precalculus

[polar phoenix]

the only numbers im given is t5 and t8 its asking me to solve for the common difference and find t1 but i think my solution is wrong

so common diff =3

yeah ts is def wrong, i just dont know where to start this off

[rough leaf]

yeah so if common diff is 3

and as the t values increase the numbers increase

if you wanna go down

js subtract 3

like t4 wpuld be like 13

[round geyser]

What’s this about

Is it arithmetic sequence

[rough leaf]

?

[round geyser]

You notice

They’re inverses

And it can relate addition to multiplication

[polar phoenix]

yes

wait so i am right but wth

the solution my teacher used looked so complicated

can someone tell me if my way of doing it is correct or was i lucky? i still cant understand how she solved it compared to the way i did it

[unreal plover]

can someone explain why at 0.5 where the red graph's slope is 0 the blue graph (red's derivative) isn't 0

[placid fiber]

,w d(sqrt(3-x)+sqrt(2+x))/dx at x=0.5

[obsidian monolith]

Results provided by WolframAlpha

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[placid fiber]

The parentheses

[unreal plover]

yeah i figured that out thanks anyways

[long ruin]

How much math i need to know to start studying precalculus

Algebra 2 and geometry/trigonometry is enough??

[karmic brook]

it does but it is also correct, and the prefered way of doing so, ecspecially when showing working. i would look at trying to learn this way as it is easier and more consistent

[viscid thistle]

Trigonometry and functions are enough

[round geyser]

Is this easy

What’s Type I, II, and III about

[fervent elk]

I dn why, but logs were like one of my first lessons this school year and don't remeber a single thing

[raw hill]

<@&268886789983436800>

[torpid rain]

I think I know what logs are

Doesn't it have to do with stuff like 10^n?

So log(70) would be 7, right?

[golden abyss]

no... log(70) is some irrational number

[torpid rain]

Oh

[golden abyss]

log(100) is 2 because 100=10^2

[torpid rain]

Ohhhhhhhhh so it's square roots but instead of finding the number you're finding the exponent

[golden abyss]

yeah

[torpid rain]

Ok I get it

[round geyser]

Type I is like an onion one layer and one layer

[round geyser]

Log in a log

[round geyser]

Why does this won’t work in Rational root theorem

[echo sinew]

Because that polynomial has no rational roots 🤷‍♂️

The rational root theorem always works, otherwise it wouldn't be called theorem (as long as its hypotheses are satisfied, of course)

[uncut mulch]

rational root theorem helps identify candidates for rational roots if they exist

the only candidates are +-1
and since neither of those are roots,
you can conclude that polynomial doesn't have any rational roots

[median kite]

Hey guys

I wanna ask something

[tender quest]

Asking the actual question right away is more likely to get responses.

Asking "Can I ask...?" or "Does anyone know about...?" doesn't give people enough information to decide whether they can help, and answering can feel like a promise to help with the actual question, which they might find themselves unable to.

[median kite]

Suppose two points on a graph f(x) and f(x+h) then the slope of the line formed by them is given by f(x+h) - f(x)/(x+h) - x which gives f(x+h) - f(x)/h, when we look at this formula of slope of secant line of points f(x+h) and f(x) is literally the formula for average change, when apply limits like, limit h->0 f(x+h) - f(x) / h makes points infinitely closer because the gap between them starts shrinking and we often solve questions and after cancelling h in denominator, we substitute h=0 as a shortcut now when h is literally 0 the points coincided on the graph as difference between them became zero. How does this gives instantaneous change and why is it said that instaneous change is limit of secant line as x tends to 0 which is tangent, when tangent needs two points because it's a line and there is literally one point now because both coincided.

Am I assuming smth wrong

Is tangent not line?

[stiff sorrel]

Tangent does need two points, yes, instantaneous change, however doesn't reduce these two points to a single one but rather gets them arbirtarily close.

Otherwise we wouldn't get a tangent line

[round geyser]

Correct

I assume there will be 4 complete roots

[echo sinew]

Yeah, four of the 5th roots of unity

The ones that aren't 1

[round geyser]

If I graph it

It never touches the y-axis

[raw hill]

mhm

that tells you the roots aren't real

[raw hill]

which is consistent with this ^

[valid topaz]

z^5=1

[round geyser]

Look

It has to be 4 imaginary solutions

And plus 2 positive and 2 negative

a+/-bi

[raw hill]

complex/non-real, not imaginary

imaginary numbers have a real part of zero

and the roots are in conjugate pairs if that's your concern

$e^{2\pi i/5}$ and $e^{8\pi i/5}$ are complex conjugates, as are $e^{4\pi i/5}$ and $e^{6\pi i/5}$.

[obsidian monolith]

Civil Service Pigeon

[round geyser]

What the advanced way to solve it

[valid topaz]

very mathy

[rough leaf]

finallu i get this integral stuff

u sub and ibp fibally make some sense

[median kite]

Arbitrarily small but we literally substitute 0 after cancelling h in denominator

That's just makes them one single point

[golden abyss]

why is the limit 8.75 as n approaches infinity

[valid topaz]

umm, it's not 8.75

it's 7+π^2/6

comes from basel sum

[golden abyss]

??? answer key says 8.75?

huh the basel sum makes sense, ig the answer key is wrong

[valid topaz]

does it have a solution?

cause it's wrong

[golden abyss]

no, but the answer key does have typos sometimes

[valid topaz]

ah, i see

[golden abyss]

I didn't know about the basel sum so i couldn't solve it

[valid topaz]

oh

well now you know, ig

[willow bear]

you should not use "imaginary" where you mean "complex but not real"

if you say "imaginary number" people will think you mean numbers on the y axis in the complex plane

[round geyser]

Okay

[feral inlet]

dmamn

[hot fog]

Found an absolute gem today 🥹

[echo sinew]

1?

That's awesome though

[valid topaz]

dunno if dis should be on precalc...

[echo sinew]

Yeah it shouldn't definitely 😁

[valid topaz]

cool question tho

[echo sinew]

Indeed

It's pretty "original", compared to many others I have seen around

[valid topaz]

also, the answer seems to be -1

[echo sinew]

Mmh there's an abs value though

[valid topaz]

😭

yeah it's 1

[echo sinew]

Fun fact: just two seconds before sending this I also thought -1. But then I realised lol

[raw bobcat]

wtf is that?

[valid topaz]

limit

[raw bobcat]

i know but why integral here...so we need to solve the integral,right?

[valid topaz]

it's a gaussian

if u wanna solve it, go ahead

but it's answer is well-known

[ripe scarab]

is there a calculus section

[valid topaz]

channel? cuz yes #calculus

[charred rapids]

Cool, but this would fit better in #calculus and not precalculus

[round geyser]

U know today I found a silly questions on my pre-calc hw

25^(2x+3)=80^(x-5) with logs

[spice crown]

idk what is the final answer but it is a non negative real number

[valid topaz]

||1||

[spice crown]

well i wasnt wrong was i

[valid topaz]

i never said u were wrong bruh

i just stated my answer

[spice crown]

well i never said you said i am wrong

[valid topaz]

the english 😭

dumb it down 4 me

plz

[spice crown]

my english is prob wrong bro

[valid topaz]

mhm

[ripe scarab]

thx

[round geyser]

How would u solve this 25^(2x+3)=80^(x-5) with logs

[raw hill]

\begin{align*}
\log \left(25^{x+3} \right) &= \log \left(80^{x-5} \right) \
(x+3) \log(25) &= (x-5)\log(80)
\end{align*}

[obsidian monolith]

Civil Service Pigeon

[raw hill]

Note how it’s just a linear equation now

[round geyser]

Distribute

2x log (25) + 3 log (25) =x log (80) - 5 log (80)

2x log (25) - x log (80) = -(5 log(80) + 3 log (25))

[raw hill]

Check your signs on the right

[round geyser]

I factored out

[raw hill]

Yeah it’s good now

[round geyser]

x ( 2 log (25) - log (80)) =-(5 log (80) + 3 log (25))

x = (-(5 log (80) + 3 log (25)))/(2 log (25) - log(80))≈-15.356

[round geyser]

Btw on the last unit I got 101

[raw hill]

[round geyser]

It’s just graphing log functions, solving log and exponential functions

Log’s domain will be inside like log(x-5) then D: x>5

Exponential’s Range will be outside of a functions like 2^x + 1 which R: y> 1

These three units are the one I got 100 on

[raw hill]

nice

[snow dawn]

This problem is ungodly.

[uncut mulch]

not that bad

[rough iron]

anyone got resources to self study for ap precalc exam

[snow dawn]

I would recommend Delta Math, as is provides some great practice problems for almost every unit. And for FRQ's just work through the ones that AP provides. Also Organic Chemistry tutor makes many great videos over almost every aspect of AP Precalc.

[rough iron]

is there any way you would recommend pacing the units and such

also it says i need a class code

[snow dawn]

How long do you have?

[snow dawn]

That's for schools, I believe there are also options for independent learning

[rough iron]

till may 12

thats when the exam is taken

[snow dawn]

k, so thats about 100 days

I recommend spending about a month on Polynomials and rationalizing functions. If you are really good with fundamental algebra then it could be even less. Then Exponential and logarithmic should not take you that long, so ,about 2-3 weeks. Use most of your time on Trigonometric and Polar functions. Leave about 2-3 weeks before the exam for review. Also make sure to take some mock exams in that time, and practice writing FRQ's between each unit.

Actually AP Precalc is a pretty easy class, if you understand the material then you should do well

[rough iron]

as it stands im in alg 2, so I should have a good foundation for the class rn, yeah?

[snow dawn]

That is literally perfect

That means you should already have most of unit 1 covered

Then spend extra time on Log's exponentials and especially trig

[rough iron]

I already have some fundemental practice with logarithims and trig so i think i should be set if I basically review the curriculum

ty

[snow dawn]

Nice, no problem, and good luck!

[charred rapids]

Really random but:

A common mistake is to think that
[
\sqrt{x^2} \neq x \quad \text{(in general)}.
]
What is actually correct is that
[
\sqrt{x^2} = |x|.
]

[obsidian monolith]

Rodro

[willow bear]

bad phrasing mate

it is a mistake to put an equals sign between sqrt(x^2) and x

[charred rapids]

True

[spice crown]

nice pfp buddy

but mine is better

cuz i am true and you are 0

[viscid thistle]

this is no precalc

but what have you tried

Integration by parts should be it

[round geyser]

That’s what I got

And I used AC method

[fringe phoenix]

What am i seeing

[round geyser]

I want to proof that 3/4 is not a solution

Bc I remembered I got 4 and 3/4

[fringe phoenix]

Buddy, you cant just take functions common on both side and cancel them, that's a wrong presentation

[dusky comet]

just sub it inside and get it

[round geyser]

Both logs can cancel

[dusky comet]

i agree, gotta take anti logs

[fringe phoenix]

Its not a correct presentation tho

You cant just cancel out functions

[dusky comet]

it might mess up when youre doing inequalites

[round geyser]

That’s what my notes said

[dusky comet]

cause when you have increasing and decreasing functions the inequalties can flip

[fringe phoenix]

Um x is 4 or 3/4, just put it in to check whether it satisfies the domain or not

[round geyser]

Look

Drop logs and set arguments equal and solve

[fringe phoenix]

Thats a short cut tbh

[dusky comet]

yea

[fringe phoenix]

Its not the proper method

[round geyser]

How would you do it

[dusky comet]

its still gonna be very wrong when inequalties enter

[round geyser]

I’m curious

[dusky comet]

you take antilog, like taking both side in power of the base

[round geyser]

Then the ans key said it’s 4

[dusky comet]

and then with prop of log

[fringe phoenix]

Yeah because it satisfies the domain of log

Not 3/4

[dusky comet]

you simplify

[round geyser]

(x-4)(4x-3)=0

That’s the factor

[fringe phoenix]

Buddy

This a quadratic equation, you just solve for x

But

[round geyser]

of 4x^2-19x+12

[fringe phoenix]

Initially you had log, so you have to put values and check if it satisfies the domain

Do you know about domain?

[round geyser]

ik

I’m solving and we don’t check domain

[fringe phoenix]

Yeah so just check it after putting both the values of x

[round geyser]

Cuz that’s last unit

[fringe phoenix]

What

[round geyser]

Graphing them

Graphing log functions

Now I’m solving for logs

[fringe phoenix]

Oh yeah thats also one thing

[echo sinew]

<@&268886789983436800>

[round geyser]

Huh

[round geyser]

Can we factor this?

[echo sinew]

,w factor x⁵-x⁴-1

[obsidian monolith]

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[echo sinew]

Seems so, but honestly I don't know how one would come up with that

Maybe something like "add and subtract" some quantity

[round geyser]

,w solve x^2-x+1=0

[obsidian monolith]

Results provided by WolframAlpha

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Upgrade to WolframAlpha Pro!

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[round geyser]

,w solve x^3-x+1=0

[obsidian monolith]

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[round geyser]

Here is the five solutions

[obsidian monolith]

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[edgy aspen]

How come integral 1/e^x =-e^-x

[raw hill]

https://math.stackexchange.com/questions/474993/how-to-solve-the-integral-%24\int-\frac{1}{e^x}\%2Cdx%24-step-by-step%3F

Also this would belong in #calculus

But try to look up the question first

[fringe phoenix]

Its e^-x
Not -e^-x

Laws of exponents duh

[fringe phoenix]

This bot put the cubic formula for finding out x 💀

[pale anchor]

how did you guys stop thinking degrees and start thinking in Radians

[west ice]

Remember special angles in radian

And where they are in unit circle

Tbh I don't think people can visualize where the angle is in radian a good as in degree

If I ask where is 1.246 radian on the circle it would be pretty hard to visualize quickly

[willow bear]

you can eyeball it though

pi/2 = 1.57

[willow bear]

i wish radian scale protractors were more popular tbh

[sweet coral]

what r some shortcuts for solving bayes theorem related ques in high school

[echo sinew]

Mmh I'd say identify properly all the events that come into play

[rough leaf]

i js did enough problems

[charred rapids]

Force yourself

Try to get out of your confort zone

And it will just happen

[warm halo]

Comfort zone...it's just so comfortable mann

It's like stepping on fire to cross the bridge

[valid topaz]

forced to 😢

degrees scare me now

[tame agate]

XD

[meager light]

just through using the unit circle

[nimble wolf]

ikr? lol

[nimble wolf]

Unit circle....which is the most useful part of trigonometry.

[echo sinew]

You don't have to stop thinking in degrees though
You just need to get really used a lot to calculations with radians, and that can be done by a loooot of practice.
But the most important thing that you must never forget is the proportion between rad and deg, i.e. π = 180°, or 2π = 360°, if you prefer

[valid topaz]

π = 180 is basically circumference to degrees

in unit circle, ofc

[valid topaz]

#calculus

[opaque rover]

Okay

[inland current]

yes but isnt this something the calculater was inventor for

calcaljustice

[round geyser]

This one is easy

Can we take distance formula

[charred rapids]

Have you taken derivatives?

[round geyser]

No

I thought of taking x^2 as y2

And 4 as y1

[charred rapids]

Ok but how are you going to get the m (slope) from the tangent line equation?

[round geyser]

I haven’t learned it

[charred rapids]

But does eqn mean equation?

Or what is the exercise asking?

[viscid thistle]

how are you supposed do without derivative

[charred rapids]

Yeah that was also what I was wondering

[round geyser]

Distance

Then we plug these back in

[charred rapids]

But does the exercise ask the equation of the tangent line?

[round geyser]

You know I did closet coordinate

Get the minimums x value and square it

[viscid thistle]

hum..

somehow if you get slope of tangent at that point

[charred rapids]

I dont think your supposed to know how to do that exercise @round geyser , even if you think so

[round geyser]

With Desmos

As part of pre-calc word problems

[charred rapids]

Oh but are you supposed to have hekp from desmos?

Help*

[round geyser]

It didn’t say it

But I can

U know that’s should be in calculus

[round geyser]

Can I look

And make line equation

Take (0,0) and (2,4)

[charred rapids]

No , thats incorrect

The tangent line doesnt pass through the origin

And even if it did, it would have been a complete coincidence

As I said, youre not supposed to know how to do this exercise

[round geyser]

I literally misread it as

Closest coordinate

[icy verge]

https://vt.tiktok.com/ZSap8HxpT/

[soft harbor]

i am new to calculus , so how do i calculate the lim h(x) ?

[nimble wolf]

this will be a good problem in #calculus

[soft harbor]

oh ok

[round geyser]

Hey

[echo sinew]

Teacher, leave us kids alone

[amber valve]

When you approach x from positive to 0, h(x) approaches -5. When you approach x from negative to 0, h(x) approaches 5. Since the graph excludes h(0)=-5 (idk how to write not equal to), a reasonable estimate for the limit of x => 0 for h(x) is 5

[celest mortar]

thats calc buddy

[dusk vortex]

I need help with pre calc 11

[raw hill]

<@&268886789983436800> ^ also check #help-10 as well

[dusk vortex]

Can u help gng

[marble quest]

dont offer payment for help

[dusk vortex]

Why bro I need help gng

It’s bad

😭😭😭

[marble quest]

people will help you for free within the server if youre patient

[dusk vortex]

Okay bet

Can u help

Me gng

[marble quest]

no im dumb as shit

wait for someone capable

also you already have a help channel where people are helping you

[dusk vortex]

Wym

[marble quest]

are you not being helped in https://discord.com/channels/268882317391429632/903430332324405288

[dusk vortex]

Yes

Not being helped

[marble quest]

you havent even asked a question

[dusk vortex]

Bro

Look

That shit

[dusk vortex]

Yo

[ionic temple]

if u mean #6
its D

square the binomial (x-8)
when u do that u will put it in the form (a^2)+2ab+(b^2)

that'll give u (x^2)-16x+64

multiply ((x^2)-16x+64) by 8

when u multiply everything out, u get 8(x^2)-128x+512

lastly u subtract 3, and ur left with ans D

[frosty folio]

Quick question if anyone's on

isn't really a math problem more of like a learning problem

[hollow widget]

Just ask the question

[long bane]

guys idk how to graph multiplicities, cross, and tangent(or bounces) like it looks simple but im just rlly confused how to graph it

[round geyser]

Tell me what’s the parent graph

That you are in

[long bane]

Uhhhh can I give u an example ??😭 its like polynomial -2x^7+12x^6-18x^5
I just don’t know how to graph multiplicities, the tangent, and cross things

[deft spade]

Does anyone know how to graph the first function

[copper ledge]

Squaring the function gives us a parabola but also notice that g(x)=2√(1-x) so g(x) must be always postive so remove the portion below the x axis

[deft spade]

Wait how come it’s always positive

[copper ledge]

Cuz sqrt always returns a positive value right?

[deft spade]

OHHH RIGHTT righty TY!!!!

[round geyser]

Can you factor

[round geyser]

the lowest power is 5 so we can take it out

Here you see

Your degree is odd

And LC is Negative

[round geyser]

Take this

[eager meteor]

prove binomial theorem pls

cudnt understand any of the stuff on the web

thx!

i will also make a post there

[lean rapids]

you may as well just use desmos
desmos is soooo time-saving

👌

[carmine orbit]

through what permutation and combination?

[copper ledge]

Stop tagging my answer please

[carmine orbit]

https://youtu.be/t7qXYdql--o

[rough leaf]

ts sucks

you don’t need to know it

[round geyser]

Use pascal triangle

[rough leaf]

exactly

[lean rapids]

binomial theorem is pretty much pascal but in words
I've only ever used it in a practice question and proof of the derivative power rule

[still brook]

help

[fallow roost]

What is precalculus?

[raw hill]

As per the channel description ^

@fallow roost

[sour tinsel]

Quick question - if we're converting polar functions to rectangular coordinates, what is the proper or best way to convert $r=\theta$ to $f(x,y)$? Just naively going for $\pm(x^2+y^2)^{1/2}=\tan^{-1}\bigg(\frac{y}{x}\bigg)$ seems to have domain and range issues. Is there a way to get more than just a small portion of the solution?

[obsidian monolith]

Concomitant

[raw hill]

Did you look it up? If so, did you find this? https://math.stackexchange.com/questions/1138696/conversion-from-polar-to-rectangular

[sour tinsel]

Ah thank you - I had complely forgotten about the atan2 "function." Cheers!

[fringe phoenix]

Sin t = -1/2
Is this correct?

[sour tinsel]

Why the negative?

[fringe phoenix]

Um in that question, t is in 3rd quadrant and in unit circle sin t will be the y coordinate

Y coordinate is negative in 3rd quad so yeah, i mean this is what i thought

[sour tinsel]

To me, it looks like t is indicating the blue curve as the angle, which wraps clockwise to yield the point P(x,y) in the first quadrant. I don't think t is in the third quadrant, but I might be mistaken. It's not entirely clear. If it is what I think it is, then sin(t)=1/2, positive, and P is in Q1.

[fringe phoenix]

It says P is in unit circle, and in the fig P is on the blue curve, i guess t is also on that blue curve and is a point on unit circle

[carmine orbit]

pascal triangle is not the formal proof

its either induction or combinatorics

[round geyser]

Ok

[lean rapids]

calculus 101

crash course

[knotty dove]

I'd recommend always looking for patterns. It's moreso intuition but it's good to think "does this look familiar" "what should be / not be expected" and things like that

[stoic sleet]

does anyone have suggestions for better understanding/memorizing the cross product formula?

[hollow widget]

Well what is the cross product meant for

[stoic sleet]

to find the area of the parallelogram formed by two vectors right?

[round geyser]

Guys what’s A=Pe^kt

[echo sinew]

Some letters smashed together

Jokes apart, it's an exponential function

[round geyser]

Yes

[round geyser]

I’m doing that

How can we relate to example “The radioactive gas radon has a half-life of 3.8 days. Haw long will it take 1 gram to decay to .2 grams?”

[echo sinew]

Well, k = -ln(2)/3.8

Now, if I let T be the requested time, we have the following:

So 0.2 = 1•e^(kT) => kT = ln(0.2)
=> T = ln(0.2)/k

[round geyser]

Hint

Use two equations

One for k and one for t

[echo sinew]

Huh? Who is the OP? 🤔

I'm confused

[mental badger]

uh...i think the question belongs here? anw, in the resolution, sorry its in portuguese but we can just take the 4 to the other side? even if theres limits there?

[raw hill]

Limits fall under #calculus, not #precalculus. But note that the limit is just a number and you still just have an equation, so the usual rules of equations still apply.

[mental badger]

oh i see, i saw this section is pre-uni thats why i put it here 😭

[round geyser]

Can we covert moths to years

[echo sinew]

Sure

1 year = 12 months

[round geyser]

Someone said years to months is easier

[hollow widget]

Because it tripled every 9 months so converting years to months will help you see how many times it will triple

[round geyser]

Ok

First equation is for the k

[still brook]

can someone help with sections 3 & 5?

[carmine orbit]

limits should be precalculus no?

[raw hill]

limits are sometimes covered in precalculus as an intro to calculus (since limits are the "first part" of calculus)

[carmine orbit]

yeah we were taught in precalc that why i asked

[still brook]

help pls

[carmine orbit]

i mean are you aware of which function is positive in which quadrant

[still brook]

yes

[carmine orbit]

so where is the problem

[still brook]

i dont understand what csc 0 means

like

[carmine orbit]

cosectheta

1/sintheta

[still brook]

how does that help

[carmine orbit]

so cosectheta is positive in 0<theta<pi/2 since all functions are positive in that range

[still brook]

so everything is positive except b

bc the negative sign is in front of tan

[carmine orbit]

ill brb

[carmine orbit]

yes

now check for the second condition

[fluid sorrel]

Show ur work

[lapis hound]

if u simplify Q F) from section 3 it becomes 1

[old wren]

https://discord.com/channels/1468598276461957122/1468948226638876909

follow thí link pls

it has a few questions

[echo sinew]

???

[old wren]

???

[warped shuttle]

any ideas ?

lim x->(-2) (f^(-1)(x)) is basically
f(x) = -2

[vocal harbor]

This question is best when you send it in #calculus

[opal gust]

let g(x) = 1 / y(x) = 1 / (1 / f(x)) = f(x), so to obtain f(x) you can just reciprocate all values on that graph

then you can reflect g(x) over y = x to find f inverse

or you can find the value of x that approaches y(x) when it reaches -1/2

[deft tangle]

guys i need help with simplifying trig identities

the problem is (cscx-secx)(cosx+sinx) and idk how to factor it cuz we had a sub today and they didn’t teach this

[willow bear]

you're trying to factor that which is already factorized?

do you have like, the exact problem statement on hand? cause right now it's not clear what you're trying to do at all

[deft tangle]

oh so the question is to simplify as much as possible

[willow bear]

ok, and have you made progress with that simplification so far?

[deft tangle]

because like yk how csc can be rewritten into 1/sin

so far i have (1/sinx - 1/cosx)(cosx + sinx) and idk how to move in

[willow bear]

simplify the subtraction of fractions

at this point it will be good if you can show some paper

[deft tangle]

do i do it like FOIL method?

[willow bear]

no, not what im asking you to do.

simplify $\frac{1}{\sin(x)} - \frac{1}{\cos(x)}$ into one fraction. and as for the other $(\cos(x)+\sin(x))$ bracket, i want you to \textbf{not touch that} for the time being.

[obsidian monolith]

Ann

[willow bear]

understood?

[deft tangle]

oh ok

[willow bear]

also will say once again i will want to see your work on paper.

[deft tangle]

wait i think i got it

[deft tangle]

i got 2cot(2x) is that correct or am i completely off

[willow bear]

,w simplify (1/sin(x) - 1/cos(x))(cos(x) + sin(x)) - 2 cot(2x)

[obsidian monolith]

Results provided by WolframAlpha

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

Requested by ann_dec

[willow bear]

correct

[deft tangle]

okay, tysm!

[valid topaz]

what is that format 😭

and the emojis

[woven thicket]

Chatgpt one

[valid topaz]

i know, the emojis gave it away

no one would do that

so i shall ping moderators

<@&268886789983436800>

[drowsy vessel]

Using chatGPT to 'help' people

[tired pilot]

If they wanted chatgpt slop they could just use it themself

[round geyser]

Hey

Ann

@woeful parcel can you do 2x^3+3x^2-3x-2=0 by rational root theorem and P/Q?

[winter comet]

<@&268886789983436800>

[chilly magnet]

Use the cubic formula

[round geyser]

Show us

[chilly magnet]

I don’t know the cubic formula. Its really long. One moment

[round geyser]

Okat

[winter comet]

Do not shoot a rabbit with a cannon 😭

[chilly magnet]

Wdym

[valid topaz]

he says ur solution is very well thought out and cardano's method is the best : )

[chilly magnet]

Ok well they both work

[valid topaz]

cardano is better

[chilly magnet]

I didnt know cardano til 5 minutes ago

[chilly magnet]

Heres the cubic formula. Feel free to try it urself. But im not gonna spend 5 years on this

[round geyser]

So that’s why we should use rational root theorem

[chilly magnet]

Yea

[round geyser]

Good as I learned the shortcut

P/Q

[chilly magnet]

Even quadratic sucks

[river drift]

rational root theorem doesn't work on every cubic whereas cubic formula does

ofc you wouldn't want to start with cubic formula when other methods would do

[round geyser]

Great

[chilly magnet]

Its just too long. Thats why nobody does cubic equations

But quartic formula is even worse

[round geyser]

Yep

[round geyser]

How about Complex zero theorem

[chilly magnet]

Whats that

[valid topaz]

no quintic and above 😢

galois

[chilly magnet]

Yeah. Its impossible.

[chilly magnet]

There are some cases where it can be solved but it’s been proven that there is no formula that can solve every quintic equation. Only some can. Actually the general quintic can be solved with elliptic modular functions.

[chilly magnet]

No idea what that is lol

[round geyser]

Here

[chilly magnet]

Hey

[round geyser]

That’s the theorem I want to show u

[sonic loom]

how to get access to the calculus channel

#calculus

[raw hill]

You can probably just follow the channel in here, no?

[fiery trellis]

can any1 share how to learn calculus urself

like a proper roadmap or smth coz khan academy thing is pretty confusing they got tons of diff playlists

[jovial thorn]

read a calculus book

[dreamy echo]

You can start off with essence of calculus by 3Blue1Brown... its not too detailed but good for a start

Then Any good theory book... Any good author works tbh

I personally learnt calculus through JEE exposure

[plush orchid]

Can anyone help with trig and complex numbers? I missed the lecture cause i was sick!

[willow bear]

$\cos(z) = \frac{e^{iz} + e^{-iz}}{2}$

[obsidian monolith]

Ann

[willow bear]

also $e^{i \theta} = \cos(\theta) + i \sin(\theta)$ (works for $\theta \in \bC$ generally, but is of interest for $\theta \in \bR$)

[obsidian monolith]

Ann

[plush orchid]

Ah i forgot about old Euler, thank you

I found one for hyperbolics, although after working it out I’m left with an unsimplified answer

My homework says it’s ‘(5*j)/4’

[raw hill]

Your first line shouldn't have an $i$ in the denominator.
\begin{align*}
\sinh z &= - i \sin iz \
&= -i \left(\frac{e^{i^2 z}-e^{-i^2 z}}{2i} \right) \
&= -i \frac{e^{-z}-e^{z}}{2i} \
&= -\frac{e^{-z}-e^z}{2} \
&= \frac{e^z-e^{-z}}{2}
\end{align*}

[obsidian monolith]

Civil Service Pigeon

[obsidian monolith]

Civil Service Pigeon

[plush orchid]

AH thank you :0

[raw hill]

[plush orchid]

Blessed by the pigeons

[valid topaz]

hiii csp

[plush orchid]

Wld the negative not also effect the 'i*pi/2'

[raw hill]

typo, mb

[plush orchid]

cool, i got a final answer although I just cant seem to simplify it

[median spear]

yes u can

[plush orchid]

motivation 😁

[median spear]

Whatt

u want to simplify this;

[median spear]

..

[plush orchid]

its okay, trying it again

[median spear]

$e^{-ln2}$

[obsidian monolith]

ʟօӄɨ

[plush orchid]

thats just 2

-2 even

[median spear]

Can be written as 1/2

[median spear]

Nahh

[plush orchid]

yeah 1/2 😭

I think I’m doing something dumb idk

[median spear]

What's the question

cuz sryy but ur handwriting is not readable

[plush orchid]

Wait no not that

Part B

[raw hill]

I have no idea what I'm reading here.
\begin{align*}
\frac{2e^{i \pi/2}-\frac{1}{2} e^{-i \pi/2}}{2} &= e^{i \pi/2}-\frac{1}{4}e^{-i \pi/2}
\end{align*}

Converting to trigonometric form, we have that:
\begin{align*}
e^{i \pi/2}-\frac{1}{4}e^{-i \pi/2} &= \left[\cos \frac{\pi}{2}+i \sin \frac{\pi}{2} \right]-\frac{1}{4} \left[\cos \left(-\frac{\pi}{2} \right)+i \sin \left(-\frac{\pi}{2} \right) \right]
\end{align*}

[obsidian monolith]

Civil Service Pigeon

[plush orchid]

The final answer is 5j/4

So confused aboout it

[raw hill]

read what I said above

[ionic temple]

woah thats cool

[plush orchid]

Still not sure how to get to the final answer :/

I dont think i wrote this down

[ionic temple]

if i remember correctly
when u do a taylor series for e^i*pi u get the terms for the taylor expansions of cos and isin

so u can turn both exponentials into trig functions, plugging in pi/2 and -pi/2 where x would go

then u just evaluate the trig functions themselves

[still brook]

im crine how is ts wrong

[low wasp]

@everyone i need a discord server for physics so bad

or chemistry

[river drift]

#old-network

[low wasp]

thanks !!

bruh i have to wait 24h to get verified

[marsh lantern]

Whole number I assume

So 45 degrees

[still brook]

oh

[marsh lantern]

I always assume when they say that, go for nearest integer

Unless it specifies the place

[lost cliff]

yoo

do you know

percents

[carmine orbit]

60x1.414 is approx 85

[still brook]

hey maybe respond a little earlier

but A for effort

[carmine orbit]

🙏

[worthy kindle]

Euler's formula

[plush orchid]

Ah right I will give this a try, thank you :p

[lost pike]

half

[glacial flume]

Dont u just do pythagoras then sine rule

[rough leaf]

wait what happens to r if it was like r^5

cuz using de moivres thm ik whta happens

but not when using eulers formula

[red rose]

@rough leaf

[worthy kindle]

r is supposed to be a constant representing the modulus, you could imagine that r^5 = r` and just assume that's the new r, for exams if the power is too large and calculators aren't allowed, just leave it as it is r^5

[sinful pike]

this is the same as cis(theta) right

that's what im being taught

[river drift]

euler's formula can be stated as $e^{i\theta} = \operatorname{cis}\theta$

[obsidian monolith]

cloud ☁

[rough leaf]

wait

i meant z

cuz isn’t z = cos x + isinx

[plush orchid]

aye

[dreamy cobalt]

guys I have the best idea for a year 12 jacket

tell me if this is a good idea

one one side

cos

other side

wait no

one side tan

other side

arctan

hehe

get it

cuz tan inverse

and the jacket inverse too

[echo sinew]

🤔

[worthy kindle]

yea in that case you distribute the power, r becomes r^5 and by de moivre's theorem you shove the 5 with the angle, cos 5x + isin5x

[plush orchid]

yous also good with diff equations with complex numbers?

[plush orchid]

THis worked thanks

[plush orchid]

I have a question on a help forum

[lost yoke]

what is cis?

[river drift]

cis is an acronym for cosine + i sine

,, \operatorname{cis}\theta = \cos\theta + i\sin\theta

[obsidian monolith]

cloud ☁

[willow bear]

cis is also the antonym of trans

which lends some room for puns

[rough leaf]

yeha but i’m confused how does that work with the euler way

[ionic temple]

np

i wxplained terribly tho 😭

[dry verge]

I thought hyperbolas were related to ellipses when I first saw the formula, so I looked it up and they're all related.

I forgot if I ever learned what conic sections were in school, but this is actually pretty interesting.

Conic sections sound familiar... maybe I picked it up during my trigonometry class or calculus class years ago.

[reef inlet]

I am trying to prep for a test I have, but I forgot my text book in school.
Can someone send me pics of 4.6 and 4.7 problems from Ron Larson 3e Precalc with limits book?

[round geyser]

How would u do this

[willow bear]

calculator

it's extremely unlikely that there would be a nice algebraic solution here

[echo sinew]

Also, is this the original question? Like, this thing without anything else written? Neither "solve this equation"?

[frail rampart]

I kinda need help with the lower one the upper one I’m fine with i was wondering why the k is at the back I’m pretty new to this

[cold drum]

Because it's being multiplied by the sum

[daring spoke]

k will unlimately get multiplied to every number as you write so thats why the constants (in this case k) are just put at the back of summation

[round geyser]

It’s Solve this equation

[steady sentinel]

change sqrt to 1/2 power, bring 5 into log to get x^5, and convert to exponential form?

something like that

idk

[round geyser]

Yea

I thought of square it

Making it 10th power

[urban kernel]

I mean I guess, use the fact that 2x + 8 >= 0, which implies x >= -4. For the logarithm to be defined, x >= 1. Now, we try nice values. Suppose, x = 1. The LHS is √10, and the RHS is well 0. Obviously, the LHS is bigger than the RHS. For x = 2, again the LHS is √12 (around 3.46) and the RHS is around 3.15. The LHS is bigger. For x = 3, the LHS is √14 (3.74) and the RHS is 5. Now, the RHS is bigger. Note that the LHS and the RHS are both continuous (the limit of any function as x -> c is simply the function evaluated at c for all points c in its domain), we can use the immediate value theorem to conclude that these two graphs must intersect between x = 2, and x = 3.
Judging by the options, ig (d) 2.2.
Not sure otherwise.

Can't think of an analytical 'nice' solution. I think it could Lambert W function, but the solution seems VERY messy.
https://en.wikipedia.org/wiki/Lambert_W_function

[round geyser]

Ok

[urban kernel]

It might help to see it graphically. Once you know one or two values, and the general shape of the curve, it is easy to plot it. You will get an idea of continuity, roughly the area where graphs intersect, and all.
In the image, the green curve is the LHS, and the red curve is the RHS. They roughly intersect between 2 and 3, around 2.2.

This is reference.

[patent owl]

any tips on indifinite intigration

[urban kernel]

Indefinite Integration is just like the inverse operation of differentiation (It's called an antiderivative for a reason). Just think "differentiating what function would give you the integrand." In MCQ questions, you can make do with his technique (differentiate the options). Substitution and Partial Fraction Decomposition are the most common techniques used in indefinite integration. Learn integration by parts and all. Some trigonometric substitutions (Like weirstrass), and euler's identity, you are all 'set' I guess.

[frail rampart]

Thanks

[rugged spruce]

I HAVE A QUIZ FRIDAY AND NEED UR NERDS HELP. It will be on complex zeros and the fundamental thermos of Algebra, Rational functions with vertical and horizontal asymptotes, AND Real zeros of polynomials

[ancient inlet]

attach your image

[rugged spruce]

It’s a bunch of lecture notes so I was wondering if u guys know any good YouTubers that I can watch perhaps on these topics

[ancient inlet]

organic chemistry tutor

khan academy

[rugged spruce]

Ok let’s see

[ancient inlet]

send a picture of your homework ig

[rugged spruce]

This is complex zeros and the fundamental theorem of algebra there lecture notes but I don’t understand any of it

[ancient inlet]

ok so

for the 1st part

you know that you cannot factorise a^2 + b^2

you would get imaginary number or aka complex number in the form of a +bi

[rugged spruce]

Is their a vc?

We can join

[ancient inlet]

im sure there is

[rugged spruce]

They only got the event stage how they gonna make a math discord without vc

[round geyser]

I’m here

[ancient inlet]

dm

[rugged spruce]

Ok ok imma add u both to a gc so we can vc

[round geyser]

Take my notes

[round geyser]

Dm me also

[rugged spruce]

Acccept friend request

[round geyser]

Accepted

[round geyser]

Okay

[terse echo]

can someone help me understand why they use a [ for the point -13.674? i thought you only use brackets when including the point, but on that point it seems like they are equal

[lilac lance]

The question asks for f >= g so inclusive of when they are equal

[terse echo]

ooooh shoot i just didnt read that part, thanks

[sterile trench]

I'm not in precal yet but people have told me its suuper hard

is it true?

[lone temple]

i saw this on a problem in youtube, applying logarithm definition into limit like this is still right?

[lilac lance]

Yes due to the limit property lim (x->c) of f(g(x) = f( lim(x->c) g(x))

Limits can come inside composite functions

As long as both the limit of f and g as x approaches c exist

[velvet cape]

It's not too difficult, at least in my experience

Trigonometry is a nightmare for me tho 🙏

[rare ferry]

What is considered precalc

[carmine orbit]

topics like function, continuity differentiability limits

[rare ferry]

What are the prerequisites to it.

[carmine orbit]

well you can start immediately

[uncut mulch]

basic algebra, geo

pre-calc builds on top of that

[carmine orbit]

yeah

topics like sequence and series binomial theorem may help

[gloomy vector]

I really like your notes! Should I take notes like this for these theorems once I study them and color code them? I have a really good memory though.

[round geyser]

Sure

[carmine orbit]

jake i havent seen you say more than 5 words ever

[round geyser]

Yea

[carmine orbit]

you nonchalant like dat

[round geyser]

I’m now doing basic SOHCAHTOA

With Cec, Sec, and Cot

[floral veldt]

I think something must be wrong

Because if U sub n=2 in both sides it doesn't work

[valid topaz]

#calculus

[floral veldt]

Mb

[round geyser]

Can we factor

[hollow widget]

Yes

[round geyser]

What will happen if I factor x^26

[hollow widget]

Do you think that is the best to factor

You can see the factor as x^24/x^24 + x^25/x^24 + x^26/x^24

[round geyser]

x^2+x+1

Ik if we factor x^24

We can get negative exponents

[dry verge]

We could do a more complicated answer, but it would be more inconveniant than doing x^24(x^2+x+1). And mathematicians like to be as lazy as possible.

Now that we have x^2 + x + 1 = 7, it's much easier to factor and solve.

[hollow widget]

Ok and you can find x

and make sure it’s not 0

But it won’t be

[round geyser]

Let’s cross multiply

and rearrange the term

From greatest to least

x^26+x^25+x^24=7x^24

x^26+x^25-6x^24=0

Set y=x^24

It’s now

y^2+y-6=0

(y+3)(y-2)=0

y=-3, 2

No -3 since the radical can only have a positive ans

it’s 24th root of 2

[uncut mulch]

is incorrect

[willow bear]

(x^24)^2 = x^48 ≠ x^26

[willow bear]

up to here you're fine

what you say afterwards is bull

confidently wrong too

[bronze canyon]

x^26 = x^24*x^2 = x^(24 + 2)

final answer's correct tho

[weak moth]

Why does this work I was playing around with first principle and stumbled onto this

[jovial thorn]

I had to use l'Hôpital for this

[dawn breach]

is that a^h or a^n

[weak moth]

a^h

[willow bear]

$\lim_{h \to 0} \log_{\frac{h}{x}+1}(a^h) = x \ln(a)$

[obsidian monolith]

Ann

[willow bear]

is this what you wrote? @weak moth

[weak moth]

Yes

[weak moth]

I still don't understand where e comes from

[dawn breach]

I don't really understand what are you trying to do, derive your limit result? you can just inverse steps

[weak moth]

I was trying to find the derivitive of log{2}x using first principle

[willow bear]

why not just write log_2(x) as ln(x)/ln(2)

[weak moth]

Cause using first principle i always got h as a denominator I still don't understand how it plays out but I spend way to much time on this already

[obsidian monolith]

Ðɾαɠσɳ

[round geyser]

Ann

[round geyser]

I get x^24(x^2+x-6)=0 and x=0,-3,2

[willow bear]

now it's correct, except that one of these values has to be excluded based on the original equation.

[round geyser]

24 factors of 0 1 factor of -2 and 3

[round geyser]

Discard 0

It’s 2 and -3

[willow bear]

how did -4 and 3 come now

[round geyser]

(x+3)(x-2)

I mean 2

[willow bear]

-3 and 2.

[round geyser]

Okay

Is it 24 factors of 0 1 factor of 2 and -3?

[round geyser]

Right

[willow bear]

if you look at the polynomial equation you arrived at then yes.

but you discard 0.

[round geyser]

Yes

So it’s 1 factor

[remote imp]

does anyone have precalc notes

im gonna try for the testout

but my notes are kinda unorganized

[round geyser]

Okay

[round geyser]

What are u doing right now

[rare ferry]

What’s the difference between ln and log?

[raw hill]

https://math.stackexchange.com/questions/90594/the-difference-between-log-and-ln

you can google this

In short: ln is always base e

log usually means base 10, base e, or base 2 depending on context

[rare ferry]

I think I get it

[raw hill]

[round geyser]

log_10 vs log_e

[chilly magnet]

Log 10 uses a base 10 system but log e or ln uses base e (~2.718281828459045236360)

[uncut mulch]

.

[rugged spruce]

Asian people are so smart bro thanks that one Asian tutor that helped me I can finally say I am getting the hang of Precalculus. Literally just search the topics on google they teach u better than ur professors

[marble quest]

dont stereotype

[rare ferry]

But what if it’s true?

[jaunty widget]

Can you really not not least of which, comments like that lead to unreasonable pressure on people just because of their race

[rugged spruce]

I meant it in the most respectful way but I get what u guys are saying

[flat relic]

Hi, I'm currently doing calculus, but I found out that many students learn parametric equations in precalculus. Even though my calculus textbook covers them a little bit, should I go back and learn them more in-depth?

[true geode]

What your textbook is covering should at best be the "in-depth" you're referring to

They do appear in Calc II as a part of differential calculus, apparently, so I'd suggest having a brief look:
https://tutorial.math.lamar.edu/classes/calcii/parametriceqn.aspx for instance

alt: https://math.libretexts.org/Bookshelves/Precalculus/Book%3A_Precalculus__An_Investigation_of_Functions_(Lippman_and_Rasmussen)/08%3A_Further_Applications_of_Trigonometry/8.06%3A_Parametric_Equations

[flat relic]

Thank you!

[terse elbow]

can someone pls help me with number six?

i’ve been stuck for quite a while 😔

[polar torrent]

have u tried anything yet?

[terse elbow]

i’ve tried to turn everything into cos

but i don’t see how it can become sine

[polar torrent]

have you?

how did you do that?

i assume by separating them into two terms?
LHS= cosx+1

[terse elbow]

yes

[polar torrent]

and now to get sin

why dont we bring in cos²x

since we need to get sin²x

and to do that we can multiply and divide with 1-cosx

[terse elbow]

how do u know it’s 1-cosx and not cosx-1?

[polar torrent]

it doesn't really matter, you can do either

if you go for cosx-1 you would have to multiply another minus sign at the end

our goal is to get the numerator to the form (a+b)(a-b)=(a²-b²)

if you go for 1-cosx, you'll end up with 1-cos²x=sin²x.
if you go for cosx-1, you'll end up with cos²x-1 for which you would have to multiply a minus sign and get 1-cos²x which is equal to sin²x.

[terse elbow]

yess thanks i got it!

so wtv the final answer shud be determines what we multiply it by?

to save some time

[polar torrent]

yes

[terse elbow]

thanks sm

[polar torrent]

you're welcome

[round geyser]

Ik

n! can be n(n-1)(n-2)(n-3)…

Then what’s (n+2)!

[uncut mulch]

apply the same definition

[round geyser]

2-1, 2-2?

[jovial current]

Wouldn’t you rather rewrite the larger factorial this way

[round geyser]

No

[round geyser]

What’s the inverse I’m wondering

[uncut mulch]

wdym by 2-1, 2-2

[round geyser]

(n+2)(n+2-1)(n+2-2)

[uncut mulch]

where's the rest

[round geyser]

Like this

[round geyser]

I don’t know what u mean

[jovial current]

There’s still more stuff

[round geyser]

What’s the inverse of factorial

[jovial current]

After (n+2-2)

[round geyser]

I’m not sure

[uncut mulch]

are you saying that
(n+2)! = (n+2)(n+2-1)(n+2-2)

[round geyser]

(n+2-3)

Like this

[uncut mulch]

anything more after that?

[round geyser]

I’ll put eclipses

[uncut mulch]

and when do you stop

[round geyser]

Infinity

[uncut mulch]

no

[jovial current]

there’s something simpler

[uncut mulch]

it doesn't go on infinitely

[round geyser]

I’m not sure

Since I never learned it

I’m just interested in this and would want to try

[uncut mulch]

simple example,

how would you calculate
3!

[round geyser]

3x2x1

[uncut mulch]

yes

[round geyser]

It goes to 1

[uncut mulch]

yes

[round geyser]

Aka itself

[uncut mulch]

wdym aka itself

[round geyser]

Nvm

It’s

(n+2)!=(n+2-3)(n+2-2)(n+2-1)(n+2)?

Like this

[jovial current]

There was a thing I wanted to say but I don’t think I wanna give anything away

[round geyser]

Ok

(n+2)!= (n+2)(n+2-3)(n+2-2)(n+2-1)

[uncut mulch]

no

you keep changing what you think factorial does

[round geyser]

(n+1)!/n+1

You mean

[uncut mulch]

no

[jovial current]

Might I add a bit of insight?

Take 9!, if you rewrite it some way, then you get a number * a factorial of sth else

[round geyser]

Oh

[round geyser]

(n+2)(n+1)n!

[uncut mulch]

no

why are you stopping at n

if you were asked to calculate
7!

upon edit, yes

[round geyser]

okay

(n+2)(n+1)n!/n!

Now

May we cancel n! out?

[uncut mulch]

yes

[round geyser]

Then it’s a quadratic

(n+2)(n+1)=20

n^2+3n+2=20

n^2+3n-18=0

[fading gyro]

i was trying to do this question out of stewart's precalculus and i felt stuck and a little frustrated because although when i checked the answer listed in the book i had reached it at some point, i veered off doing more and more steps to try and factor it further off of nothing but the idea that it seemed like it wasn't completely factored. is there some kind of check i can do to know for certain if i am actually done?

[surreal crescent]

I would start by thinking about the relationships between functions and this as the sum of two fourth degree polynomials

What was the question asking?

[fading gyro]

"factor completely"

the answer was ||2(x^2 + 4)^4(x - 2)^3(7x^2 - 10x + 8)|| but even though i reached that i kept trying to reduce it further

[surreal crescent]

Know where to stop with factoring can be hard

The first easiest thing to know is that if everything is in one term and all are the products of two terms, it’s good

If you have another polynomials like you do here, check if you can factor it, and if not, then stop

Factoring is really just a tool in your toolbox, and it only needs to function as much as it serves you, so I would ask your teacher/prof their specific rules for where to stop when factoring

[tight wagon]

i’d look at each factor you have, and see if you could reduce each one. This sounds really obvious but i don’t think there’s any other way of knowing where to stop.

When you have that answer, you should look at each factor and ask (as @surreal crescent is saying) “is it useful to factor this any more?”. For example, the first factor in the answer is 2, is it useful to factor this any more? Even if it was a composite number like 10, it wouldn’t be useful to factor it into 5 and 2. Then check the next factor, (x^2 + 4)^4. Again, is it useful to factor any more? This term can’t be factored any more (obviously, as it’s the answer to a “factorise fully” question) but if you were doing working for a different question not solely focused on factorising, this question is what you should keep in the back of your mind.

If you’re trying to factorise fully, the only algorithm you can really follow is:

• factor out a term
• check if any factors present can be factorised further
• if they can, go back to the start. If they can’t, you have finished

There’s no other way to really check it’s fully factorised (afaik).

And, again as Root has said, if you are not trying to fully factorise, but are just factorising while within another question, you just need to keep asking if it’s useful to factorise any further (if you even can factorise any further)

holy yap (i hope this helps)

[fading gyro]

yeah, that makes sense! thank you :)

[surreal crescent]

Yeah, understanding how to manipulate an equation becomes even more important when you get to calculus, where equation form can determine ease or even ability to solve a problem

[round geyser]

Yes

[round geyser]

You will need to know how to factor difference of squares, sum of cubes and difference of cubes

And foiling

Since in functions like composite functions you will need to do that

[round geyser]

Will there 3 solution

[gusty schooner]

Hi can you please help me. I am currently learning how to graph ALL of the trig functions and I am currently struggling. Are there videos that HELPED you to graph trig functions easier? Thanks

[valid topaz]

aren't there 3 solutions?

oohhh wait u said that already

[round geyser]

Yep

Ik it can be 3

[valid topaz]

||7,3,1||

[round geyser]

7,3 is based on their exponent rules and 1 is just observing

[round geyser]

Three ways; Equal bases, Zero exponents, Opposite bases

[valid topaz]

wait 1 is from setting the bases to their negatives, if their bases are even

not that easy to spot

[plain bone]

Hi

[serene dagger]

hi, how was your weekend?

[plain bone]

Good so far

[plain bone]

What abt u?

[grim rock]

hey

[valid topaz]

<@&268886789983436800>

[carmine orbit]

what was it

[valid topaz]

some guy trying to sell a scam

[carmine orbit]

ohhh

[grim rock]

hi guys

[outer canopy]

Yoo

[floral dome]

do yall know what i should study for my unit 1b test idk what to do

this is basically what my study guide looks like but i have no idea what im looking at

[civic moon]

Study limits at infinity

[floral dome]

ok thanks

[civic moon]

Im only basing this off what u sent btw

U shuld probably just study limits in general as a whole tbh

[floral dome]

okok

and this is also some stuff my teacher put as review

[alpine bluff]

ur doing stats and calculus in the same class?

understand the notation first

what does something like this mean in english?

if u were to translate it to english terms

[rough leaf]

isn’t that

h(x) approaching 2 from the left

[alpine bluff]

yep

approaching 2 from the left so approaching x = 2 from the left