given notation?

[plush quail]

Getting the domain

[willow bear]

do you know the domain of the raw square root function?

i.e. of just sqrt(x)

[plush quail]

I have to determine the domain

Thats all thats given

[willow bear]

you did not answer my question

if you don't know the domain of sqrt(x), then you should just say "no, i don't know the domain of sqrt(x)"

[plush quail]

No i dont

Idk the sqrt (x)

[willow bear]

do you mean "I don't know what you mean when you say sqrt(x)" or do you mean "I don't know the domain of sqrt(x)"?

[plush quail]

I think i did 5 right

More so need help with 6

Actually no if i did 5 i can do 6

Number 10 is what i truly dont understand

[willow bear]

ok...

do you know how to work with equations of straight lines in general?

[plush quail]

No

[willow bear]

do words like "slope-intercept form" ring any bells?

or stuff like y = mx + c?

[plush quail]

It sounds familiar, but i couldnt tell u what mx or c is from

[willow bear]

https://youtu.be/Ft2_QtXAnh8 https://youtu.be/yoHs1h5qtuQ

give these a watch, and perhaps also search youtube for "organic chemitry tutor straight line equations" and watch the other vids that show up.

and/or also do that but for khanacademy instead of ochem tutor.

[lapis bramble]

never trust cha.gpt 😅

it is not a good study tool

[deft turret]

I did it y’all

I got into ap precalc

The 2 weeks of staying up until midnight was worth it

[winter comet]

ayy

[winter comet]

chat gpt actually made a mistake??? 🤣

unless it was talking about limits

then you ignore that kind of domain restriction

idk tho

[lapis bramble]

yep I asked it to help me simplify and it say'd that -64=-8

[winter comet]

oh

thats not even what i saw LOL

thats funny

[lapis bramble]

well cuz it canseled out x-64 and x-8 meaning their the same and also implying that -8=-64

[winter comet]

yeah

but they couldn't cancel it out anyway unless they noted that z != 8

thats what i saw originally

i didnt even notice the 64 💀

[lapis bramble]

lol

[burnt wolf]

I was just expressing my rage over AP Pre-Calc 😞

I had no problem with it until I found out everything from Matrices and up aren’t on the exam, and they don’t cover any actual Calculus content. Like not even like an introduction to limits or anything. Half of the questions from practice tests I could answer just from my Algebra II knowledge!

[winter comet]

no literally thats why i transferred out of precalc in the first two days lol

i realized i already knew everything they were gonna teach

so i just self studied the other stuff that they werent gonna teach

precalc is so undefined some of the classes are practically algebra 2 and some are actually purely stuff for calc

[river drift]

if a class taught calculus it would be called a calculus class...

[burnt wolf]

Exactly! I’m taking it over the summer since I basically know all the material, I don’t wanna spend an entire school year reviewing stuff I already know..

[burnt wolf]

AP Precalculus should at least introduce the topics in the first 2 units of Calc AB/BC or else what difference does it really make?

[winter comet]

i think they teach basic limits sometimes in precalc

[burnt wolf]

I thought they did…but they don’t even talk about it in the AP 😭

[river drift]

the point of a precalculus course is to cover everything you need to know before a calculus course, not to teach the "first part" or "introduction"

[burnt wolf]

I already know limits too…

[winter comet]

i remember i looked at the syllabus for precalc and i was like "bruh i thought this was algebra"

[burnt wolf]

Yeah, a regular Precalculus course. If the AP just teaches the same stuff and doesn’t even slightly delve in Calc AB stuff, then what makes it different?

[winter comet]

i thought precalc was matrices, polar coordinates, conic sections mebbe, vectors, limits, probability? idk

[river drift]

what makes an ap course, an ap course, is that it offers college credit (in this case, for the type of precalculus course taught in college). nothing really more or less than that

[burnt wolf]

The AP exam doesn’t cover matrices, vectors or limits

[burnt wolf]

But AP Precalc doesn’t even give you credit at most colleges given that Precalculus is a high-school course

AP classes are typically either the first semester, or the first-year of a college course

[river drift]

most colleges do offer a precalculus course for students who do not yet have the pre-requisite knowledge for a first-year calculus course

[burnt wolf]

Yeah, but again, most colleges won’t give you credit for AP Precalc given that as you said, it’s only taken for those who don’t have the prerequisite

[river drift]

for non-stem majors some colleges allow for their pre-calculus course to count for whatever math requirement they have. it's certainly not useful to every student, although that's also the case for any ap course

i think the broader issue is that the range of topics a "precalculus" course may cover is extremely wide, depending on the level of knowledge a calculus course expects you to know vs what they expect to teach you

[burnt wolf]

I suppose sure, but I’m just saying, there’s nothing that makes it “college-level” like an AP course is supposed to be

[river drift]

i don't know enough about the ap precalculus curriculum to know whether it is "college level" (which a precalculus course can be, as evidenced by the fact colleges offer them), but nothing about a course being "college level" implies it must teach calculus

[burnt wolf]

Not straight-up teach Calculus, but introduce some of the topics at a slightly more basic level perhaps. Cause like I said, AP Pre-Calc doesn’t even cover matrices, vectors, or “functions involving parameters”

And those topics are not that difficult, at a Precalculus level, matrices are super easy to wrap your head around

[sinful halo]

@steep estuary

[chrome ether]

AP precalc is not college level.

that is the big thing that just makes me go "why are they even making this a thing" 😭

a lot of colleges will likely just not take it for credit, period

[viscid thistle]

Ap calculus?

[chrome ether]

ap PRECALC

not ap CALC

(which is also hardly """"college level"""")

[viscid thistle]

Nah i know but like I was asking whether or not ap Calc Is college level. Follow up question, what can be considered college level then?

[chrome ether]

calculus is barely scratching the surface of actual undergraduate mathematics

[winter comet]

i think calc 1 is where undergraudate starts

which would technically be 'college level'

[viscid thistle]

How do you scratch undergrad maths from both ends?

[winter comet]

what does that even mean 🤣

[chrome ether]

the meatier parts of the undergrad curriculum begin when you start to move away from just endless computation and move towards proof-based stuff

[viscid thistle]

Ahh I just love smth that's meaty like proof.

[chrome ether]

truth be told

the really substantial stuff is more at the graduate level

[viscid thistle]

I tried proof, understood some of it, but like it's quite conceptual I felt.

[chrome ether]

the most important undergrad courses are probably abstract algebra and analysis

[chrome ether]

.... that is a completely vacuous statement

[viscid thistle]

What's Complex anal?

Masters?

[chrome ether]

some undergrads take it

[viscid thistle]

Ahh I see

[chrome ether]

the point of undergrad is more to

develop the mathematical maturity you need for intensive graduate level research

[willow skiff]

No it's a standard upper undergrad course for pure maths

[viscid thistle]

I probably won't even be picking maths in undergrad since I am going into bio but still it's quite interesting.

[willow skiff]

Cool

[deft turret]

At my school you have to be the top score’ers to get into ap calc

But that only applies to honors im2 I’m in regular im2

But there was space for ap pre calc so she gave it to me

But I do know that pre calc won’t be accredited as much as ap calc since college doesn’t teach or calc

Pre calc*

[burnt wolf]

Oh, all you have to do for me is to get an C or higher in AP Pre-Calc, or an A in regular with a teacher recommendation. And that’s for BC, it’s even more lenient for AB

[viscid thistle]

Hey guys

Need your help

I need to calculate the derivative of this function

I dont understand this moment

Why x^5 became x^2

[river drift]

they factored out an x

,, x^{5/3} = x^{3/3 + 2/3}

[obsidian monolith]

pnoןɔ

[viscid thistle]

I still dont understand it

[river drift]

,tex \exprules

[obsidian monolith]

pnoןɔ

[viscid thistle]

I understood it but another way

Anyways thanks

We can also figure it out this way: cube_root(x^5) = cube_root(x^3x^2). And then we basically get x^3 out of the root. xcube_root(x^2)

[viscid thistle]

Whoa, are there cmds for other types of formulas aswell?

[clear scarab]

,tex \integrationrules

,tex \intrules

,tex \help

[warped cipher]

you need preamble for that

[willow summit]

any one help me i have high school additional math exam in this year

[willow bear]

post the question(s) that you need help with

[viscid thistle]

Guys

Why (a^x)' = ln a * a^x

Can it be proved?

Or i should just remember it

Generally, i dont understand what a natural logarithm has to do with it

[willow skiff]

$a^x = e^{(\ln a) x}$ and the chain rule

[obsidian monolith]

south

[willow skiff]

Next time ask in #calculus

[viscid thistle]

What is precalculus then?

What do you discuss here

[willow skiff]

Trig and functions

Like logs, exponential functions, rational functions, sin, cos, tan etc

[viscid thistle]

Usually if i have smth regarding to trig, i go to trigonometry channel

[agile sparrow]

Let f : (0,1)   be the function defined as f ( ) x n  if 1 1,
1
x
n n
 
     
where n   . Let
g : (0,1)   be a function such that 2
1 () 2
x
x
t
dt g x x
t
    for all x(0,1). Then
0
lim ( ) ( )
x
f xgx 
(A) does NOT exist
(B) is equal to 1
(C) is equal to 2
(D) is equal to 3

[modest bay]

😧😧😧😧

[agile sparrow]

sorry

im so sorry

[willow bear]

give us a picture.

[agile sparrow]

shit

not this

[viscid thistle]

So high.......

[agile sparrow]

the one in #calculus

[viscid thistle]

Bad joke mb.

[agile sparrow]

wat

[proven void]

,, \text{Find } a \text{ and } b > 0 \text{ such that the } \lim_{x \to \infty} \left( \frac{\sqrt{ax^4 + bx^3}}{x} - x \right) = 4.

[obsidian monolith]

milanesa de pollo

[willow bear]

you don't have to do a million \text{} btw

you could have typed the same thing like this:

Find $a, b > 0$ such that $\lim_{x \to +\infty} \left( \frac{\sqrt{ax^4 + bx^3}}{x} - x \right) = 4$.

[obsidian monolith]

|Ann⟩

[willow bear]

also show all your progress thus far.

[proven void]

is it bad to say status = 1

!status

[tender quest]

What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[willow bear]

no, it's much better than leaving the status query on read.

for step 1 i suggest rewriting this thing as a single fraction

then you can do some bullshit with conjugates.

[proven void]

what now?

the conjugate of the numerator ?

[agile sparrow]

Take x^4 common

From root

That would give x^2

Take both x^2 common

And cut one x

[willow bear]

recommend against manipulating the limit and dragging it all along with you

just work algebraically with the function inside of it

as mentioned earlier, do some conjugate bullshit

namely multiply top and bottom by sqrt(ax^4+bx^3) + x^2

[agile sparrow]

the limit is tending to inf

That would make it messier

[willow bear]

not any more so than any other limit with roots like this.

[agile sparrow]

It's useful when tending towards 0

[proven void]

I dont get it

[willow bear]

trust me it is still helpful here

[willow bear]

mb because two people are trying to tell you different things

and so high is doing it in a somewhat hard to read way

[willow bear]

i have to go sleep tho so maybe it will have to be the blind leading the blind

[agile sparrow]

take x suare common

[agile sparrow]

a has to be 1

[proven void]

why?

[agile sparrow]

i solved it ]

in my mind

it could be not right

but it should be

my mind calc sucks

after u take x^2 common

u'd be left

ith]

x(underroot[a+b/x]-1)

underroot a+b/x-1 has to be 0

to create inderminant form

put the limit x = inf

b dissapears

and a=1

and b is 8

[tender mist]

I had 0 context before seeing you say that, so for a moment I thought you were saying that calculus sucks, lol

[river drift]

i made it myself, so you can get such commands by making them

[proven void]

I dont get it dammit

[north anchor]

what’s the formula for an alternating series like this? i cannot remember lol

[river drift]

alternating geometric series don't have a separate formula from other types of geometric series

[agile sparrow]

lets talk tomorro

[agile sparrow]

a/(1-r)

1/5

1/26 is tat the ans

[north anchor]

ok ty!

[north anchor]

oh that’s an s not a 5

[agile sparrow]

oh my fucking gad

[river drift]

!nosols

[tender quest]

As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.

[willow bear]

this is still a geometric series

$\frac{(-1)^n}{s^{2n+2}} = \frac{1}{s^2} \cdot \paren{-\frac{1}{s^2}}^n$

[obsidian monolith]

|Ann⟩

[proven void]

Represent the following sets on the line. Write them as intervals or as a union of intervals
${ x \in \mathbb{R} \mid |x| > 3 }$

[obsidian monolith]

milanesa de pollo

[proven void]

sorry if its badly translated wtf should I do here?

in general if |x| < 3 that means
-3 < x < 3

but how is it for |x| > 3 tho

[willow bear]

well

|x| is the distance from x to 0 on the number line, yes?

[proven void]

yeah

[willow bear]

well

|x| > 3 means x is further than 3 units from 0

this should enable you to draw it on the number line

[proven void]

-3 > x > 3 maybe?

so strange smh

but I only know because if |x| < 3 then -3 < x < 3

[willow bear]

no

[proven void]

then prolly |x|>3 means -3 > x > 3

[willow bear]

i said draw on the number line

[willow bear]

no, stop trying to mash symbols together

do as i say

dont play 5D chess with my instructions

again

draw a number line

mark the points 3 and -3 on it

then color all of the points which are further than 3 units away from 0

[proven void]

I dont get it

What about it

[willow bear]

erase the gd -3 > x > 3

but yes the diagram is correct

thats what you were supposed to do

thats what i asked you to do

by the way -3 > x > 3 would translate to (-3 > x) AND (x > 3), but there's no point which satisfies both inequalities at once -- you cannot be under -3 while also being above 3.

[proven void]

That was the lesson learnt that the two side inequality is wrong

How to write that as union of intervals?

[willow bear]

well

how many intervals do you see marked on your diagram?

[proven void]

not sure

[willow bear]

look at it.

ok let me reproduce yours in somewhat better visual quality.

how many intervals are colored?

don't overthink this, i know for a fact that you're able to answer this.

im not asking you to do the whole thing at once. just do what i say.

do what i say nothing more. nothing more. dont do more than i ask for.

[proven void]

two

[willow bear]

right.

let's look at the first interval, on the left.

how do you write this in interval notation?

[proven void]

not sure if the ifinity is inclusive but
(-infty , -3) would be my guess, since you cannot include infinity

[willow bear]

you never include infinities.

but yes, (-∞, -3) is correct.

now look at the second interval, on the right.

what's that in interval notation?

[proven void]

(3, +infty)

[willow bear]

right.

[proven void]

I think I understood, tysm

[viscid thistle]

,tex \exprules

[viscid thistle]

How do you make em?

[river drift]

https://www.overleaf.com/learn/latex/Commands

read the pins in #latex-help for a more general introduction to latex

[proven void]

how can I graph $f(x) = cos(2x + \pi)$ just from knowing how to graph $g(x) = sin(x)$ and $h(x) = cos(x)$

?

[agile sparrow]

@proven void

[proven void]

wassup

[agile sparrow]

What's senx

[obsidian monolith]

milanesa de pollo

[proven void]

typo, mybad

[agile sparrow]

Draw the graph of cos2x

and flip it upside down

[proven void]

,w plot cos(2x)

[obsidian monolith]

Results provided by WolframAlpha

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

Requested by xbz24

[proven void]

how do I flip it

[agile sparrow]

nah

I thought u meant on paper

Like for general understanding

Do u know how to plot cos2x=

[proven void]

no

[agile sparrow]

Draw cosx and shifts the point by half

Life π/2 becomes π/4 due to the 2x

And vice verse

cos(2x+π) the pie makes it shifts to the 3rd quadrant

thats why it flips

[proven void]

how do I

[agile sparrow]

anyway

[proven void]

graph

that

[agile sparrow]

cos2(x+π/2)

[proven void]

?

[agile sparrow]

2cos^2(x+π/2)-1

[agile sparrow]

Im checking it for myself

I don't see the use of sinx graph in this

Also

meh

[proven void]

you dont need to use sin

its only if needed

I dont get it tho

[agile sparrow]

Well I hope u understand the graph

[proven void]

how to I sketch cos(x) ?

[agile sparrow]

Shift sinx

Graph

Making it start from 1

as cos 0 = 1

where 0s the x and 1 is the y

And at π/2 it's zero

Use desmos for reference

[near sentinel]

how do i do this

1/512 is the answer

its not in the Q

[willow bear]

$\lim_{x \to 0} \frac{(16+x)^{1/4} - \frac{1}{32}x - 2}{x^2}$

[obsidian monolith]

|Ann⟩

[willow bear]

did i read the problem correctly?

@near sentinel

[proven void]

how to do this?

we got to clear the x^2 from the denominator no?

how to doe

[willow bear]

i think i'd rather wait for OP to give confirmation that i have the problem written correctly

[proven void]

@near sentinel

[willow bear]

assuming that it \textit{is}, i would rewrite $(16+x)^{1/4}$ as $2(1 + \tfrac{1}{16}x)^{1/4}$, and then expand that into its taylor series up to and including the $x^2$ term.

[obsidian monolith]

|Ann⟩

[willow bear]

failing that (such as if op doesn't have access to taylor series), we could resort to l'hôpital's rule.

[proven void]

what are the conditions that need to be met so lhopitals is possible to be used?

[proven void]

2^4 = 16 I didnt noticed

taylor series?

[willow bear]

i mean, by asking this question you admit to not knowing what l'hôpital's rule is.

but the most important condition is that the limit is in either of the two forms 0/0 or ∞/∞.

here it is 0/0.

[proven void]

oh yeah its 0 numerator as well didnt noticed, again 2^4 = 16

[willow bear]

https://en.wikipedia.org/wiki/Taylor_series

(1+x)^p in particular has a known and simple(ish) taylor expansion.

[proven void]

? how so

[willow bear]

$(1+x)^p = 1 + px + \frac{p(p-1)}{2!} x^2 + \frac{p(p-1)(p-2)}{3!} x^3 + \dots$

[obsidian monolith]

|Ann⟩

[willow bear]

this could also be obtained as a generalization of the binomial theorem (the generalization being that the exponent no longer has to be a natural number)

[proven void]

was tough to see at first glance, but this is covered in two weeks for me

whats next?

[near sentinel]

yupp

[willow bear]

ok then i repeat my suggestion:

[near sentinel]

yeah

[willow bear]

expand (1 + x/16)^(1/4) into a power series up to and including the x^2 term

do some simplification

the answer will just pop out

if you don't have access to power series, then apply l'hôpital.

[near sentinel]

here's what I did, I rationalised the numerator and got until {16+x - (x/32 +2)^2]/32x^2

[near sentinel]

idk both of these 🫠

wait whats power series

[willow bear]

can you show your work for rationalization

[near sentinel]

i think ik it but with a dff name

[near sentinel]

sure

[willow bear]

taylor? maclaurin?

does either of those names sound familiar?

[near sentinel]

yeah ik taylor

[willow bear]

then that is what i meant

[near sentinel]

ohkk

my teacher did teach us aa method uding binomial expansion, but im not very good at it

[willow bear]

yeah binomial expansion is exactly what i am suggesting here.

$(1+x)^p = 1 + px + \frac{p(p-1)}{2!} x^2 + \dots$

[obsidian monolith]

|Ann⟩

[near sentinel]

ohkk

If you dont mind can you please explain it to me

[willow bear]

explain what

[near sentinel]

the method; i tried it but i couldnt get rid of the x/32 and 2 term

ill send pic of my working

if i apply DS, im left with 1/x^2

[willow bear]

rewrite $(16+x)^{1/4}$ as $2(1 + \tfrac{1}{16}x)^{1/4}$ and then as $$2 \paren{1 + \frac{1}{4} \cdot \frac{x}{16} + \frac{\frac14 \cdot (-\frac34)}{2} \cdot \paren{\frac{x}{16}}^2}$$

[obsidian monolith]

|Ann⟩

[willow bear]

you are not expanding far enough

you need the x^2 term from the expansion too

[near sentinel]

ohkk

ill try doing that

thank you!!

that way there will remain a term of x in the denominator

[near sentinel]

this is what im getting

the x^2 term got cancelled

but the ans is wayyy off

ok not wayy off

close

but not accurate

where am i going wrong

hmm

i can try that

[proven void]

I got x^3 from the expansion and Is still not enough

did you managed to solve?

I still cannot

I dont think any other more terms will work because it cannot get cancelled with the denominators x^2

so will be zero, without providing any more preciseness, e.g. the third term of the expansion is x^3 something so it will result in something multiplied by x and since x is approaching zero . . .

if someone manages find out the answer ping me

[proven void]

also I dont get it your prof is saying the answer is positive but when I try do it I also get the middle x^2 being negative

mm

whatever

[near sentinel]

nope 😦

[near sentinel]

yeah, it only works till 3rd term, we were told to ignore the rest of them as it approaches to zero

[proven void]

💀

[near sentinel]

exactly, and even the ans is coming diff

[proven void]

@willow bear , any idea?

[near sentinel]

its actually a build up question

ill send the prev q

like its implied you have to solve it like the prev question

[proven void]

did you rationalized for 3?

or binomial expansion/ taylor series around x = 0

?

why is 4 repeated twice

but with different functions

[near sentinel]

thats a diff question, its like a hw assignment and our sir is too lazy to change index

[near sentinel]

yupp

i got the answer for 3

by rationalisation

but im supposed to do it by binomial exp

and the ans comes either way

[willow bear]

,w lim[x -> 0] x^-2 * ( (16+x)^(1/4) - x/32 - 2)

[obsidian monolith]

Results provided by WolframAlpha

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Requested by ann_dec

[near sentinel]

💀

[willow bear]

did your prof proclaim that the answer is positive?

[near sentinel]

its a hw question; so i dont know until tomm

ill ask

and confirm

[proven void]

I got the same number

[near sentinel]

🫠

i think i should ask my teacher

cause smth is wrong

with the question

imo

[proven void]

really nice question tho

[near sentinel]

or the ans key

[near sentinel]

yeahh

[proven void]

whatever

[hot kayak]

Help

[hoary mountain]

What to do?

[willow bear]

this is unreadably small