precalculus

[trim fable]

and ye im just confused on how u would do that for a quartic function

the domain and range

[fluid shore]

What do you mean? It's still under a square root.

[trim fable]

oh true..

tho like

if i was supposed to draw it

[fluid shore]

Okay, don't repeat the same thing over and over again with 'domain and range'

It's under a square root

[trim fable]

yes

[fluid shore]

When is a square root defined?

[trim fable]

when

its greater than 0

[fluid shore]

What's greater than 0? Be precise.

[trim fable]

x>0

[fluid shore]

Are you sure?

[trim fable]

yeah

wait

[fluid shore]

when is $\sqrt{4-x^4}$ defined

[trim fable]

when

[obsidian monolith]

Abhijeet Vats:

[fluid shore]

??????????

[trim fable]

lol

ye x has to be greater than 0

well

[fluid shore]

Okay, go and google this. Learn it properly and then do the harder problems.

[trim fable]

oh

uh ok

[fluid shore]

What if $x = -\sqrt{2}$

[obsidian monolith]

Abhijeet Vats:

[trim fable]

yeah

[fluid shore]

Is the composite function still defined?

[trim fable]

hmm

no

idk tbh

[fluid shore]

You need to go and learn this a bit more carefully. Don't do the problems until you have covered the theory.

[trim fable]

ok

ill google it and try to

watch vids then

[fluid shore]

Yeap

I'm not turning you away, it's just that it's pointless for me to give you a 10 minute lecture on this

[trim fable]

dont worry 😛

its fine

that makes sense

[fluid shore]

🙂

[trim fable]

tho

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

u know the way they did it

like mapping it out

would u say thats a better way then just graphing?

[fluid shore]

It looks analytical, as opposed to being graphical. Certainly sounds like a better sort of approach

It's not like you HAVE to do it that way but it's good to know how to do it that way, especially if you're doing something math-related at uni

[trim fable]

oh

tbh the thing is

im good at math but

i feel like with my teacher she really just

tells us what steps shes doing and not really

explaining the reasoning or like

idk i get confused with her

so i usually mess up the thinking questions

[fluid shore]

But those form the meat of what math is about. It's okay, there are many resources online that you can rely on. Use them to learn the material and work through the problems

[trim fable]

ikkk

and i love mathhh

tho ik my calc teacher would be good

coz i have him for data rn so yeah

ill be able to understand and discover more

which im excited for and ye

well cya ima work on this

[fluid shore]

Well continue with your work and keep the passion alive

It'll help you quite a bit in the future

[trim fable]

hey tho

i left 6 for now so i can focus on it tom

but for #7

there can be more than 1 answer write so is it fine that i said

that for a
g(x)=x^2
f(x)=sqrt(x+6)

@fluid shore

[fluid shore]

Looks correct

[trim fable]

ok yay

thanks

[viscid thistle]

is this pre calc

this looks hard

[valid violet]

Just make g=id and h=f

@trim fable

[fluid shore]

^id means identity, in case you're not sure

[trim fable]

i have a question

nvm

wow

[languid crane]

Ok

[trim fable]

didn't even

notice that

lol

till now

tho why is the answer 2?

@languid crane

[languid crane]

Well

[trim fable]

?

[languid crane]

I’m not sure what it’s asking

[trim fable]

same tbh

[fluid shore]

Well, it seems to be the case that the car is running most economically when the rate of gasoline consumption per unit time is at a minimum

[uncut mulch]

not sure what you plugged in,
but if you plot c(v(x)) on a graphing program, you should observe that the is a min at x=2

[trim fable]

oh yeah

it was

[trim fable]

@fluid shore

i have a question

if ur there

how do u know when the data would be validif u take the range of one function and set it as the domain of another function

[trim fable]

anyoneee

[willow bear]

....

what even are you talking about

[trim fable]

oh

hi

domain and range of

composite functions

so u know how u take the inner function and u set its range as the domain of the outer

im talking about that

[willow bear]

what

[trim fable]

and checking if the values are valid

[willow bear]

no, i don't understand what you're talking about

[trim fable]

oh

one sec

like this

the range becomes the

domain of the

function

and use that to find the domain and range of fog

[willow bear]

...

what

is that diagram even supposed to be

i mean

[trim fable]

oh

like an example

of what im saying

[willow bear]

ok i guess i sort of understand what you are getting at? sort of?

what is your question again

[trim fable]

how to

find domain and range of composite functions

i just dont know when to restrict them or

when the values of the inner would be valid in the outer

[willow bear]

$x \in D(f \circ g)$ if and only if $x \in Dg$ and $g(x) \in Df$

[obsidian monolith]

Ann:

[willow bear]

this should be fairly self-evident

[trim fable]

oh

so all real

for both?

oh also question

for fog

would the domain of g(x) be the domain of fog and range of f(x) be the range of fog?

[clever inlet]

Think about what function composition actually does

[willow bear]

no, at least not in the sense that "find the domain of this function given by a formula" questions use the word 'domain'

[trim fable]

ye i really do get composite functions its just

this that i dont get

[willow bear]

the domain of f . g is a SUBSET of the domain of g, and the range of f . g is a SUBSET of the range of f

[trim fable]

and i have a quiz tom and

its the last unit before exams and

im really stressed

[willow bear]

...

[trim fable]

?

[willow bear]

maybe the first order of business would be for you to calm yourself down first

[trim fable]

oh sorry

my bad

[wide niche]

is this a specific review question you're working on?

or just trying to grasp a concept?

[trim fable]

concept

of finding the domain and range

of composite functions

using graphs like how i tied to explain^^

[wide niche]

ok. so tell me in english (not math-ese), what is a composite function?

[trim fable]

a function that idk how to explain it would

depends on be the right word

so like two different functions where one of their output would depend on the other as their input

[willow bear]

that kinda falls flat imo

[trim fable]

oh ok

sorry

[willow bear]

a composite function is what happens when you take two functions and make the output of one the input of the other

[trim fable]

ye

ok

ok so

can u do 6 b with me

or a

as an example

to guide me

i got a fog

but not a gof

so can we do that?

[willow bear]

do you want me to do one of these points or all six

oh

you said 6b

[trim fable]

one

[willow bear]

my bad didn't see

[trim fable]

yeah

but actually

6a gof would be good

to start with coz

[willow bear]

okay

[trim fable]

its prob easier and i wanna get the

[willow bear]

let's go with that

[trim fable]

concept

ok

[wide niche]

I made a visual diagram to maybe help make life easier for you?

[willow bear]

...

can

[trim fable]

aww thanks thats sweet

[willow bear]

we proceed

[trim fable]

yes

we can

[willow bear]

okay so 6a: we have $f(x) = 3x, g(x) = \sqrt{x-4}$ and we want to find the domain and range of $g \circ f$

[obsidian monolith]

Ann:

[trim fable]

yeah

[willow bear]

naturally we'll want to find the domain first

well

[trim fable]

yes

[willow bear]

ok

before that

$(g \circ f)(x) = \sqrt{3x-4}$

[obsidian monolith]

Ann:

[trim fable]

so the outer is 3x this time and

inner is sqrt(3x-4)

right?

[willow bear]

no

[trim fable]

oh ok ill

just let u

talk

sorry

[willow bear]

you got that reversed

[trim fable]

oh rip

by outer i meant

[willow bear]

ok so as i said

[trim fable]

first

ok

[willow bear]

$x \in D(g \circ f)$ if and only if $x \in Df$ and $f(x) \in Dg$

[obsidian monolith]

Ann:

[trim fable]

d(f) is domain of

f(x)?

and g(x)

is all real numbers?

[willow bear]

D(f) is the domain of f

[trim fable]

ok

yeah

[willow bear]

i claim that $Df = \bR$ and $Dg = [4, +\infty)$

[obsidian monolith]

Ann:

[trim fable]

where did that

4 come from

tho

[willow bear]

$g(x) = \sqrt{x-4}$

[obsidian monolith]

Ann:

[trim fable]

ok

[willow bear]

in order for $g(x)$ to be defined, you need $x-4 \geq 0$

[obsidian monolith]

Ann:

[trim fable]

yeah

tho yeah

i have a question about that

so like u know linear functions how

their d and r is always all real

so like for b

the domain was x>-1/3

whys that?

and not like all real

[willow bear]

we'll get to that in a moment

[trim fable]

ok

so question

i should always

write an inequality?

to solve for it?

[willow bear]

...what

[trim fable]

like

how u did

x-4>=0

[willow bear]

=

[trim fable]

so then x>=4

yeah

sorry

[willow bear]

=

you should not imprint rules of the form "i should always do X" in your head

[trim fable]

ok

[willow bear]

you should instead understand what is going on and, if you write inequalities, where they come from

[trim fable]

ok

[willow bear]

do you understand now why $Df = \bR$ and $Dg = [4, +\infty)$?

[obsidian monolith]

Ann:

[trim fable]

yeah

coz u used

g(x)

[willow bear]

ok great

[trim fable]

tho

the square root

what did u do to that

[willow bear]

what?

[trim fable]

oh nvm

nvm

that was a dumb question

u look at the inner part

so ye

[willow bear]

_<

ok

[trim fable]

lol

sorry

[willow bear]

so can we continue now or are you gonna spew more shit and get me to validate it

[trim fable]

uh

go on

sorry about that

i just overthink things

which leads to dumb comments

[willow bear]

so as i said before, we need x to be in the domain of f and f(x) to be in the domain of g

[trim fable]

yeah

[willow bear]

that first condition is automatic, given that the domain of f is the whole number line

[trim fable]

ohh

that would be my first question

to check for

is domain of f : a whole number?

[willow bear]

....what

[trim fable]

nvm..

lol

go on

[willow bear]

so now our other condition is $3x \in [4, +\infty)$, or $3x \geq 4$

[obsidian monolith]

Ann:

[willow bear]

or x >= 4/3

[trim fable]

oh

[willow bear]

so we get that $D(g \circ f) = [4/3, +\infty)$

[obsidian monolith]

Ann:

[trim fable]

so u compared it to

g(x)

[willow bear]

i didn't compare anything to anything

[trim fable]

the

well no

i mean

u used the domain from

f(x)

[willow bear]

all i worked with was the two conditions i laid out.

[trim fable]

and used that as an inequality to see

if it is valid?

in g(x)?

[willow bear]

_<

that phrasing, god

[trim fable]

uh

what about it sorry

well im just using terms my teacher uses like

she says "check if its valid"

so i really have no clue on whether on using the right terms

[willow bear]

h

[trim fable]

im really sorry if im not

[willow bear]

i think your teacher's wording is not good

[trim fable]

umm same tbh coz

sheee

really

confuses me

[willow bear]

i mean the underlying idea is simple

[trim fable]

ye

[willow bear]

if you think of functions as input-output machines

[trim fable]

but like shes teaching this course for the first time soo

[willow bear]

anyway

[trim fable]

maybe thats why shes not really using correct terms idk

[willow bear]

okay, so i guess now we are to find the range of this thing

which really is as simple as applying the composite function to what we got as the domain

so perhaps you could graph sqrt(3x-4) in desmos

or maybe even realize that it's monotone increasing

[trim fable]

ok

[willow bear]

either way its range is $[0, +\infty)$

[obsidian monolith]

Ann:

[trim fable]

ok so

what should i look at specifically for

3x-4

[willow bear]

what

[trim fable]

its linear so

the range is all real

but u know how

[willow bear]

no look

[trim fable]

do i restrict?

?

[willow bear]

you already HAVE your restriction. it's the domain. [4/3, +infty). that's where x can be such that g o f is defined.

[trim fable]

oh

ok

[willow bear]

grgh

[trim fable]

aww

sorry

nvm then i guess

[willow bear]

here lemme

[trim fable]

ill try to figure it out myself then

?

[willow bear]

nvm

[trim fable]

sorry

hey actually

@willow bear

whys that not all valid?

[willow bear]

the leftmost bit of [-4, +infty) on the second graph goes beyond the domain of g

like g(-4) is undefined for instance

[trim fable]

why tho

oh

coz theres

no point there?

so all points have to be

located on the

outer function?

[willow bear]

hurrrrrrrgh wow wording

[trim fable]

sorry

well goodnight

[hexed bolt]

Is there like a formula to solve these?

[willow bear]

no and you shouldn't chase formulas anyway

but it might help if you make a picture

[hexed bolt]

Okay

[viscid thistle]

math

[summer monolith]

What is this 3(y+x)^2 doing in both parts of the equasion?

[willow bear]

wdym

also, it's equation, not equasion

what exactly do you not understand about what's being done here @summer monolith

[summer monolith]

Nevermind

I've got it

[viscid thistle]

How do I do it

[willow bear]

,rccw

[willow bear]

is... that meant to be a zero

[viscid thistle]

Yes

@willow bear

[empty fulcrum]

Solve for x or y in either equation and then substitute into the other one

[viscid thistle]

I did

I got 5.25 for y

But idk if it’s right

[empty fulcrum]

Also a 0 like that looks a helluva lot like an empty set tbh

[willow bear]

5.25?

[empty fulcrum]

Can I see what your work looks like?

[willow bear]

please don't use decimals

[empty fulcrum]

Must've thrown it right into a calculator

[viscid thistle]

Yep

[empty fulcrum]

You should try solving it without one

[viscid thistle]

Can’t

[empty fulcrum]

Just use straight algebra and chug through

[willow bear]

like hell you can't

[empty fulcrum]

Nah you totally can

[willow bear]

x = -sqrt(y)

substitute into the other equation

get a quadratic in y

[viscid thistle]

I did

[empty fulcrum]

Doesn't it just turn into 4y^2=88

Then y^2=22
y=(sqrt)22

[willow bear]

no

x^2 isn't y^2

x^2 is y

[empty fulcrum]

oh fuck I'm dumb lol

Look at me making dumbass errors

Okay anyway

3y^2+y-88=0

[viscid thistle]

So?

[empty fulcrum]

From this point you should be able to solve for y

Then you can just throw y into the old old equation and grab x

[viscid thistle]

That’s what I did

And I got 5.25

[willow bear]

how

show your work

[viscid thistle]

I used my quadratic formula silver

solver*

[willow bear]

you used a calculator.

why couldn't you have done it by hand?

[empty fulcrum]

It sounds as if we're being pushy but it's just good to get into the habit of doing it by hand

You'll get a more exact answer this way (assuming 5.25 is rounded, anyway), and instructors are gonna want to see the work anyhow

[viscid thistle]

:0

[willow bear]

also yeah don't round

[empty fulcrum]

Even if you do round at the end, you'll want an exact answer for when you solve for x

So you're gonna want that icky looking y value to finish the problem

[green zenith]

How do I know what the range of (f+g)(x) with f(x)=x and g(x)=sqrt(x-1)?

[willow bear]

graphing most likely

[green zenith]

I know how to look for ranges from type of functions and from shifts

[willow bear]

or the realization that f and g are both increasing and thus so is their sum

[green zenith]

So in this case, we know that it is the sum of functions that are increasing so I know its going to infinity, is that right?

Are there any techniques to find the range aside from graphing?

[tawdry olive]

Can someone explain factoring by grouping?

[heady harness]

just the algorithm?

[fleet yew]

@tawdry olive yeah

Basically you factor

By grouping

[uncut mulch]

eg
x^3 + x^2 + x + 1 = x^2( x + 1) + (x+1) = (x^2+1)(x+1)

[winter isle]

how to memorize identities

i've got most of them down, but i can't remeber the pythagorean identities and which functions are even/odd

for even/odd identities

[jaunty mason]

Is it true that the sum of the squares of the roots of a polynomial can be used to determined if the polynomial has complex roots?

[winter isle]

huh

[valid violet]

let's ask professor google

[jaunty mason]

Yeah of course I tried Googling first

Didn't help

[willow bear]

i don't think so

the sum of the squares of the roots of $(x^2+1)(x-9)$ is equal to $79$, and that alone doesn't really tell you anything.

[obsidian monolith]

Ann:

[willow bear]

despite it being positive, the polynomial has complex roots

[jaunty mason]

Exactly

In my textbook it expects me tell the nature of the roots from it

[valid violet]

didnt find anything useful googling

[willow bear]

can you show the problem exactly as stated

[jaunty mason]

The sum is equivalent to $\frac{b^2 - 2ac}{a^2}$

[obsidian monolith]

Tāhā:

[willow bear]

can you show the problem exactly as stated

[jaunty mason]

where as the actual way to tell if a polynomial (at least a quadratic one) is from its discriminant

[willow bear]

can you show the problem exactly as stated

[jaunty mason]

$b^2 - 4ac$

[willow bear]

can you show the problem exactly as stated

[obsidian monolith]

Tāhā:

[jaunty mason]

Excuse me

[willow bear]

can you show the problem exactly as stated

[jaunty mason]

bro

[willow bear]

can you show the problem exactly as stated

[jaunty mason]

stop

I'm showing alright

STOP

[willow bear]

you've not shown me the problem exactly as stated

what is given and what is asked for?

[jaunty mason]

I am in the process of showing it

There are multiple problems

[willow bear]

take a screenshot of them all

[jaunty mason]

k

fuck this dick broke blurry camera fuck

might wanna retake a few

@willow bear

[willow bear]

#7?

[jaunty mason]

yep

[willow bear]

so what did you get for S_n

[jaunty mason]

Expects me to use S_2

[willow bear]

for n=1,2,3,4

...

what

it doesn't say in the question that you are expected to use S_2 and nothing else, does it?

[jaunty mason]

You get impatiently easily

It says in the answer scheme/ page / whatever

[willow bear]

can you show the answer scheme

[jaunty mason]

Not the answer scheme but https://cdn.discordapp.com/attachments/363224154469826562/665066610264113192/20200110_103508.jpg

[willow bear]

and your value of S_2

oh, so S_2 is less than zero.

[jaunty mason]

Yep

[willow bear]

yeah, that does guarantee you at least one pair of complex roots.

[jaunty mason]

I don't see how

[willow bear]

if there were no complex solutions, i.e. if they were all real, then the sum of their squares would be nonnegative

[jaunty mason]

Consider the equation $x^2+ 3x + 3 = 0$ here $S_2 > 0$

[obsidian monolith]

Tāhā:

[willow bear]

and?

[jaunty mason]

S_2 being gretaer than zero implies that the eq does not have complex roots while in reality it does indeed have complex roots

[willow bear]

no it doesn't

[jaunty mason]

?

[willow bear]

no, the implication you just asserted doesn't hold

and at no point did i claim it does

[jaunty mason]

it literally has complex roots

how does it not?

[valid violet]

"If it is raining i have an umbrella open' does not mean that seeing an open umbrella implies rain

[willow bear]

at no point did i say "if S_2 > 0 then all roots are real"

and you just presented a counterexample to this

[jaunty mason]

Well that would be the logical deduction would it not

[willow bear]

no

just because A implies B does not mean (not A) implies (not B)

[valid violet]

P implies Q does not imlply not P implies not Q

People use umbrellas for sun, for example

[willow bear]

i said

"if S_2 < 0 then complex roots exist"

you have to understand that, in making this claim, i am not saying anything AT ALL about the case where S_2 ≥ 0.

[jaunty mason]

O I GET IT

if S_2 < 0 then the eq definitely has complex roots but if it's >= 0 then it could be either way?

@willow bear Is that right?

[willow bear]

yes

[jaunty mason]

Yeet

While we're on the topic, I'm trying to build an algo that determines Sn for any integer value of n, but there's a hitch

Let me draw the flowchart, one min

[pale vault]

Anyone know the steps to find the range of this?

[valid violet]

Do you know derivatives

[pale vault]

nope

the question said estimate the range

is it impossible without Calculus?

[valid violet]

Where start by looking at it when x is very nearly 1

And when x is 0

And when the numerator is 0

[pale vault]

so when x is 1

[valid violet]

Nearly 1*

Actually the numerator can't be 0

[pale vault]

ye thats the domain

[valid violet]

What's the domain?

[pale vault]

x all real, x=! 1

[valid violet]

So x=2 is defined?

[pale vault]

i dont know

[valid violet]

x=2 gives a nonreal value

[pale vault]

it was given as a f(x)/g(x)

[valid violet]

I assume you only want real numbers

[pale vault]

yes

[valid violet]

The domain is limited by what's allowed in the denominator

[valid violet]

Since the numerator is just a polynomial

[pale vault]

thats the exact question

b.)

[valid violet]

,rotate

[valid violet]

So you need the domain

[pale vault]

yes

[valid violet]

What is it?

[pale vault]

but its just x!=1

[valid violet]

x=2 gives a nonreal number

[pale vault]

oops

meant x< 1

thats the domain

[valid violet]

Okay cool

Do you know limits

[pale vault]

no

[valid violet]

Okay

[pale vault]

only ration functions asymptotes

[valid violet]

What happens when x is very nearly 1?

[pale vault]

it goes to infinitly closer to 1

but never touches it

[valid violet]

So that implies what about the range

[pale vault]

i have no idea

[valid violet]

It includes numbers arbitrarily large

What about numbers very close to 0

[pale vault]

no idea

it goes closer to y = 0

[valid violet]

Its not obvious. Try multiplying and dividing by sqrt(1-x) to see if the answer becomes more apparent

[pale vault]

the answer key states the "approximate range is: y > 1.7"

i have no idea how you get this answer

[valid violet]

Its not obvious. Try multiplying and dividing by sqrt(1-x) to see if the answer becomes more apparent

[jaunty mason]

Here I wrote a piecewise function instead of a flowchart @willow bear
a is the set of the coefficients of the given polynomial
n(a) is the number of coefficients and is thus equal to the degree of the given polynomial
The first coefficient is a_1, the last coefficient is a_n

https://cdn.discordapp.com/attachments/488104216678760469/665076853496741918/424633184505036802.png

[willow bear]

your notation's a bit confusing

also uh

the number of coefficients is 1 above the degree of your polynomial

not equal to it

[jaunty mason]

fuck

Well, n(a) is still = number of coefficients in that case

But S(0) will be equal to n(a) - 1 then

Anyways it has a problem, it fails for polynomials of degree >= 5

It's easy to see why

[willow bear]

ok alright so like you're trying to make a general formula for the sum of certain powers of the roots

[jaunty mason]

I wouldn't put it that way

Actually let me just redo this, there are further mistakes

[willow bear]

wdym not put it that way

is that not exactly what you're doing

[pale vault]

what do i do from here?

[willow bear]

what are you trying to achieve exactly

[pale vault]

find the range of

https://cdn.discordapp.com/attachments/363224154469826562/665075743004295218/205675981845954560.png

b.)

Ann you there?

[jaunty mason]

Well, I'm trying to build a program that just calculates S(n) for me, not sure in what sense that would involve a general formula?

I mean, I guess you could call it that idk

https://cdn.discordapp.com/attachments/488104216678760469/665081132915753000/424633184505036802.png

This looks more ok I guess

It's a recursive approach

I mean there should be a way to deal with this iteratively, not like this is non-primitive-recursive

Might as well explain the reasoning behind it, take the cubic $ax^3 + bx^2 + cx + d = 0$ that has roots $\alpha$, $\beta$ and $\gamma$

[obsidian monolith]

Tāhā:

[jaunty mason]

$a\alpha^3 + b\alpha^2 + c\alpha + d = 0$ \newline
$a\beta^3 + b\beta^2 + c\beta + d = 0$ \newline
$a\gamma^3 + b\gamma^2 + c\gamma + d = 0$ \newline
thus $aS_3 + bS_2 + cS_1 + dS_0 = 0$

[obsidian monolith]

Tāhā:

[jaunty mason]

You can mulitply that through (by powers of x) to obtain different values of S, multiplying the equation obtained above would give us $ aS_4 + bS_3 + cS_2 + dS_1$

[obsidian monolith]

Tāhā:

[patent beacon]

Is S(n) just not αⁿ?

[jaunty mason]

$\sum\alpha^n$ , yes

[obsidian monolith]

Tāhā:

[patent beacon]

Oh I see you're adding the lines together

[jaunty mason]

yep

[patent beacon]

S(3) is the sum of the cubes of the roots of the polynomial in question okay gotcha go on

[jaunty mason]

Right, so we can conclude that $aS_{n } + bS_{n - 1} + cS_{n - 2} + dS_{n - 3} = 0$

[obsidian monolith]

Tāhā:

[jaunty mason]

Rearranging that to obtain $S_n$: $\newline S_n = \frac{-(bS_{n - 1} + cS_{n - 2} + dS_{n - 3})}{a}$

[obsidian monolith]

Tāhā:

[jaunty mason]

Generalizing that for the polynomial of (x - 1)th degree, we get:

$S_n = \frac{-\sum_{i = 2}^x (a_i \cdot S_{n + 1 - i})}{a_1}$

[obsidian monolith]

Tāhā:

[jaunty mason]

Thank you for listening to my TED talk

Wew, that make sense?

[willow bear]

can i offer some advice

typesetting wise

avoid putting sigma summations in fractions if at all possible

it'll look way nicer if you write

$S_n = -\frac{1}{a_1} \sum_{i=2}^n a_i S_{n+1-i}$

[obsidian monolith]

Ann:

[jaunty mason]

Got it

The limit of the sum is x, not n btw

[willow bear]

what's x

[jaunty mason]

The number of coefficients

[willow bear]

[jaunty mason]

The degree of the polynomial + 1, if you'd like

[jaunty mason]

Ight imma zzz, @ me if you have any ideas

[steel sequoia]

How do I explain analytically "Does a function have to be defined at a value in order for a limit to exist at that value?" I know it doesn't have to be defined because cases of holes but and I explained it graphically and numerically but am unsure how to analytically

[willow bear]

you can answer the question of "Does <thing> have to happen?" with "No, here's an example where it doesn't happen."

[steel sequoia]

I guess the issue is I don't know what example to use, I've used graphs and charts already so I need an equation but don't know where to find one

[willow bear]

don't overthink it

you don't need your function to be too complicated

you can have a function that's equal to 1 everywhere except at x=4, and have it be undefined at x=4

surely, then, you agree that the limit of this function as x->4 will be 1?

[steel sequoia]

Okay I think I got it

Thanks mate :))

[verbal leaf]

In addition or subtraction of functions

Do you take on the domain of the “more restrictive” function? (Idk how else to word it)

Since a defined and undefined addition or subtraction isn’t possible?

So ie if I had domains -5<x<3 and -2<x<8

The new domain would be -2<x<3 right?

[willow bear]

what is R.O.C supposed to stand for

[verbal leaf]

Rate of change

[willow bear]

...

blink blink what

[verbal leaf]

Wair sorry m’y bad Im mixing two units together

Ignore the rate of change part

[willow bear]

can

you maybe post the problem you're doing

[verbal leaf]

Just focus on the combination of functions

I can’t it was a test, I just wanted to go over what I did cuz I was foggy. But there were two different functions and it said what was their combined domain and range

In the case where f(x)+g(x)

[willow bear]

ok, so one function had domain (-5, 3) and the other (-2, 8)

yeah the domain of their sum will be what you wrote

[verbal leaf]

That’s an example I made but yes

Okay thank u idk why I was freaking so much

[willow bear]

why would x be a cos(180-θ)

a.k.a. -a cos(θ)

here θ is depicted as obtuse to its cos is negative, and so a cos(θ) is negative too

[green zenith]

Ahhh so cos(theta) in an obtuse angle is equals to -cos(theta). So a(-cos(theta))=cos(theta)? @willow bear

[willow bear]

no???????

[viscid thistle]

Theta is the angle not cos(theta)

[green zenith]

Yeah sorry

[viscid thistle]

Also it's not always an obtuse angle

The theta

[green zenith]

I think its cos(180-theta) = -cos(theta) like ann said

Is that right?

[viscid thistle]

Yes

Nah it's right

[green zenith]

Okay now i get it

Thanks

[spark cliff]

um this is probs precalc I don’t rlly know but what’s a regular period

never mind

[fleet yew]

@spark cliff a regular period is reason to celebrate

[summer monolith]

Whaat is continous positive number? And why would we want to treat the number of people as such in this case?

[willow bear]

it's a bad way of wording the fact that you're allowing n to take on fractional values, i.e. having it be a continuous (real-valued) variable rather than discrete (integer-valued)

because that makes it easier to model

[summer monolith]

I'm kinda ignorant in the sorts of numbers. Need to brush up on those. That was a bit premature question I guess.

[viral imp]

when changing from the graph of y=log(x)

to the graph of y=log(|x|)

I don't get why there is symmetry on the y axis, after all there are no negative x values

[willow bear]

y = log|x| allows negative values for x just fine

log|-1| is defined

[viral imp]

right, clear, thanks a lot

[green zenith]

I need help

I am to find the other 2 x in the given interval

[willow bear]

the other two what

[green zenith]

The x

Oh

Not the x i guess

I dont know what to find

I already have the answer given by the book which is 8pi m

Im confused

[rigid beacon]

I think it's asking if there is an x in the interval [pi/2 , pi] where sin(x) = 3/5

[green zenith]

But the answer is 8pi m

Its not in the interval of [pi/2,pi]

Isnt it?

Am i missing something here?

[stuck lark]

$\forall m\in\bZ,\quad\sin(8\pi m)=0$

[obsidian monolith]

RokettoJanpu:

[stuck lark]

so we don't know what you mean

[uncut mulch]

it may help if the image wasn't cut off

[green zenith]

Okay ill take another one

Voila

[uncut mulch]

But the answer is 8pi m
that doesn't make sense, especially now that we have the whole question

[stuck lark]

it never hurts to start off drawing right triangles within the unit circle

[green zenith]

Ahhh now i understand the question

Yes it doesnt because i wasnt looking at the wrong thing

Sorry

Im a dumbass

It is to look for the other 2 trig functions

[willow bear]

okay...

so you know the fundamental identity of trigonometry, right

[green zenith]

Yes

[willow bear]

sin^2(x) + cos^2(x) = 1

this identity right here

just to have that on the table

[green zenith]

Yes im familiar

So i can use that to solve it

[willow bear]

there's something else you'll need

and that's the knowledge of when the trig functions are positive and when they're negative

which if you have a picture of the unit circle in mind is somewhat obvious

[green zenith]

Yeah their position in the quadrants

The ASTC

[willow bear]

blurgh.

i really hate that mnemonic, personally.

but ok whatever floats your boat ig

[rigid beacon]

never heard of that mnemonic

[willow bear]

those two things should be enough to get the cos from the sin or vice versa. and once you know both of them, tan is easy too.

[green zenith]

I just read it recently and using the image of the quadrant alone is useful enough

Okay I got it now.

Thank you

[vapid torrent]

When you throw the two dice simultaneously n times, where n is a large number, how many times do you expect to get 2 as the sum? What about the other numbers (3-12)?

isn t it just n/12 for all

[heady harness]

no because then that would imply that the larger the number the greater the probability, when it is really normally distributed

[frank mason]

@vapid torrent Did you do a probability table

[vapid torrent]

wouldnt the probaility be the same for all numbers though

[frank mason]

No

[vapid torrent]

your right

just realized

[frank mason]

I know

Draw a table of 2,3,4,5,6,7,8,9,10,11,12

Then you can find E(X)

[vapid torrent]

but how would u do it in relation to the n number of times

[frank mason]

A large number of times just means the expected value

And the expected value is found by multiplying the value by its probability for each of the values and adding them together

since rolling the two dice is a discrete random variable which number you’ll get

@vapid torrent you get it now?

[vapid torrent]

not really

[heady harness]

A large number of times just means the expected value
law of large numbers

[frank mason]

@vapid torrent do you want me to draw a table for you and send a pic?

[vapid torrent]

no its fine

[frank mason]

i’m bored out my mind dude

u sure??

[vapid torrent]

what do u multiply the probabilities by

ok sure

ty

[frank mason]

wait

My whiteboard doesn’t have enough space

@vapid torrent and id rather attempt to explain it

[vapid torrent]

all good

i found the probabilities, 1/36, 2/36, 3/36,4/36,5/36,6/36 ,5/36,4/36,3/36,2/36,1/36

not sure how the n comes in

[frank mason]

Yeah you’re doing it right

Do they all add up to 36

yes

[vapid torrent]

yep

[frank mason]

@vapid torrent okay so draw those above the value they correspond to

so 1/36 = 2 yes

etc

[vapid torrent]

yes

[frank mason]

You know when you want to take an average from a frequency table you do f(x) then total it then divide by the total amount @vapid torrent

So for instance#

x, freq, fx

2,3,6

3,4,12

so you have 18 total, divided by 7 which is your total frequency to give 2.57

[vapid torrent]

so i just do 1/all the probabilities?

[frank mason]

No no no

but you understand that right

[vapid torrent]

yes

[frank mason]

So your value is your X, your probability is your frequency

And what do all probabilities add up to

[vapid torrent]

average freq = 1/av return period

1

[frank mason]

All probabilities add up to 1

So

The name for large values of N is expected value for X

[vapid torrent]

yes

[frank mason]

And E(X)= Sum of all your values times their respective probabilityp

So 2*1/36 + 3*2/36 and so on until 12*1/36

Sorry to explain it but I’d rather you@know how to do it than just get the answer

[vapid torrent]

i dont get So 21/36 + 32/36 and so on until 12*1/36

[frank mason]

Fuck sake

The formatting broke

@vapid torrent look again, make sense now

you’re basically doing the f(x) column then dividing it by the total (1)

but with values and probabilities

[vapid torrent]

okay but isnt that the total

what do i do with this

[frank mason]

What number did you get and what do you mean isn’t that the total

Yeah it’s the total, but you divide the total by the number of things and in this case it’s 1,

[vapid torrent]

i got 7

number of things?

do u mean totsl probability, which would laways be one ?

[frank mason]

Yeah

Probabilities always add to one

So you know how you found your total by multiplying the values and their probabilities and adding them?

Then you divide your total by the sum of your frequency don’t you

and the sum of all your frequencies(probabilities) is equal to 1 always

[vapid torrent]

but what is significance of the number 7

what do i do with it

[frank mason]

It’s your answer

It’s the expected value for X

[vapid torrent]

how

they just said you rolled it n times

and n is a large nujmber

[frank mason]

when two dice are thrown simultaneously n times, what is the long term value for n

Yes I tried to explain before

[vapid torrent]

so its 7 for all numbers?

[frank mason]

What they’re trying to hint at is

Large values of n ‘should’ ‘in theory’ give the expected value

Which is 7

What grade maths are you in?

[vapid torrent]

Ok but logically how does that make sense, what if n is a billion, you are obviously rolling more than 7 2s?????

9

[frank mason]

@vapid torrent i’m confused what do you@mean

[vapid torrent]

dude

its rolling n times, n is a large number of times

[frank mason]

I know, I do statistics

[vapid torrent]

how can the expected value of rolling a 2 be 7

you have a 1/36 chance of rolling sa 2

[frank mason]

I’ve read your question wrong

Fuck sake

i’m so sorry let me re read

[vapid torrent]

When you throw the two dice simultaneously n times, where n is a large number, how many times do you expect to get 2 as the sum? What about the other numbers (3-12)?

you were finding the most common number

which would be 7, since it has the highest prob anyeways

[frank mason]

that’s not why it’s 7 though

not bc it has highest probability

@heady harness can you shed some light?

the probabilities he has are correct, but how many times do you expect to get 2 as the sum? 1/36th of the time?

@vapid torrent i think it’s just 1/36, i’m sorry for making you do the extra work man

it’s 1/36*n

[vapid torrent]

but the n number of times would want an answer with the n vsariabkle in it right

[frank mason]

Because say you rolled the dice 36 times you’d expect one two wouldn’t you

it’s 1/36*n

and for the other numbers it’s their probability times by n

i’m sorry tho dude

[vapid torrent]

ok ye

[frank mason]

you didn’t do too much extra work and now you know how to find expected value lmao

[vapid torrent]

nw

ty

[frank mason]

No problem lmao,

[urban moat]

does someone know what this is

[clever inlet]

well

we have to make some assumptions

look at those last 3 terms

what type of sequence is it?

[urban moat]

geo right

[clever inlet]

yep

what's the common ratio?

[urban moat]

3

[clever inlet]

yep

does it make sense

that we can write the nth term of the sequence as 3^(n - 1)?

(assuming n = 1 is the first term)

because we are multiplying by 3 to get new terms in the sequence

does that part make sense?

[heady harness]

the ratio is 1/3

[clever inlet]

ok yeah

that's true

i was thinking of doing the question backwards

cause right to left will have the same number of terms as left to right

[heady harness]

true

[turbid tangle]

16

[heady harness]

yes

log_3(14348907)+1

[viscid thistle]

Well, that seems convoluted.

3
9

27
81

243
729

And so on

Note these are all 2 numbers with the same amount of digits.

2 numbers for 1 digit. 2 numbers for 2 digits. 2 numbers for 3 digits. Then there will be 2 numbers for 4 digits.

The last one has 8 digits.

So there will be 15 numbers + 1, so 16

There’s only one number of 8 digits in the sequence, btw — not two, so its 15 numbers, plus the 3^0

[willow bear]

243*

[meager otter]

How can I prove that sinx < x by using the unit circle and area calculation? I create two triangles in the first quadrant, one with the area sinxcosx/2 and the second bigger one with the area sinx/2. How can I use this in my proof?

[earnest nymph]

https://youtu.be/mZiPdyHyUvE

[rain mulch]

Can anyone help me explain how this work

[spare hull]

what is c?

[rain mulch]

I think it is combination

[spare hull]

yeah but

what is c? what does it mean? what is it equal to?

[rain mulch]

IDK

[spare hull]

can you tell me where you got this from?

[rain mulch]

From YouTube

[spare hull]

is it in a textbook?

where exactly?

[rain mulch]

Trying to learn binomial expansion by using this

[spare hull]

send a link to that video

[rain mulch]

U want to dm ??

Or in here?

[spare hull]

right here

[rain mulch]

https://www.youtube.com/watch?v=3Yw--3L7bCg&list=PLTzMaA6R8ce1-9cGaxx3nZpOnizYTHmax&index=2

Get to 10:00

????

Y u delete ur comment?

[willow bear]

uhhhhhhhh

look up "binomial coefficient" @rain mulch

[heady harness]

yes, that is what you solved for on the piece of paper

[rain mulch]

Hm

[night wigeon]

i got y=1/2x

altho it was wrong

[undone pawn]

show work

also I see no C

[night wigeon]

Excuse the bad handwriting

i forgot to show that they give y and x

[willow bear]

uh

what

happened in the fourth line

[night wigeon]

i tried integrating 1/4x to ln(4x) when i tried it first time

and it gave me the wrong answer

so i assumed you had to take out the 1/4

This was my original working

[undone pawn]

ok but that doesn't explain a random square root being introduced

[night wigeon]

ah its not a square root

its the integral thing

[undone pawn]

bruh

[night wigeon]

i made the top line too wide cause i was rushing

[uncut mulch]

$\int$ looks very different to $\sqrt{ \ \ }$

[undone pawn]

your integral thing needs work

[obsidian monolith]

ramonov:

[night wigeon]

oh your right

ill fix that symbol now then

where else did i go wrong

i cant figure it out

i tried a different question

and im still getting it wrong

where am i going wrong

[serene heath]

integral of 1/3x

[uncut mulch]

()

[night wigeon]

i havent done integration in a long time is there a integration sheet with all the different integrals i can use to refresh my memory