precalculus

[hasty lily]

Yeah it's initially defined for positive integers only

then they introduced another rule so this is fine ig

[willow skiff]

0! = 0 * (-1)!

checkmate

[hasty lily]

The book talked about factorial only for non-negative integers though

[fringe ridge]

hm

23! has 23 digits

i think

[hasty lily]

22,23,24

[urban lily]

The fact about 22 is an interesting one yet.

[hoary iris]

The greatest mathematician Ramanujan's birth date itself has an interesting property in factorials 😁😁

Is negative set exists?(I.e,, can number of Elements of set can be negative)

[hasty lily]

No

[wicked sail]

Can anyone of u tell me from where i learn complete coordinate geometry

[willow skiff]

Khan Academy

they have material on conic sections as well

or the book Algebra and Trigonometry by Stewart

[obsidian monolith]

A2

[empty vapor]

Can anyone explain to me the solution here? This is begginer stuff but I already don't know how

Also is this the right channel for this?

[night herald]

its just a way to represent multiple functions --> (f+g)(x) = f(x)+g(x), (f/g)(x) = f(x)/g(x) given that g(x) != 0, the condition is for the second case, just to be clear

[empty vapor]

Thanks man

[willow skiff]

I would not advise you to use that notation though

it's just more confusion when you could just write it out properly

[empty vapor]

Thanks I'll keep that in mind, I'm just copying whatever is on the board

[raven swift]

how should ı learn to precalc which channels ı need to watch ?

[uncut mulch]

you can start with khanacademy for the basics

[gritty dirge]

can anyone help me

nevermind

[viral dock]

proffesor leonard

youtube channel

[gritty dirge]

yo can someone dm me i need help pls

[gritty dirge]

about trigonometric functions

[viscid thistle]

man

😭

Im feel like im gon cry finding focal point if the given is focus and directrix😢😢

[sullen plover]

can someone help me with an trigonometric identity question?

[night herald]

Yeah sure

[night herald]

What's the question??

[gritty dirge]

Nevermind

[night herald]

I got some time

did you figure out by yourself??

[gritty dirge]

yeah thanks though

[sullen plover]

you still there?

[night herald]

Yeah

[sullen plover]

thx

hold a sec

i tried too many times but i just couldnt get the right answer

I'd really appreciate any form of help

[night herald]

what is t?

tanx??

[sullen plover]

yes

[raven swift]

bro thanks but ıt has not a limit and regularity

does not precalculus include limits?

[viral dock]

no

ben de Türküm bu arada

(gizli)

[raven swift]

:DDD

reis calcülğste limit lazım deilmidir ? ben depreme denk gelen senedeyim limit türev integral fgörmedik ne önerirsin

[night herald]

and g(x)

??

[viral dock]

calculuse limit lazim

onları öğren

precalculus 11 e kadar zaten

o konuları biliyorsundur

[raven swift]

reis sabahtan beri araştırıyorum en önemli bilgiyi verdin 11e kadar olduunu yeni öğrendim 😄

peki 11e kadarbu sonrası için nereyi önerirsin?

[viral dock]

xd

calculus başlayabilirsin

[raven swift]

yabancı bir yer varmıdır ?

[viral dock]

nasıl yani

[raven swift]

limittir türevdir integraldir

youtube kanal

[viral dock]

hee

onerdigim kanal var

bence kitaptan calis

thomas kalkülüs yaz

çıkar

[raven swift]

saol kralll

[viral dock]

ingilizcen iyiyse stewart calculus okuyabilirsin

gayet iyi anlatımı var

[viscid thistle]

anyone got time to help me find Focal point(P) in parabola with only focus and directrix given?😭

[night herald]

You mean the vertex of the parabola??

[viscid thistle]

yess😭

[night herald]

Do you have any specific question??

it might be easier to explain that way

[viscid thistle]

Determine the equation of the parabolas

Standard eq and gen

im jst really having hard time finding vertex if the focus is given😭

[winter comet]

the vertex is between the directrix and the focus

[raven swift]

bu 12. sınıfımı içeriyor

[elfin cargo]

Yeah, the vertex is halfway between the directrix and the focus

[winter comet]

yea just realized between doesn't necesarily mean the direct middle lol

but yea its the middle

half way

middle is also vague 😭

[night herald]

I got a idea

[viscid thistle]

so if the distance is 3 the vertex is somewhere in 1.5?

distance between focus and directrix

[night herald]

Its general too

[viscid thistle]

thank youu i get it now😭

[winter comet]

sure, but the important part is the coordinates

[night herald]

So, you have been given coordinates of focus(say (h,g)) and equation of directrix(say Ax + By + C = 0), right! Now we know that the vertex lies rigth at the midpoint of line joining of focus and foot of perpendicular of the line drawn from focus to directrix. the coordinates of midpoint between two points --> (x1,y1) and (x2,y2) is ([x1+x2]/2,[y1+y2]/2), so what you really need is the coordinates of foot of perpendicular

For that

[winter comet]

so if the disance between the focus and directrix is 3, then the vertex is the point 1.5 below the focus and 1.5 above the directrix (along that line)

[night herald]

First, suppose the foot of perpendicular to be something(say (l,m)), now find the slope of line joining focus,(h,g) to (l,m)

the slope of directrix is A/B

Now we know the product of slopes of two perpendicular lines is -1

hence, (A/B)*(slope of line joining focus,(h,g) to (l,m)) = -1

this will give you a equation in 2 variables(variable are l,m)

for the second equation

we know that the point, (l,m) has to lie on the directrix(Ax+By+C = 0), hence Al+Bm+C = 0

this is the second equation

Now find the midpoint

which will give you the vertex

Let me know if i wrote something wrong

[viral dock]

limit türev integral 12. sınıfta var

geri kalan üni 1

[raven swift]

anladım 1200sf falan baya çalışmam lazım sanırım 😄

[viral dock]

evet

üniye mi gectin

[raven swift]

hazırlık bitti

ibilgisayar müh.

[viral dock]

anladım kolay gelsin

[raven swift]

kafam çok karıştı ayt mantığında çalışıyordum gitmedi öyle

ingilizce kaynaklara bakmaya başlamıştım

sende iyi denk geldn

[viral dock]

zaten lisedekine benzemez lisans kitapları

hatta muhtemelen yine roman okur gibi okuyacaksın

ama kendin ogrenirsin bir sure sonra

[raven swift]

sen matematkmi okuyorsun

[viral dock]

hayır

11. sinifim

[raven swift]

nerden biliyorsun bunları ozaman :DDDD

[viral dock]

merak var diyelim

[raven swift]

helal olsn

hedef nedir

[viral dock]

bilkent fizik veya itü mat müh

[raven swift]

helal olsn kardeşim devam et

[viral dock]

teşekkürler

[viscid thistle]

hello

[viscid thistle]

is this correct?

dnt mind the erasures ik tht alottt

[night herald]

Yeah good job

[stray locust]

Hi

Can someone please help me

[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/

[fading monolith]

Just ask

[runic fiber]

ask bro

[viral dock]

bro left the server 💀

@runic fiber eth ya mı gidiyorsun

[runic fiber]

yok kuzeyine

almanya

nicknameden mi asokdmfdlwkf

[viral dock]

evet ordan

hdidneihsiajs

[normal gale]

hi

[wary timber]

hi

i want to learn how to do precalc

someone teach me pls

[violet fern]

im looking for a tutor

[viscid thistle]

hi

[viscid thistle]

hey u there?

When ur graphing and u pass the origin point do u also need to count it? or skip the origin point

when ur counting i mean to find directrix or any other that needs counting in order to find

And wht if u pass thru the origin point

Im confused if i have i count it aswell or skip it

[tender mist]

I can't teach, especially because I'm still taking classes, but you can use a pdf online, should be plenty of pre-calc books you can at least study off of. If you're new to self-study, it might seem strange at first, but I tend to find it makes it easier to learn the nuances to a subject quicker

And then you can ask people questions whenever you're stuck
(I don't know who teaches good classes, so that's my own answer at least, hopefully someone knows better than me)

[viscid thistle]

bwvshshehd

[night herald]

I don't understand the question.. Count what?? ask with a example

[viscid thistle]

nvm 😭😭

i got the answer hehe

but i have other question tho

wait

kemme

[night herald]

Yeah sure

[median raft]

1+1

[night herald]

1

[median raft]

omg

[night herald]

Its binary

[median raft]

i have been looking for this answer

since ages and you r the one with it

u r the only person who knows maths

[viscid thistle]

here i was lost on the signs

[viral dock]

Proffesor Leonard youtube channel would be good choise

[viscid thistle]

i changed the sign of neg16 and 32 i shouldnt have done it?? cuz idk😭

[night herald]

what do you want to ask??

[viscid thistle]

If i did the right thing or no😭

[night herald]

Yeah, why do you ask

[viscid thistle]

hi guys, i suck at math, alot. im gonna take pre cal for gr 11. I just need to learn algebra to have an understanding of pre cal right?

[ember veldt]

no shitake

[final pier]

Hi guys, just kind of confused on the "Function, yes or no" part, like what if the 2 separate lines have the same X or whatever, does that not make it a function?

my teacher isn't exactly good at teaching let alone teaches at all

🙏

[night herald]

Don't do it if you don't have interest in it

[night herald]

Do you know what are relations?

[final pier]

uhhh yeah

[night herald]

So, a function is a relation in which each x value is related to only one y value

[final pier]

I'm just confused like after graphing the line and finding the points of like F(3) and so on, what does it mean by is it a function, like is it talking about both lines or just one

[final pier]

yeah I get that

[night herald]

These type of functions are known as piecewise function, they differently based on given conditions on specific domains

Like for example

If you are considering a x value greater than -2 than you will plug in the value in -2x-1

[night herald]

And if you are considering a x value lesser than -2 than plug in the value in x+5

[final pier]

okay

I'm just like

confused on like the

yes or no function

I get everything else ik how to graph it and everything

but like if one of the equations which is presented in the piecewise function and one turns out to be a parabola which then contains a X or whatever would that be "no" on the function part

wait parabolas are functions

bro I'm overthinking this

😭

a piecewise function is a fucntion which is made of multiple seperate functions right

so like, yes its a function

[night herald]

Ok, so after graphing, look if any one value of x is giving multiple y values, and do this for the entire domain,(if it's asking about function, yes or no? It's talking about both collectively)

[scenic coral]

if the domains don't overlap and each piece is a function, yea, it would be a function

[final pier]

okay

[final pier]

okay

[night herald]

Yeah you can say that

[final pier]

but if a more than one X value has the same Y value on the graph it makes it not a function

and since its collectively talking about both lines, that means that it wouldn't be a function then right

[night herald]

Read again what I said before, a function can give same y value for multiple x values

Just look at a parabola of x^2

[final pier]

x^2 is a function right

wait nvm

it is

[night herald]

But $y^2 = 4ax$ is not a function of x

[obsidian monolith]

Black_Gold

[final pier]

wait but, if you have like (2,3) and (3,3) on a line then it wouldn't be a function because they have the same Y right

[night herald]

It would be a function, yes they have the same y but for different x

[final pier]

OH it can have same Y but not same X

[night herald]

If it would have been (1,2) and (1,3) then it would not have been a function

[final pier]

okay

so then

okay 😭

yeah I get it

oh and what if like 2 separate lines have the same X values would they both still be functions in a piecewise function

or like

like if a seperate line has the same X value, and so does the other one in a piecewise function would it still make it a function collectively

idk if that makes sense or not 😓

but i get everything else now

[night herald]

So usually from what I have read, you have to enter separate disjoint intervals of x, but if that does happen then it won't be a function

[final pier]

see like both these lines have the same X value in a piecewise function would it still be like a function 😭

[night herald]

Ok i will tell you one thing which may clear all your doubts

[final pier]

okay

[night herald]

No

[night herald]

So there is test known as vertical line test, in this you take a line parallel to the y axis

[final pier]

well like, I mean like the lines, they both contain a X value that the other line also has, so it wouldn't be a function

[final pier]

okay

[night herald]

Then you move that line horizontally over the domain of whole graph

Keeping it parallel to y axis

[final pier]

like the green one contains (3, 21.0176) while the purple one contains (3, undefined) so would the entirety of the piecewise function be a function or not since the separate lines contain the same X values

[night herald]

If the line intersect the graph at more than one point at any time then it won't be a function

[final pier]

okay

[final pier]

alr so both lines are functions

[night herald]

Yeah exactly

Wait I'll try to show you

[final pier]

so its not a function 😭

[night herald]

Yes

[final pier]

okay

so at any point that both lines contain the same X value then the entirety of the piecewise function is now not a function

so the entirety of the piecewise function wouldn't be a function but the lines could still be functions

like this right

😭

[final pier]

which means this piecewise function isn't a function

[night herald]

Yeah

[final pier]

okay that makes sense

thank you for the help 😭

[final pier]

oh do you still wanna show me 😭

[night herald]

Yeah wait a bit

[final pier]

and this piecewise function wouldn't be a function since both of the separate lines contain the same X value

[night herald]

yeah they wouldn't be a function(but make sure both lines does have that value of x in domain, many times they set the inequality with just greater than and less than signs, not with greater than equal to or less than equal to)

[night herald]

see the video

it may be laggy

[final pier]

okay

[final pier]

okay

[final pier]

okay so I use the line to make it pass the check

[final pier]

okay

[final pier]

okay I get it, so I would use the check on this and since the separate lines both overlap at the X value 2 that means the entirety of the piecewise function as a whole is not a function, but the lines by themselves pass the check which means the lines by themselves are still functions, but the piecewise function as a whole is not a function

[night herald]

yeah exactly, good

[final pier]

okay, thank you for the help I appreciate it 🙏

[viscid thistle]

dad wants me to take it because it has alot of oppurtunites in the future

[night herald]

There are lots of opportunities in every field, ik its hard to convince your parents, but your life would be a living hell if you don't like your subjects

[winter comet]

i think you should take it

if you can self study ahead, its probably more enjoyable and easier

but yea math is in a lot of fields

you dont even really have to be inherently 'good' at it for it to be useful at times, as long as you relatively know what you're doing eventually

learning is the most important thing XD

[viscid thistle]

Yeah i will, im just slow cuz my head is empty 💀

[viscid thistle]

Khan academy works right?

[winter comet]

yeah that should work

[subtle nymph]

Are we allowed to post ap calculus or calculus questions here

[winter comet]

#calculus

[winter comet]

you technically could but its better to post in a relevant channel lel

[subtle nymph]

Ok thanks, it wasn’t in my channel lists before

[winter comet]

yea, you'd prob need the undergrad role

or smthn

you can swap that

[subtle nymph]

Yes, which is why I asked
Wanted to double check c

[subtle nymph]

Oh I’m still in hs

[winter comet]

yea fs XD

[winter comet]

yeah but its about the uh

content

[subtle nymph]

Fair

I’m so struggling rn

[winter comet]

XD

[subtle nymph]

I barely understood precalc
I can’t do math classes online
Ever again

[winter comet]

😔

whats the question tho?

[subtle nymph]

I’m praying I learn some calc overnight cause

[subtle nymph]

Inverses of functions

[winter comet]

that's...not a question 🗿

[subtle nymph]

Topic

It’s cause I have a question…
On the entire topic…

[winter comet]

what's the question 😭

[subtle nymph]

Whatever is on that paper 🥲

Idk if I did it right or not

I think I got the first one right tho

Second one I’m so confused
Third one I have no idea

[winter comet]

whats the x != -1 about?

[subtle nymph]

Inverse

?

Not sure what that notation is

X ! Unfamiliar to me

[winter comet]

x not equal to -1

[subtle nymph]

Oh

Yes idk what I’m doing

[winter comet]

and also why does it say Di( -infinity,infinity)

😂

[subtle nymph]

I don’t think it’s right tho

[winter comet]

are you trying to find the domain of the inverse function?

or uh

[subtle nymph]

I’m figuring it out but at the same time not trying to look at the answer key that was alr posted

[subtle nymph]

Yes!

[winter comet]

LOL

[winter comet]

ahh, which is also the range of the original function

notice how f^(-1)(x) = 1/(7x) + 3/7

[hasty lily]

That's also not calculus

[winter comet]

when is this expression undefined?

[winter comet]

they prob doing like a review or smthn who knows

[subtle nymph]

Right now it’s still review of pre calc

But later on

[winter comet]

yea

[subtle nymph]

It will be calc

Yes

[winter comet]

but do you know when this expression is undefined?

[subtle nymph]

X is 0?

[winter comet]

yeah, so x cannot be equal to 0

[subtle nymph]

When denomination is 0

[winter comet]

well

[subtle nymph]

Right

[winter comet]

i should say for the inverse function

[subtle nymph]

Still tho

[winter comet]

the domain of the inverse function tells you the range of the original function, right?

[subtle nymph]

For inverse
Still can’t be 0

[subtle nymph]

I didn’t know that

[winter comet]

well, technically the y can't equal 0

[subtle nymph]

But on her answer key, that’s what it said 😭

[winter comet]

because the x of the inverse function actually represents the y of the original function

[subtle nymph]

Ohhh ok

[winter comet]

yeah

and thats why you have an asymptote at y=0

[subtle nymph]

Ok

[winter comet]

a horizontal asymptote

[subtle nymph]

Y≠0

F^(-1) (x) ≠0

So asymptote is y = 0

?

[winter comet]

yeah

you could(should) actually make this a dotted line if you can

and i would also put the numbers just for clarity 😂

[subtle nymph]

I haven’t even figured out the points LOL!

[winter comet]

vertical asymptote x=3/7, horizontal asymptote y=0

[subtle nymph]

Yes

Ok

Thank u

[winter comet]

well you wrote it down but i guess that answer sheet really be kickin in 🤣

[subtle nymph]

I’m trying so hard not to look at it

[winter comet]

x=3/7 is an asymptote because its where f(x) is undefined, if that makes sense

like regular f(x) = 1/(7x-3)
the denominator is 0 when x=3/7, and thats when the expression is undefined

thats why there's an asymptote at x=3/7

[winter comet]

do the instructions tell you to draw the inverse function as well?

[subtle nymph]

Yes

[winter comet]

ah

thats probably why they drew the inverse function 😂

the blue

[subtle nymph]

Yes

Exactly why

[winter comet]

XD

[subtle nymph]

Also exactly why I’m so lost

[winter comet]

why are you lost?

[subtle nymph]

I understand how to do f(x) but
I never really learned how to do inverses

🥲 and this is just the easy inverses
I still need to figure out how to do invereses for trig

[winter comet]

do you understand algebraically how to find an inverse given a function?

[subtle nymph]

Replace y w/ x

[winter comet]

yeh yeh

[subtle nymph]

Yes

[winter comet]

but then you have a new function

[subtle nymph]

The inverse

[winter comet]

so like if g(x) = f^(-1)(x) = 1/(7x) + 3/7
now you can graph g(x) = 1/(7x) + 3/7 as normal, right?

well

[winter comet]

yup

[subtle nymph]

Yes

Graph as another

[winter comet]

yeah, you know how to graph that?

[subtle nymph]

Probably

[winter comet]

😂

[subtle nymph]

It would be a little tedious but
Yes

[winter comet]

do you wanna try it without looking to the answer 🤣

[subtle nymph]

🫠 not really !

[winter comet]

fair, the draw of the answer is too much

[winter comet]

🤣 🤣

[subtle nymph]

Although I will take your teachings into mind
I did learn something new
I asked my teacher if she could go over again tmr she said ok but
She is a talker so we’ll see how it goes

[winter comet]

well anyway, the inverse function f^(-1)(x) would have a horizontal asymptote of y=3/7 and vertical asymptote of x=0 (the opposite asymptotes of the functions)

[winter comet]

👍

[subtle nymph]

Yes, understood

Thank you so much !

Have a happy cat

[winter comet]

cat :D

i like cat

[subtle nymph]

Thank you, she is my favorite

[winter comet]

😂

[night herald]

graph what exactly??

I just got here

[winter comet]

i was asking her if she knew how to graph the inverse function

[night herald]

Can you re-send the inverse function

[winter comet]

f^(-1)(x) = 1/(7x) + 3/7,
btw i wasn't asking as if i didnt know how, i was asking her to make sure she knew 💀

[spring girder]

is nonstandard and general form the same, or am I tripping?

[uncut mulch]

wdym by "nonstandard"
and what context?

[lone dove]

for the last part is it 6days and 5 hours cuz the ms doesnt show the answer

[dire tusk]

what the name of the book

[foggy valley]

I NEED HELP CONVERTING PARABOLA GENERAL FORM INTO STANDARD FORM. Because when I put my answer in desmos, and the original equation of general form of that problem. It doesn't matches. I don't know what am I doing wrong.

But mathway said the answer is

(y + 2/7)^2 = 8 (1/7) (7x - 13)

And it matches in desmos, it matches the general form. And my question is WHERE DID THE 1/7 CAME FROM???

My final answer is almost the same as what should be the answer, but I'm confuse on where the 1/7 came from?

my answer is

(y + 2/7)^2 = 8(7x - 13)

Mathaway's answer, and it actually matches on desmos compared to mine is

(y + 2/7)^2 = 8 (1/7) (7x - 13)

[late geyser]

392(7x-13) =/= 392x - 732

[still cape]

Hello guys, I need some help with graphing rational functions

[viral dock]

just ask

[still cape]

how to graph

[raw hill]

it should be $-9<x \leq 5$

[obsidian monolith]

Civil Service Pigeon

[worthy yew]

How to solve please help asap thank youuuu 🙏

[echo sinew]

You started well

Just change the concavity now

[solemn raven]

Careful with your negative signs people! Don't make the same mistake I did.

Correct:

[winter comet]

😔

[verbal trout]

Yep, happens extremely often. I usually get students I tutor to put a set of brackets around the thing they’re subtracting as a reminder that the negative distributes into more than just the first term for a non-monomial

Even for monomials, it can be helpful when you’re subtracting a negative, so the only redundant use is when it’s a positive monomial

[silent prism]

Im pretty unsure how the answer to d would be written in interval notation, because I know that the intervals in which f(x) ≤ g(x) are (-4,-2) and (2,3) so would it be written as (-4,-2) U (2,3)? If so would I need brackets anywhere since it specified ≤ instead of <?

[raw hill]

If so would I need brackets anywhere since it specified ≤ instead of <?

Yes, you need to include the values of x where f(x)=g(x).

Note sure how you got (-4,-1) and (2,3) tho

[silent prism]

Looking at the graph, the x values that result in the the statement being true

Beyond -2 f(x) ends up being greater than g(x) up to 2 so

[undone sage]

if the roots of the equation x^3+px^2+qx+r=0 are consecutive terms in a geometric series, prove that q^3=p^3*r and that this condition is satisfied for the equation 8x^3-100x^2+250x-125=0

ive done the first part but im not sure what the second part means?

assuming the coefficents remain in the same power, 250³ ≠ -100³*-125 so im not sure what it means

wait i got ittt

[brittle timber]

hi can someone help me find the inverse function for y = arctan|x|. I'm a bit confused and if it's not too much trouble to explain each step

[undone sage]

i do not get question 6

[rotund ore]

sub in x=-1 for the original polynomial, that = -4
diff and then sub in x=-1 again

i think at least

[willow skiff]

where are you stuck?

[chrome barn]

Made this to have us think about it

[willow skiff]

us?

[weak epoch]

Is this correct?

[glossy dove]

Guys what is PRECALCULUS?

[lilac cosmos]

Algebra n trigonometry

[lilac cosmos]

Yup

[weak epoch]

Epic

[glossy dove]

Ohhh

[weak epoch]

Some of precalc is functions right?

[tacit grotto]

well not exactly

it's mostly on geometry, trigonometry, and algebra

[mellow ore]

math, that includes algebra & trigonometry

[glossy dove]

Okk thanks

[lilac cosmos]

it’s easy bruv

[glossy dove]

Actually I didn't know about that before

[lilac cosmos]

Algebra is interesting

Like

U get addicted

Trigonometry as well

It might seem hard

[glossy dove]

Not relatable

[lilac cosmos]

But it’s not that complicated

[glossy dove]

Ohh

[lilac cosmos]

Just learn the basics

[glossy dove]

I know the basics of trigonometry and algebra

[lilac cosmos]

Good for u

[chrome barn]

should be

[chrome barn]

actually for (b), I haven't think of any good solutions yet

[night herald]

What b)?

[chrome barn]

B

[night herald]

Ig you can differentiate to find the maxima

Which is at pi/4

And check values at 0 and pi/2 for minimum

[night herald]

After differentiating you can also see that the graph is increasing in 0 to pi/4 and decreasing in pi/4 to pi/2

Hence minimum value can only be at end of intervals

[winter comet]

precalc is basically the "everything you need to know before taking calculus"

it might introduce topics in calculus but mostly its just a review of algebra and trig

[summer ruin]

is pre-calculus something you should learn before calculus
or is pre-calculus the precurssor to calculus?
these aren't mutually exclusive

[winter comet]

i think he meant like do you NEED precalculus before calculus

well you don't need to if you already know it from other classes

its just algebra and trig

other than that, it may introduce topics in calculus but you learn it in calculus anyway

i didnt take it :l

[summer ruin]

well yes to both

[river drift]

precalculus is a name given to classes that are meant to prepare students for calculus. all of the information taught in precalculus classes is needed for calculus, but it doesn't necessarily have to be taught in a class called "precalculus"

[plush cargo]

In the states typically you do precalculus after algebra 2. In algebra 2 you don't cover much trig. So precalculus is usually heavy on trig, vectors, matrices and misc topics with some review of algebra 2. You tend to end with some basic intro to calc with limits.

A lot of highschools just do some type of compression class where they do algebra 2/ precalculus in a year like sophomore year for the advanced kids. So they will do a hybrid algebra 1/geometry freshman year than algebra 2 precalculus sophomore year then go into ap calculus Junior year and typically ap stats senior year.

[worthy yew]

I got 100 thank you my goat

[chrome barn]

Surely the minimum occurs at the end, but the range is actually 1 to sqrt(2^3/2)

Seems counterintuitive, but the derivative is also in terms of sin and cos

[frail meteor]

yo, can anyone give me a problem for parabolas at 0,0 need some practice

[night herald]

So what?

And check

[night herald]

Yeah I know but it helps

Find a value involving pi close to the given intervals

[tiny zephyr]

Guys, why is f(x) notation used? I know what it does, but why do they use it?

[winter comet]

do u have a better suggestion? i mean its like you put x into f( ) and boom f(x)

[dusk dust]

yeah

[dusk dust]

lmaoo

[somber void]

Because Euler, apparently

[tacit grotto]

probably just for semantic purposes in my opinion

f(x) is literally just y

I guess if you're looking for a point on an equation, you use that sort of notation

[tacit grotto]

Here's an example problem I made for my precalc homework, I hope you understand how it works lol:

An engineer is making a big spotlight for a stage. He plots the stage as a 2D parabolic shape on a cartesian plane where the tip of the stage is at the origin along the horizontal axis.

The spotlight he was making needed to cover a 10m range on its horizontal axis where the endpoints of the parabolic light cone measured 25.2m in height. If the bulb of the flashlight is to be placed at a certain distance called the focal length from the lens, then what should this said focal length be in meters?

[weak epoch]

Quadratic functions r functions that "accelerate", while linear functions are functions that are constant, right?
Ex of quadratic: f(x) = x^2

[distant vine]

can sm1 help pls?

[willow skiff]

$\left(y-\left(11x-68\right)\right)\left(y-\left(-11x+64\right)\right)= c$ where $c$ is any real number

[obsidian monolith]

south

[worthy yew]

Unit 1 test today 🙏🙏🙏

[willow skiff]

$c \ne 0$, forgot to mention

[obsidian monolith]

south

[quick skiff]

can someone help

in differential eqns

<@&286206848099549185>

[daring tapir]

#odes-and-pdes

!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/

[quick skiff]

how do i solve ax square plus by square =1

[daring tapir]

!15m

[tender quest]

Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.

[quick skiff]

by eliminating arbritary constants

[daring tapir]

You probably wont get an answer here

[quick skiff]

oh

[daring tapir]

But sure

Theres a reason the ode channek exists

[winter comet]

well

you probably will

🤣

[daring tapir]

🤡 i mean yes- but alright lmao

[quick skiff]

my brains are fried

this is my last chapter

[daring tapir]

You have multiple?

[winter comet]

same

[quick skiff]

and i dont understand the basics of it

[vast totem]

rearrange it to isolate one variable, then express possible values for the other variable based on that arrangement.

[winter comet]

im stuck on the easiest problem for some reason 😭

[quick skiff]

my last chapter of my syllabus

and im getting fd in simple questions

[quick skiff]

wait i dont get something

my prof said if there are 2 arbritary consts

take derivative twice

[daring tapir]

They just wanna eliminate a and b, all they gotta do is differentiate twice

[quick skiff]

i do that in most sums

but dont reach the final ans

[daring tapir]

Skill issue

[quick skiff]

ikr

but i have only a week to master differential

[daring tapir]

Give up?

[quick skiff]

nope

tooo high weightage

[daring tapir]

Its a whole damn semester in most unis?

[winter comet]

do yall wanna check #odes-and-pdes for my dumb question, its more like a calc 2 question but man im trippin 😭

[quick skiff]

almost 2-3 questions

[quick skiff]

im not in uni yet

im writing

[daring tapir]

Lets go !noadvert

[quick skiff]

jee advanced

so i thought it might fall here

[daring tapir]

Say that bruh....

[winter comet]

!noadvert isn't a command?? ._.

[tender quest]

Please do not advertise your help channel or thread in other parts of the server. There are many people who need help, so advertising can quickly turn into spam.

[winter comet]

oh

[quick skiff]

why

what difference does it make

[daring tapir]

Mastering odes isnt even close to whats in the syllabus for jee

[quick skiff]

math is math

[winter comet]

💀

[quick skiff]

oh

i covered everything exept this

any tips on how i should start

[winter comet]

nobody is in the channel bruh, ok actually mb this was in my precalc class 100%

😭

[daring tapir]

💀 yea all jee asks is substitution/integration factor based sums

Thats not even 5% of real ode solving

[quick skiff]

but how do i start

[quick skiff]

ik

we are in high school still

cant learn it all in high school

[daring tapir]

Looks correct unless you messed up the quad formula part which am not bothering to check-

[daring tapir]

You said "master differential equations" but all right

[winter comet]

ok, i have no idea how they got their answer then 💀

[quick skiff]

like the jee version

[winter comet]

i mean i see their answer works but i dont see how my answer can turn into their answer

bruh

[daring tapir]

And then you said "math is math what difference does it make" but all right

[winter comet]

well thanks for lookin anyway 💀

[daring tapir]

What was their answer

[quick skiff]

interesting

[winter comet]

P = (ce^t)/(1+ce^t)

[daring tapir]

Oh

Oh

Fuck

[winter comet]

huh

XD

[daring tapir]

Its prolly just some bullshit they did with their C

[winter comet]

😭

yeah probably 💀

[daring tapir]

Ig they took the c to the other side instead of multiply it within the log term?

Try that once

Idont think youll get that

But try

[winter comet]

t-c = ln|P-P^2| u mean kinda thing?

[daring tapir]

💀

[winter comet]

💀

[daring tapir]

Yea

[winter comet]

alr

[daring tapir]

Looks tooo different tho still

[winter comet]

oh yeah nah that just gets P^2 - P + ce^t = 0

which basically gets you to the same place :l

[daring tapir]

Hmmmmmm

[winter comet]

i just dont see how they got to their answer 😂

[daring tapir]

💀 does your answer satisfy the eqn?

Like did you actually sub it to check?

[winter comet]

i did not lemme try

ce^t = P-P^2
differentiate both sides
ce^t = dP/dt - 2P dP/dt
dP/dt = (ce^t)/(1-2P)
i guess....not 😐

[daring tapir]

💀 how in the actual hell

[winter comet]

t = ln|P-P^2|+C
1 = 1/(P-P^2) * dP/dt
dP/dt = P- P^2

THERE it works, but somehow ce^t = P - P^2 doesn't

uhh

i doubt it but it might have to do with the absolute value

[daring tapir]

C is probably again doing some bullshit-

[winter comet]

😭

it is autonomous so it make sense that there be an e in it

[daring tapir]

I dont think so?

[winter comet]

so i dont see why ce^t = P-P^2 wouldnt work :l

[winter comet]

yeah i dont think os either, the constant covers it 💀

[daring tapir]

Exactly like it should all work

[winter comet]

im glad im not alone in being confused 😭

[winter comet]

yea bruh

[daring tapir]

😭 ill come back to this in like this time next year lmfao

[winter comet]

bet 🤣

i might ask my teacher

i might just never think about it again

or i might figure it out in the next 5 minutes

[daring tapir]

The better way

[winter comet]

real 🤣

[winter comet]

the integral of 1/(1-P) is -ln|1-P|, not positive 😔

breh

yeah ok everything works now 💀

bruh

[charred oasis]

how can i find the slope of the tangent at this point?

I thought about trying to do a limit but im confused

i got m = (1/x-3)/(x-1/3)

from doing slope formula n subbing in the y = 1/x into it

[sonic kayak]

Find the derivative and plug in the x value

[reef lily]

this has infinite solutions right?

[winter comet]

yeah, lim x->1/3 [ (1/x - 3)/(x - 1/3)] gives you the slope of the tangent line at x=1/3, or the derivative

[charred oasis]

x approaches 1/3 why?

oh

wait

am i dumb

im confused

do we need to multiply the numerator and denominator by anything

[winter comet]

you can calculate the slope of the secant line between two points, (1/3, 3), and (a, f(a)), where f(x) = 1/x, right?

[charred oasis]

right

yeah

no i get the 1/3 part

does it work if i js sub in 1/3

[charred oasis]

into this

[winter comet]

well no you have to take the limit

if you directly plug it in, you get 0/0 right?

its an indeterminate form

[charred oasis]

oh yeah

one sec

ok yea

1/3 substitution does not work

so what can i do to manipulate

in the past we could factor

but that doesn't seem to work this time

so I saw in a few examples where they multiply by the LCD

but in this case there is 2 different LCD

i think

like x on the numerator

and thne 3 on the denominator

[winter comet]

well, you know how you have a 1/x in the numerator? you could try multiplying the numerator and denominator by x to get rid of the fractions in the numerator right?

[charred oasis]

yeah

that only works for numerator

oh

oh

oh

doesn't matter for denominator

[winter comet]

you do have to factor but its harder to spot unless you do this lol

[charred oasis]

i guess

[winter comet]

yeah there are no fractions in the denominator lol

but yeah

well ok my bad

there is the 1/3

you could multiply by 3x i guess XD

[charred oasis]

no

then u get 0 in numerator

so not possible iirc

wait

i did 3x

[winter comet]

well you're still getting 0/0 anyway

[charred oasis]

and i dont think that works

yea

[winter comet]

im just saying

its easier to spot what you need to factor

[charred oasis]

is there anything i can factor

i dont think there is

[winter comet]

there is

[charred oasis]

what

wher

[winter comet]

if you multiply the numerator and denominator by 3x, you get lim x->1/3 [ (3 - 9x)/(3x^2-x) ], right?

[charred oasis]

ye

but once u factor

u dont get common terms

[winter comet]

3x^2 - x = x(3-x)
(3-9x) = 3(3-x)
right?

[charred oasis]

uh ye

[winter comet]

oops

[charred oasis]

wiat

what

no

i dis agree

i got 3x^2-x on denominaotr

which is x(3x-1)

[winter comet]

yeah

[charred oasis]

and numerator

[winter comet]

mb 💀

[charred oasis]

i am getting -3(3x-1)

ohhhhhh

holy s

[winter comet]

yeah

[charred oasis]

wait am i slow

i js factored it wrong

[winter comet]

same 💀

[charred oasis]

i factored out 3x from the denominator

somehow

[winter comet]

XD

[charred oasis]

lmao thanks

[opaque ember]

To anyone that can prevent my crashout before it happens, I'm doing this pre calc 11 problem on radicals, specifically restrictions on radicals, and for some reason the answer won't properly input onto the online form that I'm doing. The problem is attached with this message. Much appreciated if someone could help and explain the issue.

that isn't the answer that i've had for a while, but i've continually been wrong for the past 30 minutes.

[winter comet]

sorry, whats the problem?

[opaque ember]

I can't solve the problem, and when I seem to get the answer which seems correct, the AI on the google form says its wrong.

I'm trying to find out the answer, if you could help. 🙏

[winter comet]

it recognizes your answer as x e R tho right?

maybe it wants a different notation?

maybe it thinks you're saying x e R is the restriction, as in x cannot be anything in R

[opaque ember]

i've tried like {x I xER} but it still comes out as wrong

but the answer should be that x is a real number tho, right? regardless of what the fugazi AI says.

[winter comet]

hmm

[winter comet]

yeah, no restrictions

[somber void]

We just need someone who knows what the fuck fugazi AI is

[opaque ember]

So either we're entirely wrong, or the AI bugging. Most definetly the second option.

[winter comet]

yeah

[opaque ember]

bullshit AI 😂

[winter comet]

chat GPT ahh

(but worse)

[opaque ember]

I've even tried every other AI I could find on my phone and chat gpt but they all keep saying X has no restriction.

I think i'll just crashout and write an email to my teacher 🗿

[somber void]

maybe try writing in "x has no restriction"

[opaque ember]

still wrong

[safe basin]

bro wtf is this question

🤣

[opaque ember]

radical restrictions but the answer i have rn is right

[nimble flicker]

x has no restriction here
You should be able to write it as x ∈ R but Idk why it keeps being rejected

[proven night]

hi, can't we just find the derivative of y=1/x which is just -1/(x^2) and then just plug 1/3 in the derivative which will give us -9?

maybe i understood the problem wrong, but it seems like a solution to me

[charred oasis]

Oh yeah ig but I haven’t learned derivatives yet in my course

[viscid thistle]

why are so many people cooked?

[valid cargo]

guys is it possible to add 13.5^2 as a denominator in an ellipse equation?

[winter comet]

yes, the limit of the secant line that he was doing is the derivative

[winter comet]

this is a definition of the derivative, but usually people don't use this way to calculate it because they've created rules based on this definition that make differentiating much easier

i should have said f'(a) actually mb XD

[proven night]

Hello, guys, i am struggling with determining what a local extrema is and what is not. Here os thr function f(x) defined in [1;8]. ChatGPT says x=6 is a local minima, but x=1 isn't as the function decreases after x=1

It says that the only local minima here is x=6 as the function decreases before it and then increases after it

And x=1 isnt local minima by the definition as the func doesnt increase after this point

Is it correct?

The graph is simplified, points x=2, x=4, x=6 are infliction points

[willow skiff]

the local minima are x = 2, 6

[proven night]

Thanks

[simple scroll]

Hey can someone help me understand limits and continuity? I have a test on Friday and I don’t want to mess it up

[hard jasper]

yeah what part are you confused on

[simple scroll]

Determining limit and continuity with graphs

[hard jasper]

can you give me an example

[hard jasper]

I see two errors, from what point does x approach 1 from the left

right, a limit is as X approaches a value

[simple scroll]

For x-1

Or x=-1 I meant

Which problem r u looking at?

[hard jasper]

x=1

[simple scroll]

Okay

Right so open circle at positive 2

I mean closed

[hard jasper]

which means the point exists

[simple scroll]

While there’s closed circles at -3 and positive 4

So if there’s both a closed and open point, it exists?

Doesn’t matter where?

[hard jasper]

no, could you vc? it would be better to explain

[simple scroll]

Yeah

[hard jasper]

want me to call you? I see no vc's here

[simple scroll]

Yeah pls

[hard jasper]

called you

[simple scroll]

If it doesn’t work, I can call u

It didn’t come in I’ll call u

[muted aurora]

anyone has idea what asymptote is
i have no idea on it even after the instructor explained

if wrong channel then ping me imma delete that

[river drift]

an asymptote is a line which gets arbitrarily ("infinitely") close to a curve (graph of a function)

[muted aurora]

is it like between positive infinity to negative infinity for it parallel to axis?

[river drift]

well there are 3 types of asymptotes, corresponding to 3 types of lines: vertical, horizontal, and slant (diagonal)

[muted aurora]

diagonal? i have only learnt vertical, horizontal and oblique

[river drift]

oblique is the same as diagonal/slant

[muted aurora]

ohh ok 👌

[muted tide]

yeah like the horizontal line is an asymptote of the function's curve 1/x

also the diagonal line is an asymptote of the curve f(x) =x sinx

[velvet swift]

Hi, I have a question about catching up in calculus, and I've posted it here: https://discord.com/channels/268882317391429632/903486479064522752. I’d really appreciate any advice or guidance. Thanks in advance if anyone can help!

[spring cape]

i dont get what he meant in here

if p is 2. 2p is 4

and says about 2 additional points

[willow skiff]

you should write the parabola x^2 = 8y in standard form x^2 = 4py

then the focus is (0, p)

the directrix is y = -p

[willow skiff]

so if x^2 = 8y is to be the same thing as x^2 = 4py, we must have 8 = 4p

I also don't get why 2p = 4 is relevant btw

[spring cape]

idk i got confused about it, how so is 2p is equal to 4

[willow skiff]

it follows from p = 2

cause 8 = 4p

[spring cape]

alr i get it now, thanks for the help ^^

[willow skiff]

no worries

[viscid thistle]

They didn't change the variable
its still theta

They used King's rule

a property of integrals

int a to b f(x)dx = int a to b f(b+a-x)dx

[pliant cargo]

Help please

[winter comet]

do you know the law of sines?

[astral tendon]

can someone help me in #help-8 😭🙏

[tender quest]

Please do not advertise your help channel or thread in other parts of the server. There are many people who need help, so advertising can quickly turn into spam.

[viscid thistle]

a/sina=b/sinb=c/sinc

[stiff depot]

Help

[spiral sonnet]

I have calc 1 coming up in around a week and I need to review Trigonometry, and College Algebra. Does anyone have any recommended full review videos? Or any recommendations in general? I have already taken both of those classes prior I am just looking for very extensive reviews.

[cedar drift]

THATS PRECALC?!?!?!?

[winter comet]

no, its calc

precalc doesnt mean high school 😂

[uneven nexus]

ooh Lol i see, I'll delete

[somber void]

this isn't a parabola

[muted aurora]

im confused by these sentences as my ability to understand isn’t good as others
i am not asking for answers but i want someone to compile these words to me in simple english

like what are complex zeros etc

[cedar drift]

phew lol i was scare i had to do it this year

[willow skiff]

well it never said it was drawn to scale

[willow skiff]

zeros that are complex numbers, so in the form a + bi where b is not 0

that's actually the easy part
if you're stuck, go look up the fundamental theorem of algebra

[brittle geode]

the goat

[fossil swan]

find zeros

how many are imaginary, how many can there be, how many real rational zeros

[spring cape]

ye his videos helped me to learn algebra

[trim cobalt]

What grade is it

[stiff depot]

9

[alpine hedge]

Quick sanity check. If I am given f(x) = sinx on $[0,2 \pi]$ and I want to find f(x/2) do I graph the x-values on $ 0 \leq x/2 \leq 2 \pi \Longrightarrow 0 \leq x \leq 4 \pi $ then? Since we need to account for the domain change where as the range will still lie between $-1$ and $1$?

[obsidian monolith]

FredrikPiano

[arctic meteor]

Hi îs precalculus about limits?I need to calculate some limits and I don't know where to post

[arctic meteor]

${ /lim_{x /to /inf} /frac{/ln(x^2+x)}{/ln(x^4+2x)}$

[obsidian monolith]

wolly5114
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[arctic meteor]

How can You solve this without differentiation?

[night herald]

yeah ig

[night herald]

the questions not clear

just click a picture and send

[arctic meteor]

a

12 a

[orchid salmon]

can someone confirm if the answer to this question is 0

please

[night herald]

no

its 10/9

[uncut mulch]

how are you getting 0

[arctic meteor]

@night herald ?

[night herald]

I got a idea,but i don't know if its right or not

we can take x^2 common from ln in numerator

[arctic meteor]

The answer in my textbook îs 1/2

[night herald]

and x^4 common from ln in denominator

[night herald]

yeah thats what i am getting

then just separate the $ln(x^2+x) = ln(x^2) + ln(1+1/x)$

[obsidian monolith]

Black_Gold

[night herald]

and denominator as $ln(x^4+2x) = ln(x^4) + ln(1+2/x^3)$

[obsidian monolith]

Black_Gold

[night herald]

then take the power of ln(x^2) and ln(x^4) down

now divide the numerator and denominator by ln(x)

[obsidian monolith]

Black_Gold

[night herald]

ignore this

now you will have something like this --> $\frac{2 + \frac{ln(1+1/x)}{ln(x)}\}{4 + \frac{ln(1+2/x^3)}{ln(x)}}\$

well i have to say, math is easier than latex

[remote sonnet]

I don't know what the following is asking me to do. The answer in the key doesn't seem to map onto any concepts I've covered in the chapter.

Solve the following equations over the complex number system.
5. x2−4x+5=0

I don't know what solving something "over" the anything system means.

[night herald]

so, as i was saying then when you put limit x--> infinity ln x is growing bigger and bigger, and ln(1+1/x) smaller, so it will be equal to 0

similarly for the ln terms in denominator

hence you will get the answer as 1/2

hope its clear @arctic meteor

[obsidian monolith]

Black_Gold

[orchid salmon]

how

[orchid salmon]

okay i figured it out

thanks for the help

[empty tinsel]

Does the intermediate value theorem still apply to a discontinuous function

I have a problem where the theorem works but there’s a hole in the interval it’s asking me to check

[willow skiff]

no, the function needs to be continuous in the open interval

[empty tinsel]

Ok thanks

[willow skiff]

otherwise there can be a jump like in the middle image

np

[winged epoch]

i picked up a copy of gelfand trigonometry but the exercises don't have solutions and i find it very find consuming to give them to chatgpt every time i'm trying to check my answers, does anyone know if there's solutions maybe online or something that i can use? i haven't been able to find any

[blissful fossil]

Can someone help me with this question?

[night herald]

the inequality formed is --> ${\frac{500(3^t)}{3^t+9}\}\geq{300}$

[obsidian monolith]

Black_Gold

[night herald]

now solving this would finally get me to --> $3^{3-t} \leq 2$

[obsidian monolith]

Black_Gold

[night herald]

the minimum possible value of t is 3

hope its clear @blissful fossil

[blissful fossil]

im geting 2.4 @night herald

[night herald]

yeah, technically it would be 3 - log(2) base 3

i took the years to be as whole

[chrome barn]

🤔

[mystic sparrow]

hi

i AM in university and im struggling hard so like i need to go through the basics lol

logarithms is impossible for me to understand, and the book i have doesnt explain natural logarithms too well

im gonna play the piano and take a massive 5 hour break

[twilit ridge]

hey guys, i need help in solving this in inverse function 😓

[fossil swan]

set c(x)=y

then replace y with x

so (3y-1)/2+2=x

then

subtract 2 from both sides

mutliply by two on both sides

add 1

divide the whole thing by 3

i think(if i didnt do something stupid) it should be (2(x-2)+1)/3

i feel like i did something wrong though

check my work though

[muted aurora]

hello
anyone has ideas of what behaviour asymptote is? tyvm

[willow skiff]

for what kind of function?

[muted aurora]

rational

[willow skiff]

https://www.youtube.com/watch?v=XE-Z2-F3oWw

this video should help

[muted aurora]

my adhd can't let me watch a long video 😦
but anyways, thanks

[willow skiff]

no worries

[willow skiff]

here's a quick summary

[muted aurora]

thanks a lot 🙂

[willow skiff]

np

[muted aurora]

btw, end behavior asy. means slant asy.?

[willow skiff]

or horizontal asymptote, yes

[muted aurora]

wait
end behavior asy. has two forms?

[willow skiff]

no, it's just the asymptote can also be horizontal

[willow skiff]

well by end behaviour you should really specify +inf or -inf

[muted aurora]

alright

thank you

[willow skiff]

np

[scenic ravine]

i'm a senior in high school and i'm actually doing calculus not pre calc

[winter comet]

#calculus, what do you need?

[gritty ibex]

How do you graph, sketch a trig function like this without graphing calculator?

y=cos(2x)+sin(x-4)
y=sinx/cos(2+3x)

Or you just dont and let the calc handle?
In precalc, when basic and some transformation were introduced, we can graph or sketch them by hand with it, since normally its just one term like tan x or at worse its just simple trig with phase shift. But i have never taught how to graph equation that has more than one trig. Please enlighten me with your knowledge, thank you!

[somber void]

Can you graph cos(2x) and sin(x-4) separately?

If you have graphs of 2 functions, can you graph their sum?

[gritty ibex]

yes, i can graph separately by hand, using their fundamental points||(cosX starts at its extremum while SinX starts at its midline point)||then shifting, scaling them.
idk tbh. i dont understand the concept of their sum, like in terms of graph. In equation sure, its just simply cos(2x)**+**sin(x-4) , sum of the two. But how about graph..., the only thing i could think of is using, sum to product formula or sort. If i were to graph the sum by hand, i would just ended up getting graphing 2 graphs, because idk where to start.

I will go through the process of graphing sketching sin(x-4) as an example so you can see my problem. First i will determine the amplitude(a=1) ,period(b=1, period= 2pi) phase shift(-4, whole graph move right 4 units), midline(d=0) and extrema( max: 0+1, min: 0-1). After getting the necessary info, I will plot the start of the graph, x=4 which is its first midline point, and its closest extremum(4+-1/4period ), then i will go head plotting extrema the rest of midline point and extremum(+-1/2 period to its next point or 1/4 period with mid& extre alternating), in the interval i was assigned to graph, say [0,2pi]. With that i was able to graph them with sin(x-4) using midline and extremum as the backbone. Then id do the same for cos(2x).

But if i were to graph the "sum" of the two, i literally dont know where to start, cos(2x)+sin(x-4), idk what the amplitude(a=?), period(b=?, period= ?) , phase shift(??), midline(d=?) and extrema(d+-a=?) are. As my "backbone" is unclear, idk how to graph them. Adding their amplitude or midline of its respective trig equation doesnt seem right either...

[arctic meteor]

How do I solve this limit?

The one with the sine

I know that $sin^2(x/2)= \frac{1-cosx}{2}$

[obsidian monolith]

wolly5114

[night herald]

just write sin^2(x) as 4sin^2(x/2)cos^2(x/2)

[arctic meteor]

How did sin^2x become that expression?

[night herald]

half angle formula for sine