help-10

[wintry hearth]

understood till now?

[plain sparrow]

yeah

[wintry hearth]

alright

ticket close?

[plain sparrow]

sure

ty

.close

[obtuse pebble]

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[atomic compass]

Do trigonometry functions sin cos tan take in degrees or radians

[atomic compass]

?

[craggy surge]

both

[atomic compass]

How does the function know which one it is

[craggy surge]

radian is just another measure of an angle

[atomic compass]

I get it

[craggy surge]

like how we use km m cm

[atomic compass]

But the numerical value will still be different, 90 for degrees and 1,13 for radians.

So how can the function know

[craggy surge]

do you know how to convert degrees to radians and vice versa?

[atomic compass]

About

Or do you have to understand as a human and as a machine someone has to tell it which one it is

[craggy surge]

90° = π/2 radians
sin 90° = sin π/2 = 1

we have to figure out ourselves what angle it refers too

[atomic compass]

Yes and computers have degrees usually as default?

Cuz someone could technically write pi/2°

[craggy surge]

if you write π/2° that will be converted to 1.57°

calculators have a rad and deg button too

[obtuse pebble]

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[granite dragon]

1 – cos(x) is equal to 2 sin²(x/2).

[granite dragon]

can somebody show the derivation of this

.close

[obtuse pebble]

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[modest ingot]

can i dm you about how to invert colors on that pdf viewer thing?

[granite dragon]

holy crap thats complicated

I just got it through double angle but thanks for that

[spare gale]

its ez like that

[dense imp]

well this is a whole bunch of identities kinda proven together, but yeah basically you have to do kind of a geometric proof like that for one identity then you can derive the others from that more or less :p

but if you really want a proof that assumes you know nothing about sin cos other than their definitions this pretty much does it

[spare gale]

cos(2x) = 1 - 2sin^2(x)
cos(x) = 1 - 2sin^2(x^/2)
2sin^2(x/2) = 1 - cos(x)

writing math identites in pc is a pain

[obtuse pebble]

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[plush egret]

Hi

[plush egret]

Guys

I know we need to get the value of Bd

But I don’t know, how will we do that

The formula something

Can someone explain to me

The process in able to get that formula

<@&286206848099549185>

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[brisk parrot]

can u solve pls??

[obtuse pebble]

Please don't occupy multiple help channels.

[turbid hawk]

sub in pi/2 into ur x's

[brisk parrot]

can u explan n bref

[turbid hawk]

so the expression is asking u

what would u get close to

as ur x value approaches pi/2

[brisk parrot]

yeah

[turbid hawk]

what would ur answer be

so just sub in pi/2

into sinx and cos

so sin(pi/2)

and cos(pi/2)

solve

u should get 1

[brisk parrot]

so wat wud final ans be??

[heavy belfry]

plug it in yourself and find out

[turbid hawk]

no

[brisk parrot]

but the ans they gave is 0

[turbid hawk]

final answer is 0

sorry

calc error

mb

[brisk parrot]

yea thans a bunch

[turbid hawk]

np

[brisk grove]

What is even going on here

Subbing pi/2 definitely isn't correct

The limit is of the form 0/0

[dim temple]

try multiplying the numerator and denominator by the conjugate of 1 - sinx

[obtuse pebble]

@brisk parrot Has your question been resolved?

[marsh birch]

Are you allowed to do l'hopitals

[wraith wing]

multiply numerator and denominator with (1+sinx). this simplifies to (1-sin^2 x)/(cosx(1+sinx))

[obtuse pebble]

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[static marten]

that's so skibidi pog

ask an actual question

[brisk grove]

<@&268886789983436800>

I don't think he's going to

[obtuse pebble]

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[dreamy acorn]

i dont understand how the cross product works

[dreamy acorn]

How to calculate the cross product of 2 vectors

[red ice]

You don't need this to understand the question

[red ice]

Use the definition that $|a \times b| = |a||b| \sin \theta$

[warm shale]

south

[red ice]

From that you can show the cross product is distributive

[dreamy acorn]

i mean i dont understand how did they come out with these steps

[red ice]

$a \times (b + c) = a \times b + a \times c$

[warm shale]

south

[red ice]

Yeah so do it slowly: $|4a \times 2a + 4a \times 5b - b \times 2a - b \times 5b|$

[warm shale]

south

[red ice]

But $a \times a = |a| |a| \sin 0$

[warm shale]

south

[red ice]

The angle between a vector and itself is 0, so theta = 0

[dreamy acorn]

OH

oh yes

[red ice]

Yeah and then $a \times b = -(-b) \times a$

[warm shale]

south

[dreamy acorn]

yeah i understand now thank you 南哥

[red ice]

Lmao you're on the Chinese server haha

That is me yes

[dreamy acorn]

Im zero

[red ice]

Wow

[dreamy acorn]

lmaoooo

anyway

thx for your help

.close

[obtuse pebble]

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[solid musk]

If you have the fraction 5x / 7x , could you simplify it further by cancelling out the x on the top and the bottom

[raw torrent]

ye

[red ice]

Okay yes, that's an example of why that works

It's cause $\frac{a}{c} \frac{b}{d} = \frac{ab}{cd}$

[raw torrent]

wuh

[warm shale]

south

[red ice]

Okay that's better

[raw torrent]

what does that mean

[red ice]

Let me find some sort of explanation

[solid musk]

then why cant you simplify x / x +2 any further

[red ice]

You need to multiply by 1 in order to not change the thing

[solid musk]

huh

[red ice]

So hopefully you can see that $\frac{1/x}{1/x} = 1$

[warm shale]

south

[red ice]

anything divided by itself is 1

[solid musk]

yhh

i get that

[red ice]

So $\frac{x}{x + 2} \frac{1/x}{1/x} = \frac{1}{(x + 2)(1/x)}$

[warm shale]

south

[red ice]

Now by the distributive law on the bottom

We have $\frac{1}{1 + 2/x}$

[warm shale]

south

[solid musk]

ok now im even more confused

[red ice]

Distributive law reminder

[solid musk]

in the fraction x / x +2 why cant you cancel the top and bottom by x

[red ice]

So my point is you actually can

If you divide x by x

[raw torrent]

you are dividing by x+2 not x

[red ice]

You also need to divide (x + 2) by x

[raw torrent]

you cant split denominator into two part

[solid musk]

ohhhhhhhhhhhh

so i cant divide the top by x itself

[red ice]

And this is what you get after doing it

[red ice]

Yeah you need to be consistent

Like if you have 2x + 5 = 8

Think of it as subtracting 5 from both sides

[solid musk]

how did u get that

[solid musk]

yhh

[red ice]

Don't think of it as 'bringing' the 5 over, that logic fails as soon as you have anything even a bit complicated

[solid musk]

yhhhh

[red ice]

So (x + 2)/x = 1 + 2/x

That's our denominator

And x/x is our numerator

[solid musk]

how is that our denominator

isnt it x + 2

[red ice]

It comes from the distributive law

[red ice]

Ah, we're dividing both top and bottom by x

And that gives us the same thing

[red ice]

Cause of this

[lost otter]

Let's take a fraction, 3/5. You can't really simplify it further since it has no common factor. 3/5 is also 3/(3+2), so if you were to cut 3 and 3 it'd just become 1/2. That's not possible tho. voila

[solid musk]

in 3/5 you would know that it is not equal to 1/2 but you cant tell that in x / x + 2

[red ice]

There you go

Cause x can be anything, x can be any number

[lost otter]

Exactly

[solid musk]

yhh

[red ice]

So if x / (x + 2) was equal to 1/2

Then 3/(3 + 2) would be equal to 1/2

and 100/(100 + 2) would also be equal to 1/2

That's what we mean by when we say something containing x is 'equal' to another thing

[solid musk]

i still dont understand why you cant cancel out the x

[red ice]

That it would be true for literally any x

[lost otter]

well

hmm

choose a number

[solid musk]

5

[red ice]

This is the correct way to do it with all the reasoning

[lost otter]

alright

5/8

simplify it further for me

[solid musk]

you cant coz theres no common factor

[lost otter]

okay

fair

[solid musk]

yhh

[lost otter]

but

i mean

5/10-2

now what

[solid musk]

would it be 1/5 + 2

[lost otter]

no

[solid musk]

wait no

[lost otter]

10-2 is in the denominator

[solid musk]

yhh

[lost otter]

$\frac{5}{10-2}$

[warm shale]

sin

[lost otter]

simplify this

[solid musk]

1/2 - 2

[lost otter]

Really?

so youre saying

$\frac{1}{2} - 2$

[warm shale]

sin

[red ice]

You need to divide the -2 by 5 as well

[lost otter]

The two just, came up

Became god

[red ice]

Cause you need to do the same thing to top and bottom

To keep it the same

[lost otter]

let him do it south

i wanna see his logic

[solid musk]

so i divide the top by 5 and then divide 10 -2 by 5

[lost otter]

idk

[red ice]

Yes

[solid musk]

so

[lost otter]

uhuh

[solid musk]

it would be

1 / (10-2) divided by 5

[lost otter]

Yeah

[solid musk]

but what would that be

[lost otter]

welp

what do u think

any personal ideas?

[solid musk]

1 / 8 divided by 5

[lost otter]

Why 1/8?

[solid musk]

coz 10 minus 2 is 8

so would it be 1 / 1.6

coz 8 / 5 is 1.6

[lost otter]

aight

now

get rid of the decimals

what do u get

[solid musk]

5 / 8

waittt

[lost otter]

back to where we started

[solid musk]

that makes sense

thank you sin

[lost otter]

you can't divide something

that subtracts or adds

because that's a sum.

[solid musk]

ohhhhhhhhhh

[lost otter]

you're adding an unknown quantity to something

[solid musk]

yhhhhhhhh

[lost otter]

you cant divide

an apple

with an apple and mango smoothie

just cus apple is there

it doesn't dissapear

[solid musk]

yhhhhh

[lost otter]

get it now?

[solid musk]

yes

thank you

[lost otter]

all g

[obtuse pebble]

@solid musk Has your question been resolved?

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[oblique pebble]

$f(\theta) = \sin \theta, \theta \in [0, \pi] $
How do I calculate the arc length of this? I got to $\int_0^{\pi} \sqrt{1 + \cos^2x} dx$, but don’t know how to go from here

[warm shale]

Michael

[oblique pebble]

Oh wait nvm, I misunderstood the task

.close

[obtuse pebble]

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[solid musk]

In the fraction x / x +2 you cannot cancel the x out as you cannot split the denominator but in the fraction (x+2) / (x+2)(x+2) you can cancel out one of the ( x +2 ) to make 1 / x+2. How does that work?

[trail cliff]

$$x / x +2$$
$$ (x+2) / (x+2)(x+2) $$

[warm shale]

Bestower

[trail cliff]

So you can cel out multiplied terms

x+2 is a multiplied term on the bottom

[solid musk]

yh

[trail cliff]

you cannot cancel out added terms

[solid musk]

ohhh

is there is a particular reason

[trail cliff]

(unless you factor out from both terms, but thats not possible here)

[solid musk]

what is there was

3x+3/6x

can you still not cancel out

as

it is added term

[trail cliff]

3/6x?

[solid musk]

no

3x+3/6x

[solid musk]

then how does this work

[nova cobalt]

If u take 3 common in the numerator then u can simplify

[solid musk]

but you cant for added terms???

[nova cobalt]

Nope

[solid musk]

and what cant u take x common in x/x+2

[solid musk]

but 3x + 3/6 is an added term

[nova cobalt]

You can't

[nova cobalt]

You can convert it to {3(x+1)}/6

[solid musk]

ohhhhhhh

which makes it a multiplied term

[nova cobalt]

Yea

[solid musk]

then

why not for x / x + 2

[nova cobalt]

Nothing you can take common

[solid musk]

omdd yhhhhh

that makes more sense

thank you

i js need to consider multiplied or added term

[nova cobalt]

Welcome

[solid musk]

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[obtuse pebble]

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[shadow lava]

trying to understand where i went wrong with this question

[stray stump]

D, is not since (0,0,0) doesn't belong to it

[shadow lava]

so the answer is C and E?

[stray stump]

explain why B is not ?

[shadow lava]

no z?

but maybe that doesn't matter

[stray stump]

try to verify the axioms

[shadow lava]

so B, C, E

[stray stump]

.

[shadow lava]

sorry, which axiom for B?

there are 8 right

[stray stump]

3

0 belongs there

addition is there

and multiplication by scalar

these are subspaces axioms

[shadow lava]

oh

ty
so I will try to understand these

the reason A doesn't work

is scalar, not all are same scalar

[stray stump]

0 is not in A

[shadow lava]

oh

[stray stump]

since 0 < 0 < 0 is not verified

[shadow lava]

you mean it has to have (0,0,0) for all 3 entries

origin

[stray stump]

exactly

[shadow lava]

got it

that makes sense

[stray stump]

0 of R^3 is (0,0,0)

[shadow lava]

for B, let's see..

even tho no Z, we will just assume Z is 0

it has x and y, so it's fine?

i guess this is also the 0 axiom?

for B

[shadow lava]

it doesn't have to satisfy all 3 right? just one

[stray stump]

It have to satisfy all 3 to be a subspace

[shadow lava]

hmmm

what does this one mean? addition is there

[stray stump]

for B, not that it is of the form x(5,-9,4) + y(6,-8,8) so it's spanned by these vectors so it's subspace

[shadow lava]

oh

why does F not work?

it's a plane, IIRC?

[stray stump]

if x=(x1,x2,x3) belongs to B and y=(y1,y2,y3) belongs to B then x+y belongs to B

[stray stump]

exactly

[shadow lava]

what does that constant actually do

because E is similar

but it doesn't have constant

the constant just moves the plane up or down, right

but which axiom does that break

[stray stump]

any plane that passes by the origin is subspace of R^3

otherwise, it's not

[shadow lava]

hmmm, i am imagining the x,y,z as a 3D plane

oh, I think I get it

origin for x,y,z is 0,0,0

but origin for F's plane is (0,-5,0) relative to the first plane

something like that

so both plane origins do not match

there is a shift of -5

[stray stump]

exactly

[shadow lava]

not sure if = -5 refers to the j hat coordinate

or not necessarily

[stray stump]

doesn't matter

[shadow lava]

but it would only be one direction right?

or would you have to use RREF to find out what direction the plane shift is

[stray stump]

it's directed by 2 vectors

[shadow lava]

oh, why only 2?

[stray stump]

write z in terms of x,y

[shadow lava]

we are talking about F, right

[stray stump]

yes

[shadow lava]

9x - 4y - 6z = -5

[shadow lava]

6z = 9x - 4y + 5

z = 9x/6 - 4y/6 + 5/6

[stray stump]

so

[shadow lava]

what does that tell us about the plane shift?

[stray stump]

(x,y,z) = (x,y,9x/6 - 4y/6 + 5/6) = x(1,0,9/6) + y(0,1,-4/6) + (0,0,5/6)

last term is the shift

first two's are the director vectors

[shadow lava]

x(1,0,9/6) + y(0,1,-4/6) + (0,0,5/6)

is there a word for this? ^

it's not origin of this plane, in comparison to the other plane, is it?

[stray stump]

the origin of this plane is (0,0,5/6)

[shadow lava]

oh the constant vector IS the origin

do we call that a constant vector? or it's still z vector

i don't see z scalar, only x and y

[stray stump]

it's affine plane actually

[shadow lava]

oh OK

[stray stump]

why do you need z? it's plane (of dimension 2)

[shadow lava]

wait but wouldn't it be x(1,0,9/6) + y(0,1,-4/6) + (0,0,5/6) = 0 for the plane equation
so - (0,0,5/6) is the origin?

i thought plane needs = sign

[stray stump]

plane equation is this

[shadow lava]

oh right

[stray stump]

we just did the above to determine the origin and director vectors

[shadow lava]

fascinating

i dont think we have learned this yet

"affine plane"

but good to know, thanks

appreciate the help!

[stray stump]

You're more than welcome

[shadow lava]

i'll try to remember those 3 axioms for subspace

i thought it was 8 in total

but it gets condensed to 3?

or maybe i'm thinking 8 for something else

also @stray stump is this question asking if w is a subspace of v1, v2, v3?

just another way to word it?

or is it asking about linear dependence / independence

if it's subspace, yes
(0,0,0) origin for all 3

if it's linear dependence i can't spot anything that would make it so

oh wait, the curly brackets mean set {}

so maybe it's asking if w is in the set for v1,v2,v3
subset?

[stray stump]

8 are the axioms of vector space, for subspace you have to verify only 3. It's an efficient way to verify a set is vector space by verifying if it's subspace of a larger vector space because any subspace is vector space.

[stray stump]

I think it's asking if w is in the subspace generated by v1 v2 v3

[stray stump]

i don't think it's this cuz this is trivially no

check if w = a * v1 + b * v2 + c * v3 by determining values of a,b,c

[shadow lava]

so I would use augmented matrix?

w is the 4th column?

[pseudo swift]

they're not talking about span anywhere here

[shadow lava]

and RREF

[pseudo swift]

they are asking the dumb question

[shadow lava]

one sec lemme post full screen

[stray stump]

really

[pseudo swift]

as avid said

{v1, v2, v3} is just the set

[stray stump]

oh okay

Span is in 3rd question

that's why i said "I think"

[shadow lava]

i just know {} notation as "set" that's why I guessed that

[stray stump]

cuz the notations change from place to another

[shadow lava]

so the obvious answer for #1 is no

i'm just looking at those numbers, 6,8,2

i don't see a 6 anywhere in v1, v2, v3 so that's a giveaway right

[pseudo swift]

sure

[shadow lava]

but what if v1, v2, v3 had 6,8,2 but in different vectors

would it still count or it has to be exact same vector

to be considered in the same set

[pseudo swift]

if w is not one of v1 v2 v3, it's not in the set

the order matters

[shadow lava]

i believe it's asking this but just want to make sure
{[1,-1,4], [2,-1,8], [4,-4,20]}
is [6,8,2] in this set

so there are 3 entries in the set

[pseudo swift]

yes

[shadow lava]

is w one of those entries

[pseudo swift]

indeed that's what they're asking

[shadow lava]

OK

kinda like arrays and array index in programming

const myArr = [1,2,3]
console.log(myArr[0])
// 1

or in this case
const myArr = [[1,-1,4], [2,-1,8], [4,-4,20]]
console.log(myArr[0])
// [1,-1,4]

same idea here with the vector set question I guess, it's like an if statement to check if one entry is the same as w

[pseudo swift]

it's a decent analogy yes

[shadow lava]

I mean, it could be Inf right

we have 0,0,0

[pseudo swift]

you said the answer before

[shadow lava]

I did?

[pseudo swift]

^

we're not talking about a span here

3 vectors is 3 vectors

[shadow lava]

OH

right right

set is real entries only in the provided vectors

span is all possible vectors, (based on 3 axioms: origin, scalars, and/or addition)

space is like the grid system itself (infinite)

[pseudo swift]

all possible linear combinations yada yada

[shadow lava]

do I have that correct? with those 3 definitions

or do I have vector space confused with "basis"?

[pseudo swift]

I'm not sure what you mean by grid system here

[shadow lava]

coordinate system

x,y,z plane

or x,y plane

whatever the dimension happens to be

also not sure if "plane" is the correct word to use there

but i think so

oh right, i remember basis now

[pseudo swift]

makes me think more of a lattice when you say that but whatever https://en.wikipedia.org/wiki/Lattice_(group)

but I got you

yeah the coordinate system is just a way of describing the vectors in your space

[shadow lava]

basis is like saying my grid system is 1mm each interval
but another person uses 10cm each interval for their grid system
both draw the same vector on their grid system, but because each interval length is different we say they have a different basis

[pseudo swift]

you have the same vector but you give them two different names essentially

cause you're not using the same units or whatever

[shadow lava]

yeah, the exact same vector can be written completely differently depending on the basis

and if you draw the vector, it may look different too

but if you match the basis (not sure what the term for this is called), it will look like the exact same vector

3B1B video on basis helped me to understand this

[pseudo swift]

yeah I figured you had this video in mind

[shadow lava]

i guess i kinda look at it like metric vs imperial measurement

or celcius vs fahrenheit

if you look on a box for cooking instructions, both will give the same temp, or same measurement

but using a different basis

so if you are cooking in USA it will use one basis
if you are cooking anywhere else in the world it will use the other basis

(dumb US system, why they gotta be different like that)

[shadow lava]

just to make sure, do I have the right kinda idea there with "basis" when it comes to vectors?

how a grid is drawn can look completely different, basis helps to find common ground

[pseudo swift]

yeah for bases it's fine

[shadow lava]

OK great

[pseudo swift]

just don't confuse a basis and the space it describes

[shadow lava]

i think i'm finally starting to understand these terms

at first i thought they were redundant but now i see why each word is needed

the only one to me that feels a bit redundant is space and dimension

[pseudo swift]

in what way

[shadow lava]

in the same way that perpendicular and orthoganal basically refer to the same thing

a 90 degree angle

vector space is infinite, in n directions

vector dimension is infinite, in n directions

but I think dimension is just a number right? that's the big different

when we say R3 it has dimension of 3

[pseudo swift]

it's just a summary of the space yes

a vector space, well it's a set of objects satisfying some axioms

[shadow lava]

or would you say the space is R3
"R3" is the dimension. (we include the R?)

[pseudo swift]

the dimension is a "number" that characterizes the space

[shadow lava]

oh.. is this referring to the 8 axioms?

[pseudo swift]

as in, "in R^3, I can describe all the elements of the space w/ 3 parameters, so the dimension is 3"

[pseudo swift]

yes

[shadow lava]

but we don't need to memorize those 8 axioms right

not very useful, the 3 axioms for vector span are more useful?

[pseudo swift]

well most of the time, you aren't gonna prove something is a vector space from scratch yea

[shadow lava]

OK lol

[pseudo swift]

it's nice to have them in mind

[shadow lava]

perfect, I think I finally am starting to get it

these terms have been a pain point for me, they start to blur into one another

but I think I can finally start to see clearly now

[pseudo swift]

it takes quite a bit of time for the blur to go away yea

[shadow lava]

btw, how would you describe the difference between perpendicular and orthogonal? or is it not that important to know their difference

[pseudo swift]

they're synonyms essentially

[shadow lava]

not exactly tho

orthogonal refers to vectors only i thought

[pseudo swift]

idk

it's a question of taste really

[shadow lava]

lol OK, i don't know why math comes up with a brand new word like that sometimes

According to ChatGPT:

`Perpendicular:

Think about two lines meeting at a right angle, like the corners of a square. That's perpendicularity in a nutshell. In geometry, when lines or line segments meet to form a 90-degree angle, they're called perpendicular.
Orthogonal:

This term is like a big brother to "perpendicular". It's used not only for lines but also for things like vectors (which are like arrows in space). In math, when you hear "orthogonal", think "right angles", just like perpendicular. But instead of being limited to 2D shapes like lines, it works in all kinds of spaces, even ones with more dimensions.
Why "Orthogonal" Exists:

"Orthogonal" is a fancy term that mathematicians use because it's more general. It lets them talk about right angles not just in flat, 2D space but also in 3D, 4D, or even more complicated spaces. It's like saying, "Hey, we're not just talking about squares here; we're talking about all sorts of shapes and spaces where right angles matter." So, "orthogonal" is like the superhero version of "perpendicular", ready to tackle right angles in any situation!`

[pseudo swift]

yeah that's pretty much my thoughts here

[shadow lava]

Orthogonal is like a "superhero version" of perpendicular lol

[pseudo swift]

perpendicular refers to the geometrical notion we're used to

but orthogonality depends on the inner product you use along w/ your space

if you use the typical inner product, then the two notions are the same

[shadow lava]

I am so blown away by ChatGPT, every day it genuinely impresses me. And I've been using it daily pretty much. For code it's such a time saver too.
I know it's still not great for solving math problems, but for abstract stuff it's really doing such a great job 95% of the time

[pseudo swift]

yeah it has a nice idea, sad it doesn't talk about inner products in the answer tho

that's what's behind the "superhero" thing I guess

[shadow lava]

I have heard some opinions on here that AI cannot solve math problems, and never will be able to, but I'm also not ruling it out, because AI is still in the newborn stages here

[shadow lava]

I think it did to begin with but answer was too long so I said "explain easily" and it reworded it

[pseudo swift]

ah ok

[shadow lava]

haha, less reading, more understanding

on the basic level to begin with

it does a great job with that too!

"explain like I'm 5" or "explain like I'm 16" tend to give very good results for both

[pseudo swift]

I haven't tried gpt very much

for song parodies it's decent

for writing assembly, it's up to a coin flip whether the code will work or not

that's the extent to which I tested it

we're getting quite far from your exercise lol

[shadow lava]

lol ya

i think one day we will have XP bars like the sims

to level up our learning

a nice visual way to track where we are at

in each subject

with custom tailored lessons to level up your skill level

anyways, until that day comes, head to the grindstone

tyvm!

.close

[obtuse pebble]

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Closed by @shadow lava

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[obtuse pebble]

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[burnt thunder]

for 4a

[burnt thunder]

i dont get the lim y

in answers

how is is positive infinity

and negative for other

[warm skiff]

if x is negative, but it's absolute value is less than 5, then the numerator would be negative, while the denominator will be negative, which makes the fraction negative

but if x is negative and it's absolute value is greater than 5, then both the numerator and denominator will be negative, which will cancel to make the whole fraction positive

[burnt thunder]

where did abs value come from

[warm skiff]

it's just to say that even if x is negative, it's still possible for the denominator to be positive because 5 is being added to it

[burnt thunder]

i dont get it

[warm skiff]

you understand why there would be an asymptote at x = -5 right?

[burnt thunder]

yes

[warm skiff]

so lets look at x approaching -5 from the right

if x was -4 for example

[burnt thunder]

positive

[warm skiff]

the numerator would be -4, and and denominator would be (-4 + 5) which would work out to be -4/1

[burnt thunder]

yes

[warm skiff]

and if it approached 5 from the left side, like if x = -6 for example

[burnt thunder]

negative

[warm skiff]

then the numerator would be -6 and the denominator would be (-6+5) which works out to -6/-1

[burnt thunder]

yes

[warm skiff]

and the negatives would cancel to make the fraction positive

[burnt thunder]

yes

[warm skiff]

that's why approaching from the left goes to +infinity

[burnt thunder]

left is negative

isnt it

no

[warm skiff]

left of -5 the graph is positive, right of -5 the graph is negtive

[burnt thunder]

wait

so why does y

said

0

for -5 from left side

[burnt thunder]

i need to understand without sketch

first

sob

[warm skiff]

because when x is to the left of -5, both the numerator and denominator are negative, which causes the function to be positive

once x crosses over to the right of -5, only the numerator is negative, which keeps the function itself negative

[burnt thunder]

wait <0 means x is leess then 0?

[dim glen]

have you study limit

[burnt thunder]

yes

sketching comes later

like next lesson

[dim glen]

actually these are very basic questions

[burnt thunder]

sorp

im not very good

at math

[dim glen]

okay i send you some resource

[burnt thunder]

help

[warm skiff]

is there a reason you can't sketch it? that would be the easiest way for me to understand if i wasn't getting it

[burnt thunder]

it comes later

also

i suck

at sketching

💀

[warm skiff]

so this part of the course you're specifically not meant to sketch, or have you just not learned it yet?

[burnt thunder]

im not 4.3 of calculus

curve sketching comes 4.5

or smthn

yes

i did sketching

in my other math course

i forgot

how to

tho

ngl

[warm skiff]

that's fine

so what i would do is just test a bunch of x values

so start with x = -1

if you plug in -1 for x in that function, what do you get

[burnt thunder]

-1/4

[warm skiff]

now try something like x = -3

[burnt thunder]

-3/2

[warm skiff]

one more at x = -4

[burnt thunder]

-4

[warm skiff]

so as you get closer to -5 coming from the right side, the function spits out larger and larger negative numbers

[burnt thunder]

yrs

[warm skiff]

like if you plugged in x = -4.999 the function spits out -4999

[burnt thunder]

yea

[warm skiff]

so that tells us as x approaches -5 from the right, y approaches - infinity

[burnt thunder]

yes

[burnt thunder]

this is limits

tight

not sketching

[warm skiff]

that matches with what it said the correct answer was up there doesn't it?

[burnt thunder]

yea it does

[dim glen]

if you study limit then you automatically learned how sketch graph

[warm skiff]

you can do the same thing for the left side of -5. start at -8 or something and get closer to -5. you'll see the function spits out larger and larger positive values

[warm skiff]

this is also true

[burnt thunder]

yeyes

[warm skiff]

but a lot of it just comes down to pluggin in points and seeing what it spits out

[dim glen]

ei first you have to clear your limits concepts then you study continuity and diffrentiabilty where limits concept used then you study application of derivative at aod you sketch and solve questions quickly

[burnt thunder]

yes

okk

[obtuse pebble]

@burnt thunder Has your question been resolved?

[obtuse pebble]

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• Show your work, and if possible, explain where you are stuck.
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[tight raven]

Is there someone can solve my problem? i cant do that, plz i need help

[obtuse pebble]

Please don't occupy multiple help channels.

@tight raven Has your question been resolved?

[tired shell]

doing it for a general tetrahedron is just a bunch of ugly matrix math, but for this special one you have this https://mathworld.wolfram.com/IsoscelesTetrahedron.html

[obtuse pebble]

@tight raven Has your question been resolved?

[obtuse pebble]

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• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
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[warped zinc]

d) –x represents a negative value

Why is this false?

[worn yoke]

what if x = -4? is -x positive or negative?

[tawny copper]

positive

[warped zinc]

what would the formula of the inverse of an absolute value function be?

[worn yoke]

a function must be one-to-one in order to have an inverse

[warped zinc]

wdym?

thx for the help bntw

[worn yoke]

in other words, each input x must result in a unique output f(x)

[warped zinc]

ye

[worn yoke]

if two different inputs result in the same output, it can't have an inverse

[warped zinc]

oh okay cool

so a circle for ex doesnt have an inverse

[worn yoke]

well a circle is not a function in the first place

but if we just took the top half as a semicircle, that would be a function without an inverse

[warped zinc]

but a parabola has an inverse?

[worn yoke]

a parabola does not have an inverse for all x

[warped zinc]

oh its inverse is the root

function

[worn yoke]

however, we can create an inverse that only works for part of the domain

[warped zinc]

okay thanks, you would restrict the domain of half the original parabola?

from the vertex

[worn yoke]

yes

[warped zinc]

'

for ex, in this example

do i always have to put plus or minus sq root

[worn yoke]

well that's saying that there are really two different inverse functions

which are valid for different parts of the parabola

[warped zinc]

https://media.discordapp.net/attachments/773534888422670346/1223738779534360617/IMG_2094.png?ex=661af28d&is=66087d8d&hm=063b900c39846f92ec2289c23dd868f7aa1486680b2fd44ffc4586f0edf45215&=&format=webp&quality=lossless

in #2, i just need to set x to -1 right?

just for clairificaiton

since x+2=1

x=-1

[worn yoke]

that works, yes

[warped zinc]

olkay coolio

thanks for the help

in 1c) the -7 rep. the y value?

[worn yoke]

if we graph it on the x-y plane we usually represent it as y = f(x), yes

[warped zinc]

i mean its kindadd weird since

f(x) is like a placeholder for y

so when its y=f(x)

its just saying its y=y

[worn yoke]

a function is a thing that takes in an input and returns an output

the input is some variable (which we call x), and the output for a function f is called f(x)

y is just some other variable, which by convention we often set equal to the output of a given function

if we wanted to, we could swap the variables around, and instead use y as the input variable, and x as the output variable, writing x = f(y). which means that f takes in the variable y as input, and we set the output equal to the variable x. this is less common but still follows the same rules

[warped zinc]

in this case

would x be the independent variable?

[worn yoke]

the independent variable is the one we use as input, and the dependent variable is the one we set equal to the output. because the output of a function depends on the input

[warped zinc]

ohh

so its the other way around>?

[worn yoke]

in this case, yes

[warped zinc]

okay so normally

in y=f(x)

x is the independent

y is the ddepenet

because y depends on x

[worn yoke]

yes, y depends on x because we are setting it equal to the the output of a function that takes x as input

[warped zinc]

these are the harder questions

i mean for 23

i would find the inverse of f(x)

then insert 1 into it

for x

[modest ingot]

sounds like a plan

[warped zinc]

how would you isolate y thio

x+3=y^2+3y

[modest ingot]

complete the squares

[warped zinc]

lemme try

ohh

soi then its

x+3=(y+3/2)^2-9/4

[modest ingot]

looks okay

actually you can do

,, f^{-1} (1) = y \implies f(y) = 1

[warm shale]

nyxie9151

[modest ingot]

no need to find inverses

[warped zinc]

im confused

why did the bot post that

[modest ingot]

i typed it

the bot rendered what i typed

[warped zinc]

oh okay thanks

i get it the way i did it

[modest ingot]

i’m telling u that instead of finding the inverse

,, f^{-1} (x) = \text{something}

[warm shale]

nyxie9151

[modest ingot]

you can take apply f to both sides

,, f(f^{-1} (x))= f(\text{something})

[warm shale]

nyxie9151

[modest ingot]

[warped zinc]

would findding the inverse work tho

?

i feerl thats the only way i understand

[modest ingot]

so you have $x = f(\text{something})$

[warm shale]

nyxie9151

[modest ingot]

if you like to work harder then sure

i’m sure this is what they want you to do

lol

and it’s also much simpler

[warped zinc]

ok ill try to understand

[modest ingot]

okay here’s an abstracted definition

for an inverse u switch the input and output right?

[warped zinc]

yes

[modest ingot]

okay so if the input of (f inverse) is “x” and the output is “something”

the input of (f) should be “something” and the output should be “x”, right?

[warped zinc]

isnt the input y

when its inverse

x=f(y)

[modest ingot]

nah we’re not worried about semantics here

read what i said again

i added some quotation marks

yeah try reading now

i just switched the input and output

[warped zinc]

okay

yes

[modest ingot]

which u agreed was the notion

[warped zinc]

i kinda understand

switch x nd y

[modest ingot]

okay to say that again

f(x) = y

[warped zinc]

y=f(x)

[modest ingot]

so f inverse will take as input y now and output x

,, f(x) = y

[warm shale]

nyxie9151

[modest ingot]

do you see that x is the input of the function f?

and the output is y right?

[warped zinc]

f(y) = x

y is the input

x is the output

[modest ingot]

,, f^{-1} (y) = x

[warped zinc]

yes

[warm shale]

nyxie9151

[modest ingot]

f inverse however takes y as input and gives x as output

[warped zinc]

yesyes

okay

[modest ingot]

okay this isn’t correct

we’re talking about f and it’s inverse

anyway if i read what i said

it should’ve been clear

now

[warped zinc]

okay so the inverse of y=f(x)

is x=f^-1(y)

[modest ingot]

no you’re using f twice

f inverse is a “new” function such that it’s inputs and outputs are switched

[warped zinc]

yes

[modest ingot]

so your statement is wrong

u just don’t switch inputs and outputs on the same function

if you have f(x) = y

[warped zinc]

now is it right>

?

[modest ingot]

then f inverse is a new function that has a nice “property” which allows switches in inputs and outputs

[warped zinc]

i needd to have the -1

[modest ingot]

just read the last few sentences i mentioned

if that makes sense to you then sure

[warped zinc]

yes i get it thanks

[modest ingot]

u typing it correctly doesn’t prove much cuz i already did the typing

okay cool

[warped zinc]

it needs to be a different function

[modest ingot]

so joe for your question

now*

[warped zinc]

its not the same function

[modest ingot]

sure better way to think about it

[warped zinc]

yes

[modest ingot]

you want to find g(1) which is basically finding f^(-1) of 1

,, g(1) = f^{-1} (1)

[warm shale]

nyxie9151

[modest ingot]

so our task here is to find f inverse of 1

[warped zinc]

yes

i understand

[modest ingot]

,, f^{-1} (1) = \text{something}

[warm shale]

nyxie9151

[warped zinc]

yes that is g(x)

[modest ingot]

,, f(something) = 1

[warm shale]

nyxie9151

[modest ingot]

u apply the inverse of f inverse to both sides

incidentally, the inverse of f inverse is just f lol

just like the inverse of f was f inverse

[warped zinc]

yes

but the other side

gets changed?

[modest ingot]

what side?

[warped zinc]

right

[modest ingot]

well the inverse function swaps its input and output

the inverse function of f inverse is f

[modest ingot]

once u have this then just swap the input and output

initially u have this^

what’s the inverse of f^(-1)

it’s f

and the input and outputs swap in the inverse function so

[warped zinc]

something is the input right?

[modest ingot]

yea

[warped zinc]

yea its

[modest ingot]

so what’s something?

[warped zinc]

f(y^2+3y-3)=1

[modest ingot]

no

[warped zinc]

fuck

[modest ingot]

f(x) = x^2 + 3x -3

notice x is the input and the function is about x

[warped zinc]

wheres the 1?

oh nvm

okay

go on

[modest ingot]

so f(something) = something^2 + 3(something) -3

so f(something) = 1 roughly translates to

something^2 + 3(something) -3 = 1

[warped zinc]

whats something

[modest ingot]

lol that’s what u have to find

basically solve x^2 -3x -3 = 1

,w solve x^2 +3x -3 = 1

[warped zinc]

its +3x

[modest ingot]

it indeed is

anyway we don’t care about the right answer

i’m imparting the main point to you

[warped zinc]

,w solve x^2 +3x -3 = 1

[warm shale]

Results provided by WolframAlpha

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Requested by regormik

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Requested by nyxie9151

[warped zinc]

bruh how u so smart

okay so to recap

g(x) is just the inverse of f(x)

[modest ingot]

no

lol

here’s the recap

g(x) = f^(-1) (x)

our question was to find g(1) so basically f^(-1) (1)

,, f^{-1} (1) = s \implies f(s) = 1

[warm shale]

nyxie9151

[modest ingot]

we use the idea that the inverse of f inverse is f

[warped zinc]

wait

[warped zinc]

the x is linked?

[modest ingot]

and for this new inverse function we swap the inputs and outputs

[modest ingot]

it always was?

[warped zinc]

i thought they were different function

functions

[modest ingot]

no

to find g(5) you’d want to find f^(-1) (5)

it’s a relation they gave u

[modest ingot]

finally f(s) = 1 gave u s= -4 or s= 1

[warped zinc]

okay ill try to readd over what you said

thanks

[modest ingot]

which are the two possible values of f inverse of 1

[warped zinc]

for all the help

[modest ingot]

hence the two possible values of g(1)

hmm sure

[warped zinc]

i dont get why we need to find the inverse of f^-1

[modest ingot]

i don’t see what’s there to read over

[modest ingot]

it’s just like an algebra step

u apply the inverse function to both sides

[warped zinc]

but its not like you can apply it to both sides

its appliedd at once to all of them

[modest ingot]

what

okay look here

,, x = 3

[warm shale]

nyxie9151

[modest ingot]

i can apply f to both sides

,,f(x) = f(3)

[warm shale]

nyxie9151

[warped zinc]

yes

[modest ingot]

that’s exactly what i did

[warped zinc]

but the inverse

?

[modest ingot]

i applied the inverse function to both sides

the inverse of f inverse is f

[warped zinc]

so its

[modest ingot]

by definition, you have that

,, f(f^{-1} (x)) = x

[warm shale]

nyxie9151

[modest ingot]

if you apply the inverse function to a function

[warped zinc]

,, f^-1(1)=x^2+3x-3

[warm shale]

Evian

[modest ingot]

wrong

[warped zinc]

YO

[modest ingot]

f inverse of (1) is some number

not an expression

[warped zinc]

so

,,f-1(1)=something

[warm shale]

Evian

[modest ingot]

a number yes

hopefully you see that

[warped zinc]

then aplpy f-1 on both sides?

[modest ingot]

you apply the inverse function to both sides

[warped zinc]

,, f(1)=f-1(something)

[modest ingot]

Inverse function

[warm shale]

Evian

[modest ingot]

,, f^{-1} (1)=something

[warm shale]

nyxie9151

[modest ingot]

you apply the inverse function of f inverse

which is f

[warped zinc]

yes

and yuou do the same to the other side

[modest ingot]

,, \underbrace{f(f^{-1} (1) )}_ 1 = f(something) \implies 1= f(something)

[warped zinc]

,, f(something)= 1

[warm shale]

Evian

nyxie9151

[modest ingot]

the left side follows from this definition

cool?

[warped zinc]

😭

i hatemath

ill just use the quaddratic forumla 🤡

[modest ingot]

bruh

look