precalculus

[heavy pewter]

what graph sort of could look like this?

[willow skiff]

it's pattern recognition

consider using Quizlet (or Anki if you're dedicated) for simple recall, with the name on one side and the picture/definition on the other

[trim shard]

ive never studied from it but stewart does it best from what ive seen

i use it for my calculus course and it clarifies all the important rules and gives pretty great examples, so it cant be a bad pick when it comes to precalc either

[willow skiff]

LibreTexts better

[remote cove]

@willow bear

I got a doubt

[echo sinew]

!noping

[tender quest]

Please do not ping individual helpers unprompted.

[remote cove]

M sorry

[willow bear]

if you wanted to follow up with me about something i said then you should say so explicitly ("I wanted to follow up / ask you again about [thing]...")

if you have a new doubt/problem/question/whatever then you should just post it

maybe i will come help you maybe i will not

you should not hold out hope for me specifically

[remote cove]

Yup got it

I usually post it

But I was asking

[willow bear]

when you post your doubt directly, you give me a chance to decide whether or not i have enough energy to explain it to you

[willow bear]

so far you have not asked anything at all

[remote cove]

I'm posting...

Why did they mention the y-x curve not the x-y curve and what is the difference? What is the purpose of introducing the rate of change at the curve that is dy/dx = 0

[willow bear]

ok nope sorry you will have to wait for somebody else

i do not have the energy for this

[remote cove]

Ok

I go ask in help ticket

[uncut star]

At dy/dx slope is 0

So it is the local maxima or minima

I am not sure if this was what you were asking

[alpine shore]

You can name a graph by mentioning what's in y-axis vs what's in x-axis e.g voltage (y-axis) -current(x-axis) relationship because we know slope=dy/dx which for this case gives resistance.

[winged epoch]

A little confused about this thing. So I understand least common denominator and how to apply it to adding/subtracting rational expressions and simplifying complex rational expressions yet there's a nuance that is confusing.

In the first problem, I can multiply the expression by (x+1)(x-3)/(x+1)(x-3) and then I can cancel out the denominators of the rational expressions in the numerator and denominators giving me the correct answer.

However in the second problem, I can't cancel anything out in the numerator or denominator. I have to multiply the terms in the numerator after I put them all over the LCD. Yet before I multiply some of the terms in the denominator can be canceled out. Why is it different?

[winged epoch]

Yet I can't do that in the third screenshot

oh well actually if someone wants to answer they can but i believe i see it

[raven laurel]

How do you eliminate T in parametric eqations?

[winter comet]

with algebra

do you have an example lol

[raven laurel]

here let me give one problem

number 60

[winter comet]

try using the fact that cos^2(theta) + sin^2(theta) = 1

[raven laurel]

ok

[novel tusk]

I wanna ask, does the inscribing shape in a circle trick work for all polynomials?

[bleak forge]

im doing this in chem rn

[oak ferry]

teach me pookie

[bleak forge]

what is the equation for decomposition of N2O4

[willow bear]

what trick

[viscid thistle]

Hi guys

Any tips to improve indefinite integration , I'm able to do easy ones but the jee level I cant

[wild bramble]

wdym integration were in precalc 😭😭😭

[viscid thistle]

I'm from India we don't have such things as precalc and stuff

[wild bramble]

oh okie

but i suggest you find the #calculus channel, because you're more likely to find help there

[viscid thistle]

Oh thank you , I didn't see it sorry

[fiery crest]

I have a precalc 2 midterm this week and I'm already really behind I don't know what to do really. It feels like the questions given on tests don't match what we did in class or on homework.

It's the first time I've had a teacher not allow notes at all during tests and I keep bombing all the tests

[zinc matrix]

i'm so frustrated, i feel like i can't understand anything and my mind is just blocking information

[whole void]

do u have an example of one?

[idle field]

I've been frustrated for the last 10 years.

and that wasn't enough to stop me

[gleaming olive]

im not sure if this will help but it seems like to me that the tests must go much more in depth into concepts only really outlined in class or in homework. for me this helps, try going through the basics of the concepts again, and try to explain the concepts to someone else. if youre able to understand a concept well, u can teach it to someone else or have a conversation abt it. then it would be good to try word problems or problems that connect together several concepts or problems that require more abstract uses of the concepts. 👍

[fiery crest]

The main problem I'm facing is not understanding fractions I think

[gleaming olive]

i see. im not in precalc yet so i ddont really know the specifics but have u tried watching vids on it and asking for help in class?

[idle field]

What grade are you at the moment?

[fiery crest]

1st year college

[idle field]

What's the difficulty with fractions?

[fiery crest]

I don't really understand how to do anything with them. I've almost never used them

[idle field]

I think this topic is covered on duolingo math, try that, or any lower level text book.

[gleaming olive]

i dont know abt precalc fractions. can u try to explain them. it should help trying to outline what u know about them anyway

[fiery crest]

I'm just talking about normal fractions

Missed all of elementary school

[gleaming olive]

ah

i see

[gleaming olive]

in terms of this are u referring to operations with fractions?

[fiery crest]

Yeah it feels like there's all these rules with fractions

[willow bear]

do you mean like just-number fractions or do you mean algebraic fractions

[dim matrix]

Fairly understand the rules adding and subtracting fraction first then move on to understanding the rule when diving a whole number with a number expressed in a fraction form. If you nail those basics down, then you will more confident on taking the exam

If your exam covers problems regarding solving problems about fractions, not rate problems (which are word problems that involves fractions), then you should focus on doing many problems.

[viscid thistle]

any precalc textbook to buy

[chrome ether]

there are tons of free texts online

openstax, libretexts, etc surely have something

[whole void]

wdym “precalc fractions”

[heavy pewter]

fractions are fractions what

precalc fractions doesn't really mean much if you don't explain

[viscid thistle]

In terms of textbooks, any one will do?

I sort of don’t like the way some textbooks are, where it doesn’t feel like it’s enough info

Or they omit something that other textbooks have, weird quirks like that

I’m more curious about the why’s as I engage in the how’s

[viscid thistle]

looking for algebra and pre calculus textbooks

I briefly saw art of problem solving, openstax, and stewart textbook. But confused which one to go for

I don’t mind purchasing the book, need physical copies as digital does me no good

[cedar sparrow]

who can help me on this

[heavy pewter]

what is g(-5)?

[cedar sparrow]

example problems but i still dont get it

like i hate graphs

[heavy pewter]

ugh just think of it this way

$f\circ g(a)$ means whatever you find for $g(a)$ i.e. let's say $g(a)=k$ then that becomes the input for $f$ i.e $f(k)$

[obsidian monolith]

parabolicinsanity

[heavy pewter]

then all that's for you to do

[cedar sparrow]

okay

[heavy pewter]

is to find g(a)

then input that for f

[cedar sparrow]

alright

[cedar sparrow]

would 5. be f(g(-5)) = 2

[heavy pewter]

ehhh no?

g(-5)=-3

oh wait yes 😔

i'm kinda blind

but yes that's the answer

[cedar sparrow]

okay lemme try the next one

thanks

[heavy pewter]

$f:A\longrightarrow B \land g: B \longrightarrow C$ so $g\circ f: A\longrightarrow C$

hmmmm

[obsidian monolith]

parabolicinsanity

[cedar sparrow]

would the second one be g(f(2)) = 1

[heavy pewter]

yeah

[cedar sparrow]

ayee okay

lemme do this last one and then study on my own

[cedar sparrow]

g(f(-1))=1 for the last one?

[heavy pewter]

uhhh it's 5/2

or 2.5

[cedar sparrow]

uhm how

[heavy pewter]

f(-1)=3

g(3)=2.5

[cedar sparrow]

tysm

[heavy pewter]

happy mathing

maths is fun

[fallen swift]

hi

[silk yacht]

Real

[silk yacht]

I solved the integration but how tf do I find f(x)

this is the answer I got

but Can't find f(x)

[raw hill]

Ok this is rlly bad question design but I’ll digress on that front

Finding $f$ explicitly seems like too much effort here. Just note that $$\frac{\sin 1+\cos 1}{\sin 1-\cos 1}=\tan \left(\frac{\pi}{4}+\frac{f(1)}{2} \right)$$ and compare.

[obsidian monolith]

Civil Service Pigeon

[silk yacht]

what if I wanted to find f(x)

????

[silk yacht]

also this is wrong I believe

denomiantor would cos1 - sin1

[silk yacht]

I got the answer by comparing but what if I wanted to find the function?

should I just differentiate both the sides?

[raw hill]

Uh you could do the algebra but it’ll be ugly

[raw hill]

Eh whatever I’m not too worried about the details - as long as the concept gets across

[raw hill]

,w solve for y if |(x sin x + cos x)/(x cos x - sin x)|=|tan(pi/4 + y/2)|

[obsidian monolith]

Results provided by WolframAlpha

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

Requested by civil_service_pigeon

[raw hill]

I assume you’re working over the reals

So you don’t need to worry about the complex stuff

But it’s still not the prettiest

[crisp fractal]

Hello is this a channel for calcul ? like division and stuff ? 👀

[raw hill]

“calcul” as in “calculus” or “calculation” (arithmetic)?

[crisp fractal]

calculation i think in this case about patterns

[raw hill]

Throw it in #prealg-and-algebra ig

[crisp fractal]

About 1;4;9:16 etc

[raw hill]

Or just open a help channel ngl

[crisp fractal]

analysis

[raw hill]

Those get way more attention

[crisp fractal]

ok thanks 😄

[viscid thistle]

Hi there I need quite a bit of help

So I’m stuck on how to graph the inverse of functions

It’s 18-20 and I really have no issue with 21-23 but it’s solely 18-20

& Yes I did page back to section 6.3!

There is actually no way to graph 18

I can’t graph said set

[lethal sand]

isnt to graph inverse functoins is js switch the x and y coords

[viscid thistle]

Yes for basic ones but in this scenario Desmos wasn’t picking up my answer so what I had to do was

F(x)=-logbase9(x)

Instead of the actual relevant answer

[willow bear]

it doesn't like F^-1 as a function name.

if you clicked the ⚠️ symbol you could have seen desmos' exact complaint.

[hollow geode]

look at the error messages

[heavy pewter]

desmos doesn't like function inverses besides the library functions

don't even try

just define it as another function

[viscid thistle]

Ye I can’t lie I have no way to graph this

or know wtf it means

For comparison I tried this as well

[lean drum]

That works

[normal quail]

I'm 26 years old and self studying math too. Some friend suggested this:

1a

https://artofproblemsolving.com/store [Introduction-level & Intermediate-level]

For if you need a review of “basics”. I suggest working through Introduction to Algebra & Intermediate Algebra & Precalculus.

#############

1b

Brilliant dot org [Problems of the week Archive]

2

-The two volumes on the contest preparation section.

-Book: The Art and Craft of Problem Solving, Paul Zeitz

Have a look on these. And feel free to select only one of them (I recommend the second)

You may also check George Polya books::

Notably, Mathematical Discovery on Understanding, Learning, and Teaching Problem Solving, Volumes I and II

3

The two volumes of Tom Apostol, Calculus [main]

Spivak Calculus [have a look at the problems here while working through Apostol first volume]

---

1. Book: An Introduction to Mechanics, Kleppner and Kolenkow

2. Book: Electricity and Magnetism, Purcell [Make sure that you have the third edition with David Morin]

• https://www.feynmanlectures.caltech.edu/

I am personally self studying pre calc from AOPS.

They say AOPS is challenging compared to others.

I'm also reading the art and craft of problem solving by Paul zeitz.
Seems like a good book to complement with the pre calc textbook

[silk yacht]

OH HELL NAWW, I AIN'T DOING ALL DAT

[drifting oyster]

I don't think desmos supports $f^{n}(x)$ notation

[obsidian monolith]

Enterlessguy

[drifting oyster]

That's probably the issue

You could just define $g(x)=f^{-1}(x)$ and graph it as $g(x)$ instead

[obsidian monolith]

Enterlessguy

[drifting oyster]

He has to stop the car, that means it has an intersection at that point, ie $v(k)=0 s.t k>0 \in \mathbb{R}$ it also need to be $v(0)=0$ since the initial velocity before driving was 0. I'll try not to use derivative logic here since you probably haven't learnt it but basically when he's driving to school, the graph is most likely going to increase at a certain rate (assuming he doesn't random stop for something or suddenly reduce his velocity drastically) so that means until a certain point our velocity graph needs to be greater than 0. After that interval, since he is going to stop the car, our velocity needs to decrease and eventually hit 0. Now graph A satisfies both of these, it is increasing from the interval [0,pi/2) (I know that because its a sine graph) and decreasing from [pi/2,0). However the question says that when he is returning home, assume that velocity>0 therefore our velocity graph cannot be less than 0 for any interval of t. Therefore graph A does not satisfy this condition. So you need to find the graph which best represents these conditions. Hope that helpz

[obsidian monolith]

Enterlessguy

[viscid thistle]

Does anyone know which one is the right one?? The one on the right is the solution my teacher gave in class... but the one on the left is the solution i saw on a different book

[uncut mulch]

your teacher is wrong

also not in pre-calc territory anymore

[drifting oyster]

Let $u=3x=> du=3dx, u_{1}=3,u_{2}=9 => I=\frac{1}{3}\int_{3}^{9} \ln{u}du$

[obsidian monolith]

Enterlessguy

[viscid thistle]

The calculus channel isnt showing up for me

[drifting oyster]

Now using integration by parts we have: $I=\frac{1}{3}[u\ln{u}-u]_{3}^{9}$

[obsidian monolith]

Enterlessguy

[uncut mulch]

select undergrad in id:customize
also you can claim channels dedicated to you

via #❓how-to-get-help

[obsidian monolith]

Enterlessguy

[drifting oyster]

$I=5ln{3}-2$ that's the solution but if you're wondering about the anti derivative it's: $x\ln{3x}-x+C$

[obsidian monolith]

Enterlessguy

[viscid thistle]

Thank you

[drifting oyster]

Your teacher simplified but forgot to exclude $\frac{1}{3}$

[obsidian monolith]

Enterlessguy

[drifting oyster]

(My bad I simplified it wrong, its 5ln(3)-2 lol)

$I=6\ln{3}-3-\ln{3}+1$

[obsidian monolith]

Enterlessguy

[viscid thistle]

The one on the right had to be the same on the left
One forgot the 1/3)*3

[viscid thistle]

So the right one is the completely correct one??

[drifting oyster]

No that one is wrong

Your teacher simplified the expression ambut forgot to exclude 1/3

[viscid thistle]

That’s what I wanted to say

It’s quite evident what mistake had occurred in the right one

[viscid thistle]

Sorey i am stupid how do you do question m and z

m and z specifically

[hushed sphinx]

What do you get in (m) if you write 1-x³ as (1-x)(1+x+x²) and put on a common denominator?

[turbid skiff]

For z, make x² -5x +6 as (x-a)(x-b)

[viscid thistle]

Oh i mean the small z

Not capital Z

[viscid thistle]

Ohh you do it like that?

Not every limit is L'H

[lethal hedge]

It will turn into L'H prob

Similar principle for z)

[viscid thistle]

Not every limit is just L'H

[turbid skiff]

$(x^2)^2 - (1^2)^2$

[lethal hedge]

You do a little work by expanding and doing common denominator to turn into something that you can do L'H on like on the rest

[obsidian monolith]

╰ 𝕃 𝕌 ℂ 𝕀 𝔽 𝔼 ℝ ╮

[viscid thistle]

x^4 - 1

[turbid skiff]

Yes

[hushed sphinx]

Well, you'd need to put on a common denominator before you can use LHospital, but by then you might just as well proceed by polynomial division anyway.

[turbid skiff]

a² - b² = (a-b)(a+b)

[viscid thistle]

I always forget commen denom

[lethal hedge]

Oh wakt

Nvm

[viscid thistle]

What

[lethal hedge]

I thought you meant something else by L'H

[turbid skiff]

@viscid thistle

[viscid thistle]

What

[viscid thistle]

Oh i know

I tries that on m

Didn't work

[turbid skiff]

Consider a = 1² and b = x²
So a² = 1⁴ and b² = x⁴

[viscid thistle]

What did you think i meant with L'H other than hospriala rule

[turbid skiff]

That's different

a³ - b³ = (a-b) (a² + ab + b²)

[viscid thistle]

Difference of cubes

[lethal hedge]

If all of your limits have polynomial fractions (rational functions) , then you dont even need wasting time doing L'H, say for b) you immediately write 1/2 without doing anything not even need to calculate derivative

Well

The only thing you would be doing in b)

Is just looking at the degrees

Of your polynomials

And at coefficients near highest powers of x

Oh waot

I mightve mistaken for sequence limits oops

Wasnt there one for real functions too?

Or im tripping rn

[hushed sphinx]

That is a useful shortcut when the limit is for x->0.

[turbid skiff]

L'hopital?

Is he even authorized to use it?

[hushed sphinx]

No, the "coefficient with highest power" principle Prelude spoke about.

[viscid thistle]

90% of the limits here are lh

[viscid thistle]

Wha-

[lethal hedge]

Dont you just substitute for such limits

[hushed sphinx]

There's no such thing as a limit "being LH".

[viscid thistle]

Ohhhh when likeee

[viscid thistle]

I know

[lethal hedge]

The ones with infinity work exactly the same as the ones for sequences

[viscid thistle]

Its 5am

I shortcut

[hushed sphinx]

And limits in PREcalculus should not need L'Hospital, since by the nature of the name, derivatives are not even available yet here!

[lethal hedge]

For other cases cant u just sub in for the x?

Or im tripping

[hushed sphinx]

You can, but doing the substitution can easily be so much work that it's not a shortcut anymore.

[lethal hedge]

Ah they make it so you cant plug in that easily breh

[turbid skiff]

I think you can solve them all without it + I don't think you learned l'hobital before university

[hushed sphinx]

(Perhaps I misunderstood you. What I meant was that for a limit x->3 you can substitute x = 3+t, simplify the polynomials and then look for a limit t->0. But that is rarely a shortcut).

[lethal hedge]

Yea ik

I just misremembered which method for which but now ik

[lethal hedge]

So basically these limits is just simplifying to plug in x or substitutions or L'H spam or the common denominator and expanding stuff

Meaning that its kinda boring and cumbersome

[lethal hedge]

so b) is gonna be -4/3, e) will be -1/2 etc

When tending to 0 need look at lowest powers of x it seems

But it wont even apply in b) hm

[lethal hedge]

Shit might not even hold

Well it does seem to work for when deg(P(x)) = deg(Q(x))

But in g the limit is a bit ohio

[hushed sphinx]

well, you need the right rule of thumb for looking at the coefficients of course :-)

Oh, sorry, I was confused.

For a limit x->0 you can look at the terms with lowest degree on each side of the fraction bar and ignore the rest.

[lethal hedge]

Wait

I got an idea

Maybe a form of L'H spam

[hushed sphinx]

So in (b) we have $$\lim_{x\to 0}\frac{x^3-4x}{2x^2+3x} = \lim_{x\to 0}\frac{-4x}{3x}$$

[obsidian monolith]

Troposphere

[lethal hedge]

Where u only differentiate at highest powers of x

[lethal hedge]

U cant rly

Look at g)

Its limit is 0

But u will get 1/1 if u look at lowest powers of x

[hushed sphinx]

If I look at the lowest-degree terms, I get lim x²/x = lim x = 0.

[terse pebble]

that’s not what they’re saying

[hushed sphinx]

It's not enough simply to take the coefficients, you need to have the entire terms.

[terse pebble]

he didn’t say divide coefficients

[lethal hedge]

Oooooh

[cedar sparrow]

anyone can help me understand this, I’m horrible at understanding graphs

[willow skiff]

try doing q5 first

for the inverse function, you swap x and y for each point

so graphically, that means you draw the dotted line y = x then draw the mirror image of g(x) across y = x

after you do that, is this mirror image a function?

then that's your answer to q4

[cedar sparrow]

so it’s not an inverse

thanks

lemme try to figure out 5

[willow skiff]

you mean, g(x) is not invertible

but yes that's correct

hopefully you've found online the condition for a function to be invertible

by doing a graphical example it should hopefully click why

[cedar sparrow]

yeah bc it fails the horizontal line test.

thanks

[cedar sparrow]

@willow skiff Is this a good sketch ?

[willow skiff]

you've mistakenly reflected it across y = -x instead

[cedar sparrow]

oh lemme fix that 😭

[echo sinew]

The dashed line you've drawn is not y = x, by the way

[heavy pewter]

which is the function and which is its inverse because it looks kinda weird

[willow skiff]

yeah I can't make heads or tails of what you just did

[viscid thistle]

#advanced-probability

[cedar sparrow]

can someone check if I solved this correctly 😭😭

these are the class examples

[drifting oyster]

yes

[cedar sparrow]

thanks

[versed nexus]

That looks interesting. Does anyone mind explaining this to me. Am in grade 10 rn.

[fringe ridge]

1. Find D and R of the original function (this is being taught in earlier lectures)
2. swap all x and y in f(x), D and R. (This is mirroring the graph y=f(x) by the line y=x)
3. After step 2 you get "x = expression of y". Solve this equation into "y = expression of x"

all those steps is to make sure one x does not map to multiple f^(-1) (x)

[smoky stirrup]

Change Y TO X, SOLVE FOR Y. THATS HOW YOU FIND THE INVERSE OF F(X) (f of x)

[chrome ether]

(did you forget to turn off your caps lock)

[drifting oyster]

No I think he was just yelling

[smoky stirrup]

skibidi dop dop dop yes yes

[silk yacht]

🙏

<@&286206848099549185>

[drifting oyster]

I don't know if there's a trick here but technically you can just evaluate both sequences to the minimum required n and then take their sum ig 💀

There is some simplification that you can do with the factors of 2 though to make it "easier" to calculate

[drifting oyster]

(You could combine the sums till n=8 , factor by 2^n-8 and that will cancel out an important term since the definition of Bn relies on An which has n+1 at the end and the second sum has n in the numerator. Then evaluate the first sum from n=9 to 10. Should make the sum easier (maybe).)

Also I don't know why the question states for all n greater than equal to 2 and then starts from n=1

👍 😭

Nvm technically the recurrence starts from n=2 and both terms are pre defined on n=1 so its fine...

[fringe ridge]

This is university level problem (except if you are indian or chinese) , you should ask in competition-math or theoretical-cs.

I was reading Concrete Mathematics and found this problem similar to that book's

[drifting oyster]

(Alternatively you can derive that An here has the closed form solution : n(n-1)/2+1)

[hushed sphinx]

Doesn't really look like theoretical-cs to me.

[desert phoenix]

can't you just brute force

[true geode]

It's from the Indian National Astronomy olympiad, for what it's worth

[coarse pasture]

what is conical sections abt

[viscid thistle]

The T is just a geometric series uptill 8

[viscid thistle]

really not that bad actually

@true geode

[willow bear]

several classes of curves which can all be obtained by slicing through an infinite cone with a plane

[viscid thistle]

two cones

[willow bear]

i did not want to say "an infinite biconical surface"

but yes sure, two cones

[coarse pasture]

oh ok

[wintry bane]

hi everyone! is anyone familiar with radical expressions/equations and absolute value? i would like if someone could explain it to me. if anyone has video or article recommendations please feel free to provide them!! Thank you!

[chrome ether]

!da2a

[tender quest]

No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for everyone if you just ask your question! See https://dontasktoask.com/

[copper monolith]

How many 3 of a kind hands contain the card queen of spades in standard deck

[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/

[heavy pewter]

whats a 3 of a kind

[copper monolith]

exactly three cards of the same rank and two cards with uniquely different ranks.

[heavy pewter]

rank?

im not familiar with gambling lingo

[copper monolith]

it's poker

[fickle igloo]

@heavy pewter just google it

[copper monolith]

combinatorics

it's not

[fickle igloo]

wait ill solve it

[heavy pewter]

hmmm 3 cards from 5....

[fickle igloo]

man dont answer if u dont understandthe qs

[heavy pewter]

you might have to expand it a lot

considering there's 52 cards

[fickle igloo]

i solved it

If queen of spades in three of kind:
1 * 12C2 * 3C2 * 13 * 13 = 11286
If queen of spades not in three of kind:
3 * 13C3 * 2 * 13 = 1716
11286+1716 = 13002

Am I right?

oh sorry

the * sign made the text italic

lemme edit

[heavy pewter]

you couldve just picked 4C3 for three queens

oh wait

queen of spades

hmmm i guess it should be correct

but uhm

you didnt factor in that the remaining two have to be different

and could be either spade, diamond, clubs, etc.

maybe...

[fickle igloo]

i did

[silk yacht]

nah bro

its from JEE Mains

[silk yacht]

I am indeed Indian

[true geode]

But I'm referring to where they sourced the question (INA Jan 29 2023)

[silk yacht]

It's JEE Mains 29th January 2023 bro

I gave that exam

💀

INA doesn't stand for that😭

Idk why they wrote INA

but it's definitely JEE Mains

[true geode]

yh found the original question

It doesn't help however that the question posted here was also labelled "115"

[long tapir]

How do you find the ratio of a geometric sequence if not given the first term, nor are you given 2 terms which are next to one another? (EX: N = 3 and N = 4)

[heavy pewter]

uhm, pretty easy

lets say you are given ith term and jth term

then we know

$t_i =ar^{i-1}, t_j = ar^{j-1}$

[obsidian monolith]

parabolicinsanity

[heavy pewter]

then just take their ratios

i.e.

$\frac{t_i}{t_j}=\frac{ar^{i-1}}{ar^{j-1}}$

[obsidian monolith]

parabolicinsanity

[heavy pewter]

$\frac{t_i}{t_j}=r^{i-j}$

then just take the root of i+j

$r=\left(\frac{t_i}{t_j}\right)^{\frac{1}{i-j}}$

@long tapir something like this

ive generalised it to any ith term and jth term but you can do it for i=3, j=4

[long tapir]

Oh shit

I love how you explained that for me

Seriously thanks man

[heavy pewter]

no problem

its fun solving stuff in maths

[long tapir]

Wait

Shouldn’t it be

I-j?

[long tapir]

Because division minuses the exponents

[heavy pewter]

yes!

[long tapir]

But this is r^i+j

Which implies that it was r^i/r^-j

Which we know is not the case

[heavy pewter]

ohh yeah i forgor a little bit of laws of indices

[obsidian monolith]

parabolicinsanity

parabolicinsanity

[heavy pewter]

generalisation is always good

[long tapir]

I wonder

Side question

How did you move the i-j

Out from the r

[heavy pewter]

oh that

[long tapir]

And make it part of the left side

[heavy pewter]

just take the root of i-j

[long tapir]

Im not the best with factoring exponents

[heavy pewter]

raising the whold equation

[long tapir]

ohhhh true

[heavy pewter]

to one over i-j

does that make sense to you?

like we have x²=5y

then we raise the whole equation to a powe of ½ i.e. square root the whole thing

to leave us with x=(5y)½

admittedly thsres plus or minus as well but thats besides the point

[long tapir]

Makes sense makes sense

[long tapir]

Question

Should i be the smaller value

And j the larger value

[vast acorn]

what the heck is the antiderivative of x^4

[vast charm]

can someone help to explain to me how to find derivatives using log and e

[willow skiff]

the antiderivative is the same thing as the (indefinite) integral

because the antiderivative of a function f(x) is a function whose derivative is equal to f(x)

[vast acorn]

right, maybe I dont understand well enough to ask the right question, im trying to solve the indefinite integral of
(5x-7)^4
but im just hung up on reversing the chain rule yknow

[willow skiff]

ohhh okay

so the reverse chain rule only applies to f(ax + b)
check: you have this (where f(x) = x^4) ✅

when you differentiate this you need to multiply by d/dx (5x - 7) = 5

so to undo the differentiation, you need to divide by 5

if you want to further convince yourself, try the normal u-sub method

u = 5x - 7 and du = 5 dx, but you want to replace dx with du

[willow skiff]

the other thing is that you integrate this normally, without changing the inside
so, u^5 / 5, where u = 5x - 7
careful with brackets also

(you can add the +c on the end yourself)

[vast acorn]

ok i think this was enough to send me in the right direction thanks m8

[heavy pewter]

any i and j

doesnt matter

thats how generalisation works

[heavy pewter]

this is #precalculus 😔

[willow skiff]

no problem matey

[willow skiff]

happens all the damn time

[heavy pewter]

but honestly if you really want it

e^x stays the same

[willow skiff]

ehhh your textbook will explain, or org chem tutor on YT will also explain

it's hard without showing a problem and telling us what you don't understand

most probably, if I have to guess, is some mistake when applying the chain rule to ln(f(x)) or e^(f(x))

[heavy pewter]

you could try deriving it yourself using the definition of a derivative

as well

or something

[lethal sand]

can anyone here teach me limits

i dont really understand why some limits dont exsist

and some do

[willow skiff]

do you recall this definition for a limit to exist?
(the top one)

the limits from the left and the right must be equal

[tender vault]

Can anyone tell me what's the correct choice?

[floral sigil]

B

Bc if it’s concave down, the second derivative is negative

So the first derivative is decreasing

Therefore f(2)>f(3)

Derivatives *

[tender vault]

Okay but we don't know the concave is increasing or decrease

Like this

I mean the first derivative won't be the same as it is the tangent

[floral sigil]

You do

The 2nd derivative is the derivative of the derivative

So

If it’s negative

Then the derivative is decreasing

Just as if the derivative is negative, the function is decreasing

So f’(2)>f’(3)

[floral sigil]

What your saying is the derivative could be negative or positive which is true, but it will still be decreasing

[tender vault]

Okay thanks

[marsh cipher]

im bit confused with this, where should I start?

[full yew]

try drawing a graph with any value of m and b of the function

then find the vertices of the triangle

[marsh cipher]

so i just make up m and b and deteremine the area of triangle?

[full yew]

no, but try graphing f(x) with any m and b and seeing the height and the base of the triangle in respect to m and b

[marsh cipher]

so its like 1/2(b)(formula for xintercept) ?

[full yew]

yeah

[marsh cipher]

alr

got -b^2/2m, thanks for giving me eye opener to this question lol

[floral sigil]

when in doubt draw it

[marsh cipher]

Well yeah, I just wasnt sure what question were asking in first place, evotushon reworded question so i understand what it asked for

[vast acorn]

would the antiderivative of e^-0.1t = 1/0.9e^0.9t?

also is there a latex discord bot or something that can make these equations more readable

[river drift]

if you type a message in LaTeX it will automatically be rendered

[vast acorn]

seriously

thank you

[winter comet]

no, it would be 1/(-0.1) e^(-0.1 t) = -10 e^(-0.1 t)

and you could add whatever constant you'd like ofc

[vast acorn]

why did you put -0.1 under 1?

[winter comet]

whats the derivative of e^(ax)

[vast acorn]

ae^(ax)

[winter comet]

so whats the antiderivative of e^(ax)

[vast acorn]

hmmm

[winter comet]

when you differentiate you multiplied by a on the outside

so when you integrate, you would divide by a....right

[vast acorn]

i think that makes sense

[winter comet]

you can also differentiate your result to make sure

[vast acorn]

right but if you differentiated -10 e^(-0.1t) wouldnt you have to multiply it by the coefficient and then subtract the exponent by 1?

omg i think i get it

[winter comet]

no, you would sub if you were differentiating x^a rather than e^(ax)

yea

[winter comet]

by "sub" i meant use the power rule
not sure why i said sub lol i'm not paying attention 💀

oh i meant subtract one lol

[vast acorn]

lol

[winter comet]

XD

[vast acorn]

your good

so you mean that for x^a you would subtract 1, so what would you do for e^(ax) instead?

[winter comet]

use the chain rule, and the fact that the derivative of e^x is e^x

so d/dx (e^(ax)) = d/d(ax) e^(ax) * d/dx(ax) = e^(ax) * a = a e^(ax)

[vast acorn]

ok ok, this helps, do you know of a good resource i can look into

[coarse pasture]

y min value will be -4 at x=3/4 right??

[willow skiff]

y min is indeed -4

but it's not x = 3/4

I recommend you learn where x-coordinate of vertex = -b/(2a) comes from

[heavy pewter]

😔 pre calc but trivially solvable by derivatvies

but

you have to understand what happens in a parabola

hmmm

[coarse pasture]

no i do have some knowledge abt it but first time studying differ..

[coarse pasture]

using double differentiaion right?

[willow skiff]

no

[willow skiff]

you just set the first derivative to zero

because that means the slope equals zero, which all turning points satisfy

slope equals zero could also imply a saddle point however

like what happens with the graph of y = x^3 at x = 0

so the 2nd derivative test does check if a turning point is a maximum or a minimum or neither, but given that the 1st derivative is already zero

[coarse pasture]

yea that

[heavy pewter]

second derivative just tells you where its bent

by it i mean

uhhh

inflection intervals or whatever

[coarse pasture]

yep got it

[true geode]

Given this is precalc tho

And that this is a quadratic

I'd recommend completing the square

[willow skiff]

but yeah it's a bit dumb to apply calculus to a quadratic

the point of calculus is that you can find these things out for functions which are not straight lines nor quadratics

as an exercise it's fine though

[tight berry]

how to explain relations between coefficients in area and volume of, respectively, circle and sphere?

[willow skiff]

https://www.youtube.com/watch?v=GNcFjFmqEc8

this is really good viewing

[tight berry]

thx!

[willow skiff]

https://en.wikipedia.org/wiki/Cavalieri's_principle#Spheres

there's not as neat of an answer for the volume

depends if you already accept the formula for the volume of a cone

[hushed sphinx]

If that has a nice solution at all, the only way to attack the problem is probably to guess (based on experience and trial-and-error) what the answer is, and then try to prove that answer correct.

It must be a typo that is says f: R->R instead of f: N->R.

[willow mesa]

Anyone know a good site to learn precalc

Or is khan academy just the best

[ebon locust]

I have the same question as well

[sturdy scaffold]

Yo

Please solve for XC

[sweet grove]

y?

[hushed sphinx]

The condition doesn't menion values of f other than f(1), f(2), f(3) ...

[prisma oxide]

I need some help on this

[autumn spire]

lets see
P is on the unit circle and it makes an angle theta

so ur looking at this

and x = 2y

can u express x and y in terms of theta?

solution:
||x = cos theta, y = sin theta||
||therefore cos theta = 2 sin theta||
||therefore ½ = tan theta||
||therefore theta = arctan(½) = 0.464||

[weary trellis]

Or equivalently:
||tan(θ) = y/x
θ = arctan(y/x)
x = 2y
θ = arctan(y/2y)
θ = arctan(1/2) ≈ 0.464||

[lethal sand]

doesnt x=rcostheta and y=rsintheta

[hushed sphinx]

Yes, but the problem said "the unit circle", so r=1 here.

[lethal sand]

oh ok

[viscid thistle]

yo

anybody got the surface area and volume of a sphere through calculus?

[winter comet]

#multivariable-calculus , also wdym "anybody got"...are you looking for the derivation?

[heavy pewter]

strange to ask in precalc

[hidden birch]

or use disc/washer method

[viscid thistle]

i tried integrating with cylinders instead of rectangles where the radius is governed by the eq x^2 + y^2 = z^2

[hidden birch]

so you did the shell method?

probably not. idk

[viscid thistle]

i dont know what im doing here

[winter comet]

thats a cone......

[viscid thistle]

Nope

Integrated within the bounds of-r and r

Multiply by 2

Got the volume of a sphere

[winter comet]

using the equation of a cone.....

aight bruh 💀

[viscid thistle]

I never used it

Don’t have to

No wait you have to

[viscid thistle]

Also I don’t think it’s for the cone

[winter comet]

z^2 = x^2 + y^2 is the equation of a cone

[viscid thistle]

In 2d I mean

[winter comet]

not sure how you'd get volume of a sphere outta that

it has three variables so it lives in 3d idk what u mean

[viscid thistle]

I let z be a constant quantity for the radius

It is not a another variable

I know wrong notation

[safe basin]

😭

just say r for radius

[primal crane]

hello, this is calculus right?

[winter comet]

no this is precalculus

calculus is in #calculus

[primal crane]

thanks

[native dagger]

I think im being a bit special, or i may have just forgotten

but why is this true

nvm

i was being special

[willow bear]

what are you using "special" as euphemism for :X

[safe basin]

😭

[fiery crest]

Im really behind in my trig class and apparently I have a quiz Wednesday on material I don't understand and we don't get any notes or formulas

[lethal sand]

whATs the material?

[fiery crest]

uhhhh lemme see

right triange trig? i think

[lethal sand]

like the special right triangles?

30-60-90?

45-45-90?

or like sohcahtoa

[fiery crest]

sohcahtoa

i missed the day he talked about the test so idk exactly whats on it

[lethal sand]

do you have a textbook?

[fiery crest]

i think

[lethal sand]

did he say what sections the test is on?

[fiery crest]

we have a packet we do and he said if you can do the first 2 pages of trig youll be fine

he was annoyingly upset with me for asking

[lethal sand]

can you send me the first 2 pages of the packet?

[fiery crest]

yeah

[vast charm]

I am in pre calc but sorry for the confusion maybe I’m in a different curriculum

[drifting oyster]

What type of abomination is this

lol

[viscid thistle]

This is from pw?

@shell crest

[warped shuttle]

How to find the range of this function

((cosx - 2)(cosx - 3))/((sinx - 4)(cosx - 5))

[willow bear]

$\frac{(\cos(x) - 2)(\cos(x) - 3)}{(\sin(x) - 4)(\cos(x) - 5)}$

[obsidian monolith]

Ann

[willow bear]

this?

@warped shuttle

[warped shuttle]

Yes!

[willow bear]

god ok this looks ugly LMAO

if the bottom left bracket were cos(x)-4 it would have been much easier

what's your progress thus far

[warped shuttle]

I tried calculating the derivative to find the critical points but it would take ages

[solar olive]

i feel liek its just

ugly

[warped shuttle]

Sometimes ugly things need to be solved

[drifting oyster]

Nah just graph it on a calculator lol

[warped shuttle]

That’s not so analytical lol

[drifting oyster]

[warped shuttle]

Any thoughts ?

[drifting oyster]

I would probably just try it's derivative

But then again finding an analytical solution to that is quite hard

[vagrant raptor]

yall do derivatives in precalc?

weird

[viral warren]

can someone help me

[high pelican]

!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/

[lavish sage]

Why not simply make the bottom equal to 0?

[drifting oyster]

wow guys I'm sure this isn't scam

<@&268886789983436800>

Wow epic ping fail

[winter comet]

i'm not sure if you mean ping the moderators or ping everyone but neither one seems to be working

the mods are all asleep, nobody on the nightwatch 💀

[hidden birch]

the guy is gone

[winter comet]

what guy?

[drifting oyster]

I meant my ping 😭

[hidden birch]

the guy who sent scam stuff. He was not in the server when his message was still here. Probably sent it and left or got kicked out without his messages getting removed

[inner crag]

am learning about polar coordinates

any advice when working on it

double integrals

[warped shuttle]

That’s not so precalc

[lethal sand]

it could be

im in precalc and im learning about derivitavtaves

[willow skiff]

Different

[willow skiff]

Usually r is the inside integral and theta is the outside integral, if the question asks for the area of some sector

Also split the integral up into parts, from when the upper and lower functions swap places

Beyond it's just general integration

In particular trig subs and recognising integrals of trig functions

[vital galleon]

can someone help me do part A idk how to get the x coordinates

[karmic karma]

doesnt it cycle

sorry im still in middle high school

like

after pi/2 radians it goes down/up/up again

[inner crag]

yh

thx

[inner crag]

i thought trig identites had something to do with it thats why

i need to think simpler sometime

[rough frigate]

#1021175428326633542

[steep blade]

is help still needed

[grim flame]

The chain rule is a good method for starting calculus

[grim flame]

Do you need help? Or is it too late?

[marsh cipher]

how do i write x^2+(x^2/pi)

1 1/pi x^2?

[winter comet]

I would avoid that, you could write like (pi + 1)/pi * x^2

[marsh cipher]

should i lump up pi and change it all into one fraction

[heavy pewter]

yes it would be better for the eyes

[marsh cipher]

like this better, got accepted as well

[heavy pewter]

i guess that's fine too

[high whale]

Wish I could figure out the steps you did to arrive at that

[heavy pewter]

oh yeah

weird

i didn't look at it but this makes zero sense 😢

[dire ginkgo]

To find C u can use the scale formula prd/4 so 1/10/4 so 1/40

U can mark J as 0, 20

[elfin niche]

tips for getting good at conics in 2 days?

[heavy pewter]

practise

[elfin niche]

woah

Hey so these are my answers, this is the problem and this is the answer key, I personally think I am correct, but where did I go wrong?

[hushed sphinx]

Hard to say where you went wrong when you're not disclosing why you think your answer is correct?

E.g. why do you think 32 would look like C?

[elfin niche]

I think it is d because the eccentricity is greater than one but after looking at another assignment I think that re looking at my notes could help

[hushed sphinx]

What I immediately think for (32) is, "hmm, the denominator is always >1, so r cannot be larger than 5."
So everything that shows a curve escaping to infinity can't be it.

On the other hand, in (33) the denominator vanishes for two different angles, so there should be two asymptotes.

[covert jay]

why is the asnwer d even if the maximum favlue is at t=0.451 ? i also dont understand what roc is neg or pos means

[river drift]

the function is modeling the rate of change of the number of people

[covert jay]

how does that show that the center is at its maximum

[river drift]

at a maximum the function goes from having a positive rate of change (increasing) to having a negative rate of change (decreasing), and is 0 at that point

[covert jay]

ohh ok thanks

[prisma oxide]

Yo gango

If a function has x^2 in the numerator and just x in the denominator

Would that be a hole at x?

[torpid cape]

i think so

since its removed from the denominator

kind of like

(x^2)/x is undefined

because you cant exactly divide by 0

[prisma oxide]

No but like

It would be a hole

With a x left

In the numerator

[torpid cape]

yes

[prisma oxide]

Which is a zero

But can’t be a zero

Yeh

Also I got another question

[torpid cape]

its a hole dont overthink it (i think)

[prisma oxide]

Can we go to dm I have a couple questions before my exam tmrw?

[torpid cape]

i have it tomorrow too im doing trig stuff

just ask it here its easier for me

[prisma oxide]

Alr

[torpid cape]

lemme se

[prisma oxide]

I got 3 others that’s why I wanna go in dms

I’m gonna flood the chat

[torpid cape]

ok sure

[heavy pewter]

!noping

[tender quest]

Please do not ping individual helpers unprompted.

[heavy pewter]

but...

x²/x has a removable discontinuity

at x=0

[prisma oxide]

Can u help w this why is it b?

[heavy pewter]

its a logarithmic graph of base 4

its inverse will be an exponential graph with base 4

[prisma oxide]

How do u know it’s base 4

[heavy pewter]

because

[prisma oxide]

IM CONFUSED

[heavy pewter]

$\log _x{(4)}=1\implies x^1=4$ i.e. $x=4$

[obsidian monolith]

parabolicinsanity

[heavy pewter]

[prisma oxide]

Ok but why wouldn’t 8 work

Why 16

I still dk why it’s log base 4

How do h know?

[heavy pewter]

do you literally not see what i just did?

if it was not base 4

hpw would inputting 4

give you 1

in the graph?

[prisma oxide]

Ok bear w me

So

[heavy pewter]

so

...?

[prisma oxide]

Why do we input 4?

[heavy pewter]

because its in the graph?

(4,1)

thats in the graph

-_-

so you use x=4,y=1 in the function

[prisma oxide]

IM GONAN CRASH

why can’t I use

1,0

[heavy pewter]

you can its just

true for any base

[prisma oxide]

Ok so it’s log base 0

I’m confused

How is it log base 4

[heavy pewter]

$\forall x>1, \log _x{(1)}=0$

hmmm

[obsidian monolith]

parabolicinsanity

[prisma oxide]

If x is in the base

Chat

WHAT IS THAT 🥲🥲

[heavy pewter]

did i not show you

with ebough clarity?

[heavy pewter]

here?

[heavy pewter]

for all x>1, logarithm base x of 1 is 0

i think you should really brush up on

logarithms

and exponentials

[prisma oxide]

Ok

So

Why would 16 work

[heavy pewter]

before pursuing anything any further

[prisma oxide]

Can u show what u did

[heavy pewter]

erm

[prisma oxide]

Why wouldn’t 8 work

[heavy pewter]

because $4^2\neq 8$

[obsidian monolith]

parabolicinsanity

[heavy pewter]

$4^2=16$

[obsidian monolith]

parabolicinsanity

[prisma oxide]

Why did we square 4

Lemme think

[heavy pewter]

[prisma oxide]

HOL ON

ok so base 4

And x is 2

I mean u

Y

So 4^2

=16

Yeh

[heavy pewter]

we squared it because thats what the inverse of logarithm is...

exponential

[prisma oxide]

Ok but I’m confused how the 4 became the x

[heavy pewter]

oh no, thats true for

any x>1

that

it passes (1,0)

i just wanted to illustrate that

[prisma oxide]

So

Any log

Is base 4?

No it’s 10

What

[heavy pewter]

i meant that (1,0) is passed by any base

but

(4,1) is only passed by base 4

[prisma oxide]

Isn’t exponential 0,1

OH YEH ITS THE INVERSE

[prisma oxide]

So the base changes?

WAIT

IT MAKES SO MUCH SENSE

[torpid cape]

its because the exponent is a variable

[prisma oxide]

X HAS TO be 4

Because they cancel out to 1

U COULDVE JUST SAID THAT

JK TY THO

[prisma oxide]

?

[torpid cape]

4^1 is 4

4^2 is 16

[prisma oxide]

Yeh

[torpid cape]

you made the mistake of

4 x 1 = 4

4 x 2 = 8

[prisma oxide]

Im lowk scared for the mcq

[torpid cape]

me too amigo

[prisma oxide]

Like the questions are worded weirdly

[torpid cape]

the graph is exponential 😭

[prisma oxide]

Do u know the transformations in logs?

[torpid cape]

yes

log is the inverse of exp

[prisma oxide]

So like when is it horizontal

Certical

[torpid cape]

huh?

[prisma oxide]

Translation or dilation

[prisma oxide]

Transformations???

[torpid cape]

you worded it weirdly

[prisma oxide]

Yeh

[torpid cape]

horiz translations are inside the parentheses and are "the opposite" of what it says

[prisma oxide]

Yeh

Reciprocal

Nvm I got it

[torpid cape]

dialations make stuff wider and smaller

transformations just move the graph

[prisma oxide]

Translations

Yeh

Alr

[torpid cape]

thats what i mena

[prisma oxide]

We getting that 5

Tmrw

Cya

[torpid cape]

mhm

i have it at 8am tomorrow

[prisma oxide]

Same

[torpid cape]

this question is simply not fair

no calc section

youve gotta do the damn quad formula

in like 4 mins

[heavy pewter]

unless you don't have a calculator you could jsut.... add the value in...

[torpid cape]

y = 0

you cant plug it in

[heavy pewter]

huh

[torpid cape]

you have to zero out

[heavy pewter]

you can just make a variable vro

and add the values

see what outputs zero

[torpid cape]

wait omg

[heavy pewter]

if you have a calc (lazy route)

[torpid cape]

i dindt think of that

[torpid cape]

its on the non calc section

[heavy pewter]

well shii

you have to find it then

convert the cos into a sine

and yeah, quadratic equation solution

should give you 4 answers honestly

not just 3 atmost

[prisma oxide]

Thought it gives 2

[torpid cape]

smart guy

i thought so too

[heavy pewter]

pi periodicity

[torpid cape]

but then i realized that theres 2 for every answer

because its the entire circle

[prisma oxide]

Oh

[heavy pewter]

[prisma oxide]

Well depends on if it’s the cos or sin value

U end up with

[torpid cape]

no its 2 no matter what

[heavy pewter]

you solve with sin

you get two values

[torpid cape]

yeah with sin

[prisma oxide]

Can u not factor by grouping

?

[torpid cape]

because that cos is a trig identity

goes out to 1 - sin^2 x

[torpid cape]

its not a whole number

need to quad fomula without a calc

[prisma oxide]

U can still do u that

[torpid cape]

no xbox

[viscid shale]

Can someone exlpain why this isn't A?

[uncut mantle]

is anyone else taking ap precalc test tomorrow

[warm cove]

AP exam tomorrow for this course, I wished I'd actually studied days prior because my stress is through the roof right now

[uncut mantle]

i feel pretty confident, though polar might be the death of me

also tranformations are kinda not my best

[mystic sluice]

when the function is graphed you can see that the rate changes faster (meaning decreasing to increasing) so its not A. Rate of change means how fast the function’s output is changing with respect to the input not just whether it’s going up or down.

[mystic sluice]

yes

[warm cove]

my confidence is actually fading... I don't know what to do at this point, the study material im using is just not helping me 😭

[mystic sluice]

is it cause you already know it or you dont understand it

[warm cove]

I dont understand

[mystic sluice]

show me a problem

also if this makes you feel any better https://www.albert.io/blog/ap-precalculus-score-calculator/

[uncut mantle]

do i need to know limasons and all that

or are circles and petals good

[mystic sluice]

all that

organic chemistry tutor has quick ways to graph limacons

[uncut mantle]

thanks

[torpid cape]

Outdated

We aren’t doing unit 4

And the exam is much harder in terms of criteria

[mystic sluice]

k

[mystic sluice]

yeah unit 4 has been out of question

[mystic sluice]

not really

only very slightly based on what i see on college board

[uncut mantle]

are there any resources with practice ap questions

[dire ginkgo]

WISH US LUCK

MANIFEST THE 5s