help-17

[vast shale]

oh it’s negative 3

[drifting jackal]

Well not exactly

[vast shale]

Don’t u like take x intercept into place

Sorry I meant y

And make the x zero

Then divide?

[drifting jackal]

Knowing that it's going to be (0, ___) form, what options do you have to left?

[vast shale]

I’m pretty sure u take your intercept meaning x becomes zero

[drifting jackal]

Look at the option choices

[vast shale]

So it will be 2(0) minus 4y equals to 12

[drifting jackal]

You have 4, you can eliminate some choices because you know that the proper form is going to be (0, ___)

[vast shale]

Oh right

A and d

Since x intercept is zero

Right.?

[drifting jackal]

Are you keeping or eliminating?

[vast shale]

Eliminating

[drifting jackal]

Good

That means you have B and C left

[vast shale]

Yup

[drifting jackal]

You have the equation 2x - 4y = 12

Plug in each coordinate and simplify

[vast shale]

Mhm it’s negative 3

[drifting jackal]

So what answer choice is it?

[vast shale]

B

[drifting jackal]

Good

Now for 5, you have the coordinates $(\pm 3, 0)$

[twin meteor]

CaptainNova22

[drifting jackal]

That means if you plugged in 3, you should result in a y value of 0, correct?

[vast shale]

Mhm

So u can eliminate b by simply looking

C as well

[drifting jackal]

Exactly

[vast shale]

Hm I solved d

It’s d

ORETTY sure

U equal it to zero then u cross multiply it?

Or am I panicking sorry

[drifting jackal]

You can do that but you can also plug in 3 for x

And simplify and see what you get

[vast shale]

Mhm it’s positive negative 3 meaning d is the correct answer perhaps

[drifting jackal]

Well what happened when you plugged in 3?

[vast shale]

-3-9/(-3)

Correct?

[drifting jackal]

That's -3

I was asking about 3

[vast shale]

Oh

If it’s 3 only one answer comes out

And it’s -3

Right?

[drifting jackal]

I'm asking you for choice D, if you plugged in 3, what do you get?

[vast shale]

Yea 3-9/(3)

[drifting jackal]

And that simplifies to?

[vast shale]

0

Zero

[drifting jackal]

Yes, and do the same with -3

[vast shale]

0?

[drifting jackal]

Yes

[vast shale]

Is that for y axis?

[drifting jackal]

What do you mean

[vast shale]

Wait Nthn I’m dumb nvm

Yup it’s zero

[drifting jackal]

So the answer is?

[vast shale]

Negative plaice 3

Positive *

[drifting jackal]

No, the answer choice

[vast shale]

Oh d

[drifting jackal]

Yes

[vast shale]

TYSM

[drifting jackal]

No problem

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[reef oriole]

How to reverse this function
f(x) = x³ + 3x² -3x + √x

[vocal sleet]

@reef oriole Has your question been resolved?

[vocal sleet]

@reef oriole Has your question been resolved?

[hushed pewter]

!original

[vocal sleet]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[reef oriole]

Its just reversing function

For example if I want to reverse this
f(x) = x - 1
It will be like this
f‐¹(x) = x + 1

[hushed pewter]

It can't be done

[reef oriole]

But perplexity did it

[hushed pewter]

AI tells lies

!nogpt

[vocal sleet]

Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).

[reef oriole]

But it was working

[hushed pewter]

It was not

[reef oriole]

I tested the reversed function

[hushed pewter]

If you already have it, then why are you asking how to do it then?

[reef oriole]

I want know the way

It's this

f‐1(x) = (x-√x)^1/3

I want know the way

Ow

It's not true

Yea u was right

[hushed pewter]

Yeah

I know I was right

[reef oriole]

How I can know it can't be done

[hushed pewter]

AI puts out garbage

[hushed pewter]

Because solving it would involve solving a sixth order polynomial, which can't be done by normal means

[reef oriole]

Alr thx

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[velvet zephyr]

thought*

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[oak magnet]

Yeah, and btw i was talking about all the divisor of 24, where you add the exponents of the prime factorisation +1 at each one, ive just forgot to write D(24)

[oak magnet]

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[velvet zephyr]

ok

[vast shale]

Can someone help me

Ik how to do it but they keep giving me more problems

<@&286206848099549185>

Hello

Someone help

Nvm

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[last spindle]

do u need help?

[vast shale]

No

Not anymore

I solved it

[velvet zephyr]

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[sharp lynx]

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[vast shale]

Why is not -3?

[azure moat]

I didn't go over all the work in your picture but found that when you multiplied the two sets of parenthesis, you should have got -x^2 -6x -9 where you got +6x, this likely messed up the rest of the problem

[vast shale]

oh right, thanks

Yes, I know get that answer

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[sharp comet]

how do I do this without using a graph

[surreal basin]

you can use distance formula on pairs of points then add them together

[sharp comet]

hm

[surreal basin]

(distance formula on A and B) + (distance formula on A and C) + (distance formula on B and C)

[sharp comet]

that is alot of distance formulas

[sharp comet]

is that more efficient than the box method on the graph

[surreal basin]

the box method is still the same calculation

consider where youre getting those lengths from

if youre skipping graphing, the triangle that forms when you connect one point to another cant be seen

but you still know its lengths

you can see here the triangle is 4 across and 4 high

from -2 to 2
from 1 to 5

[sharp comet]

o distance formula makes sense

your just finding the length of 3 lines

[surreal basin]

yea so its equally efficient as the box method

[sharp comet]

ok

thx

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[tacit axle]

how am i supposed to know what the base/exponent for 512 is?

[heavy yoke]

try the first few powers of 8

[tacit axle]

alright

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[tough owl]

how would you write S = √30df as a faction exponent?

[civic otter]

What is df?

$$\sqrt{30} = 30^\frac{1}{2}$$

[twin meteor]

Alberto Z.

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[tough owl]

Question said to rewrite the radical into fraction exponent form but I'm not sure where I should put the exponents in it

would I just tack on df along with the 30? making 30df^1/2

or would I do something else with them

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[gilded peak]

Finding limits with graphs os something I need the most practice with

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[limpid garnet]

what is "formality" in mathematics

[limpid garnet]

and how do i learn more about it

[oak magnet]

Its the method of writing proof, and its an important thing, students (and more) are not neccessarly understanding the statement of a theorem, theyre must be a demonstration with a rigorous vision of the question.
Rather, developing their intuition for the matter, which is severely deficient when they start the course, should be a priority.

Rigor is used for checking the correctness of the finished product, not for creating it in the first place

[oak magnet]

http://www.refsmmat.com/articles/unreasonable-math.html

[limpid garnet]

I thank you, but formality is subject to ambiguity

this one notion is likely a restriction of the definition as you are describing it explicitly as a method for proof writing, not to mention rigor is also subjecy to ambiguity

subject*

for instance

a formal construction of a set is not a proof.

unless i am mistaken

[oak magnet]

<< Don’t worry about sets. We have them formalized. There do exist interesting questions at the extreme edges of set theory, but they have very little impact on the rest of math, and they won’t be helped by a “formal definition” of what a set is. >>
I got this from an online discussion from quora

[limpid garnet]

again, your use of formal here contradicts your explanation

well

regurgitated explanation

@blissful sentinel please pardon me for the ping, can you please help me here?

[oak magnet]

What is yours definition of formal ?

[blissful sentinel]

why me specifically 😭

[limpid garnet]

If i was familiar with a reliable; structured, extended notion of formality, i wouldn't be asking for a definition that may be within that same scope

what i was saying; is that the notion of formality is ambiguous

[limpid garnet]

you have more experience and are often present in help channels

[blissful sentinel]

Please don't ping specific helpers unless they've given you permission to, it's against the rules

https://en.wikipedia.org/wiki/Logic#Formal_logic This link may be useful to you

[sacred remnant]

if you are interested in learning logic I recomend "For all x: Calgary"

[limpid garnet]

sorry (yes i know that i've pinged you again.)

[limpid garnet]

okay, but in respect to sets; what would "formal construction of a set" mean

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[limpid garnet]

.reopen

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✅

[limpid garnet]

i've found a definition, however; i want to verify my understanding

would it be correct to consider the axioms defining properties of elements of a field (for instance, the reals) as an axiomatic system, then a formal system?

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@limpid garnet Has your question been resolved?

[limpid garnet]

<@&286206848099549185>

[woven ember]

why am i meintioned here

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[vast shale]

I don't know how to solve this.

If S is the solution set of the inequality
Consider the following propositions:
I. The sum of integer elements of S is 7.
II. s=[1;√2]
III. (2√2+1) € S
Determine from the given propositions which They are true.

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[next burrow]

my calculator got 33.3 but the website says its not true

idk what i did wrong

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@next burrow Has your question been resolved?

[dark kiln]

that's the impossible one

[next burrow]

wdym

[dark kiln]

i mean you can't know

and they let you write impossible

[next burrow]

i dont understand

[dark kiln]

it's not a hard concept, "there's 939240 candies in the warehouse, 40080 are pineapple, what's the % of blueberry candies"

[next burrow]

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[teal ore]

What is the question asking?

[hushed pewter]

Maximize the area of the garden given the amount of fencing you bought

[teal ore]

Anyway you can help me get started?

Rattling my brain on the first step

[pallid forge]

well you're worrying about two things right now, perimeter and area, both can be represented using x's and y's. I would start there

[teal ore]

Area would be 4y(2x) so 8xy

For the entire farm

[pallid forge]

right

[teal ore]

Perimeter would be 8x+14y=126

Simplified to x=(63-7y)/4 for solving x

[pallid forge]

is it not 7x+9y?

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[mystic vault]

chat

[mystic vault]

I need help

How the fuck does the e^10n not cancel out in the denominator????

[modest coral]

What is $\frac{x^n}{x^{n+1}}$?

[twin meteor]

ScapeProf

[modest coral]

(Here x=e^(10n))

[mystic vault]

ohhh fuck

i saw it right after lol

forgot to distribute

im cooked in linear algebra next semester then lmfao

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[short pilot]

Yo

[mystic vault]

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[little gulch]

is that always True ?

[rugged orchid]

No

Well

Is that P a power set or a probability

[brisk moss]

power set surely

[rugged orchid]

Then idk off the top of my head

[little gulch]

power set

its easy but the problem is that why that is true and this

false

[oak magnet]

Un ptit dessin aiderais pas mal pour voir si c'est potentiellement vrai ou faux

[rugged orchid]

That’s false because the left side doesn’t contain any subsets of A ∪ B that has things from A and B in them

Whereas the right side has all of those

[little gulch]

la premiere avec l intersection est toujours vrai mais l autre est fausse pourquoi

[oak magnet]

[little gulch]

if we say A = {a} and B={b} so P(A u B) = { o , {a}, {b}, AuB } and P(A)uP(B) = {o , A,B}

[rugged orchid]

Yeah

[little gulch]

so its false

[rugged orchid]

It doesn’t have the subset containing both the things in A and the things in B

Which in your example is {a, b}

[little gulch]

but this true right ?

[little gulch]

yes

[rugged orchid]

I don’t know off the top of my head but it sounds true

[little gulch]

yes i prove it

and that s make me confused bec one FALSE "U" but the other one TRUE

[little gulch]

ty for your help

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[formal obsidian]

Is my proof valid?

[formal obsidian]

please explain

pardon the missing conclusion

huh?

does it matter?

jeez it is very much readable

huh?

<@&268886789983436800>

you never did in the first place-

[autumn trail]

Go do something else.

[civic otter]

This is very readable, actually 😅
I don't know what standards you're used to @vital galleon

[autumn trail]

Please don't drag the issue out.

[hushed pewter]

Looks fine for non rigorous limit proof

[autumn trail]

Last warning. Go do something else.

[formal obsidian]

yup, basically ignoring ep-del

[hushed pewter]

!redir

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This channel is only for on-topic discussion. Please take casual conversation to #discussion or #chill.

[formal obsidian]

anyways thanks!

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[grizzled forge]

Need help w this

[scarlet forge]

It's a bit easy.

20 degrees, then it falls to 15. Then that's 5 degree. Then it be 10 degrees the next 5 minute.

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@grizzled forge Has your question been resolved?

[short pilot]

@grizzled forge

Use average form of Newton's cooling law

I don't remember the formula right now

@grizzled forge

[grizzled forge]

Umm we’re not meant to use that formula I think

[short pilot]

Noooo

;-;

[grizzled forge]

This is rates of change hw lol

[short pilot]

Bro he asked temperature right

[grizzled forge]

Umm it’s 11 degrees

That’s what the ans says

[short pilot]

Listen you don't average form of Newton's cooling law

Wait imma give you the formula

[scarlet forge]

ans?

[grizzled forge]

Answer

[scarlet forge]

Hold on, which grade is this.

[grizzled forge]

Year 11 maths extension 1 Australia

[scarlet forge]

It kind of relates to the mass of water.

[grizzled forge]

You prolly foreign lol

[scarlet forge]

Nah, australia as well.

[grizzled forge]

Ohk lol

[scarlet forge]

This must be heisi.

[short pilot]

delta T/delta t=K[(Tf+Ti)/2 -T0]

Apply this

Delta T change is temp. Of object

delta t time in which it changes

K a constant

Tf+Ti sum of initial and final temp of object

T0 temp of surrounding

Apply this

[grizzled forge]

Umm I don’t mean anything bad by this but I kinda wanna solve the problem using like rates of change yk? It’s just that I wanna like practice it like yk

[short pilot]

Listen first apply for initial conditions and find k

Then apply for 2nd case you will get 11

[short pilot]

Bro but how

How can it be determined simple rate

[grizzled forge]

Wait actually one sec

[grizzled forge]

That formula near the bottom, is that what u were talking abt?

[short pilot]

Bruh it's maths

Question I though phy

Do it bro then do it by this

[grizzled forge]

I solved it 😁

You don’t need to use newtons cooling for this

My ans was actually 10.83 but the actual ans was rounded to 11

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[surreal socket]

Hi guys, I have something like in the image, but I only have access to the edges (PointA, PointB) and now I want to know, which edges belong to a cell. How could I calculate that?

What I was thinking so far, go over every edge, then take all the connected edges to PointA and PointB and take the edge with the smallest/biggers angle and continue until reaching the first Edge again. And in the end eliminate all duplicates

[brisk moss]

like, you're given a dot and want to know the edges that go around that dot's cell?

[surreal socket]

yes

🙂

but i don't know the position of the dot i even need to find them

but i found, that I might need to create a dual graph for that is that right?

[brisk moss]

wdym don't know the position of the dot

[surreal socket]

I just have the edges, I want to know all the faces and their belogning edges.

[brisk moss]

i don't understand...
say i just have the edges. what are you requesting?

[surreal socket]

i want to get all the cells, so i know which edge does belong to which cell, in the process i need to figure out how many cells there are as well as what edges belong to it

[brisk moss]

ok

i thought we just wanted to make one cell before

[surreal socket]

no 🙈

[brisk moss]

hmmmmm

i have an idea let me think about it a little more though

[surreal socket]

i think i need to generate a dual graph for that, but if you got a simpler solution I'm all yours!

[brisk moss]

ok this is stupid but pick any point in the plane. say the green point. to find the cell edges you can point rays in various directions and find the first line segment that intersects them

you'd have to make a good way to choose rays

but after that pick another point that isn't in that cell and repeat

until you've made them all

also don't pick any points that are on any line segments

[surreal socket]

yea, but two issues, I have to figure out where to put the points, which might be possible trough the edge points but i dont know for sure where a cell is

also i kind of have to cast an infinite amount of rays to know where the edges are you also are missing two in your drawing

[brisk moss]

yea, i wasn't trying to draw them all gotten

it won't really be infinite, eventually you'll find them if your picking method is decent

[surreal socket]

but how many rays do i have to cast until i find the top left and right

but i think i need to create a dual graph

[brisk moss]

depends how you pick them

[brisk moss]

you could look for corners instead

[surreal socket]

yea but it seems quite complex and with a lot of pitfalls doing it with the rays

[brisk moss]

that's why i said it was stupid

[surreal socket]

but i appriciate the effort! 🙂

[vocal sleet]

@surreal socket Has your question been resolved?

[vocal sleet]

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[bleak prawn]

I dont see what I did wrong

1/(x^2 - 2x + 5)

1/((x-1)^2 + 4)

1/(u^2 + 4)

1/2(arctan((2)(x-1)) + C

[outer warren]

how did you get the last line

[tired gate]

$\int\frac{1}{x^2+a^2}dx=\frac{1}{a}\arctan(\frac{1}{a} x)+c$

[outer warren]

why is 2 being multulied to the (x-1)

[twin meteor]

Max

[warm vine]

2 should be in division^

[bleak prawn]

But what about this?

I always do a*x inside the arctan or arcsin

And get the right answer

[outer warren]

a isn't sqrt(8) here though

no

[bleak prawn]

But what about the (1/sqrt(8)) infront of it tho? wouldnt that just be (sqrt(8)) then?

[outer warren]

could you show your work?

[bleak prawn]

I just learned this formula

[outer warren]

show your steps from the beginning

[bleak prawn]

Like this?

[outer warren]

yes

[bleak prawn]

1/(x^2 - 2x + 5)
1/((x-1)^2 + 4)
1/(u^2 + 4)
1/2(arctan((2)(x-1)) + C

[outer warren]

for the question you just posted

for why you thought you should have ax instead of x/a

[bleak prawn]

I just put it into the formula, except a infront of the arctan instead of 1/a

[outer warren]

how did you get sqrt8) for a

[bleak prawn]

Thats just what gives me the right answer

[bleak prawn]

because its 1/(1/sqrt(8)) no?

[outer warren]

how did you get that

[bleak prawn]

Wait sry

[bleak prawn]

I didnt I had 1/(sqrt(8)) for a

[outer warren]

yes

sqrt(8) would be 1/that, i.e. 1/a

[bleak prawn]

But I don't understand, if I used the this formula for the 2nd question I posted it would've been wrong, no?

[outer warren]

how so

show your work

[bleak prawn]

1/(1/sqrt(8)) * arctan(x/(1/sqrt(8))

[outer warren]

you're missing some stuff

specifically the factor of -5/8 you supposedly factored out in the first step

[bleak prawn]

Why -5/8? Why not just -5?

[outer warren]

$\frac{1}{8x^2+1}$ isn't in the form $\frac{1}{x^2 + a^2}$

[twin meteor]

ℝαμΩℕωⅤ

[outer warren]

how did you even get a = 1/sqrt(8)

[bleak prawn]

Ahhhhhh

I see the issue now

OK 1 moment I will try to solve the first one

OK I think I got it

Thanks!

❤️

.close

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[vocal sleet]

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

hey rlly struggling with ii) ive got this far

not sure where to go from here

would rlly apprecaite any help as ive been on it for a while

[vocal sleet]

@last canyon Has your question been resolved?

[tawny hollow]

Not sure what steps you’re making here

[last canyon]

well i divided by sin2x-sqrt3cos2x (if theta=x) on both sides as its a common factor

[tawny hollow]

Did you then consider the case when this might be zero?

[last canyon]

it says it isnt as sinx +cosx does not equal zero

plus since its a common factor on both sides can i not just cancel it out

[tawny hollow]

How would you be able to draw the conclusion that sin2x - sqrt3 cos 2x can’t be zero from that?

[tawny hollow]

No in general you can’t if may be zero

[last canyon]

if its zero then sice its timesing both sides the whole thing is zero=zero

[tawny hollow]

Yup, which is a solution you might miss

[last canyon]

zero=zero?

[tawny hollow]

Take for example the equation x(x-1)=0

You miss a solution by diving by x

[last canyon]

ah ok

[tawny hollow]

So you’ll have to separately see when this factor is zero to get the full solutions

[last canyon]

ok

so i put the common factor=0

and see what i get

[tawny hollow]

Yup!

[last canyon]

so phi=π/6

@tawny hollow

[tawny hollow]

Good but be carful here, you’re forgetting again that cos(2x) might be zero aswell

So consider that case too

[last canyon]

why

how would i know to do that

so when i get thes questions, should i consider that either side could equal zero

[tawny hollow]

Well in this instance it won’t actually matter as the LHS can’t be zero when the RHS is

[last canyon]

why

[tawny hollow]

But it’s good to always be careful when you’re dividing by something potentially zero

[tawny hollow]

Can sin x and cos x simultaneously be zero?

[last canyon]

no

[tawny hollow]

Also you’re missing a few solutions here

[last canyon]

but if sin2x-sqrt3cos(2x)=0 both sides can be zero

[tawny hollow]

From tan(2x) = sqrt(3)

[tawny hollow]

Yeah, because the terms themselves don’t necessarily have to be zero

[last canyon]

hm ok

so 2phi=arctan(sqrt3)

[tawny hollow]

Well that you already did

[last canyon]

yeah

so it could be pi/6+n2pi

where n is any integer?

is that what u mean

[tawny hollow]

Yes the concept is right, but not quite all of them here again

[last canyon]

not the negative ones?

[tawny hollow]

So when you take arc tan on both sides, notice that any pi turn is also a valid solution

[last canyon]

how?

[tawny hollow]

Up to the restriction of phi of course

[tawny hollow]

we have that tan(x+pi) = tan(x)

[last canyon]

ahhhhhh

[tawny hollow]

So more generally you have that 2phi = pi/6 + n*pi

Where n is some integer such that phi is in [0, 2pi]

[last canyon]

would it not be 2phi=pi/3+pi

[tawny hollow]

,w arctan(sqrt(3))

[twin meteor]

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[last canyon]

yup

[tawny hollow]

You’re right!

[last canyon]

so we have phi=pi/6

and phi=pi/6+npi/2

and can n be neagtive

[tawny hollow]

Yup!

Now you could just check for which n, phi is in between 0 and 2pi and even potentially at them

[last canyon]

cool

[tawny hollow]

And there you have some of the solutions, but potentially not all of them of course for the original equation

[last canyon]

yup

ill write this down give me a wee sec

i htink it can go up to when n=3

it cannot be negative however as pi/6-pi/2

is negative

[tawny hollow]

It might be worth in the end to also check when sin x + cos x is zero to exclude them

[tawny hollow]

Yup!

[last canyon]

why? sorry i dont rlly get that part how that relates to this, do u mind explaining?

[tawny hollow]

Hm well maybe too early to mention but since we have that sin x + cos x can’t be equal to zero, we should probably check when in fact this is zero to exclude potentially illegal solutions

[last canyon]

ok

[tawny hollow]

But this step can be done whenever, so don’t worry

But as long as you don’t forget it

[last canyon]

ah ok got it

so 3pi/4 +pin

where n is any integer

[tawny hollow]

So 3pi/4 and 7pi/4 are solutions you should exclude if needed

[last canyon]

yup

so now when cos(2x)=0

and why do we do this again?

[tawny hollow]

Where was this for?

[last canyon]

here

[tawny hollow]

Right as I said above, whenever you divide or multiply by something potentially zero, we can have problems

We could miss or introduce new solutions respectively

It’s not any different from the example x(x-1) = 0

Or if you say multiply by zero you can sometimes introduce solutions, say you have the equation x = x + 1

This has no real solutions

But if we multiply by a constant zero

[short pilot]

Yo

Yo

Bro multiplying by 0 is bad whatever you do it results in 0=0

[tawny hollow]

We introduce solutions (illegally)

Since 0=0 is valid for all x

[tawny hollow]

Read before you disturb

[short pilot]

Bro

I read

But I was expressing it in a light way

[tawny hollow]

?

[short pilot]

[tawny hollow]

Huh?

[last canyon]

ok so what u are saying is if the common factor did =0

like we have found, all our solutions found are invalid?

[short pilot]

Oh trigno equations

[tawny hollow]

Not quite, this is the other way around

You started with an equation that can be zero on both sides

That is valid

[short pilot]

for safety whenever you find solutions in trigno equations satisfy in the original equations

[last canyon]

ahhh ok i see

[tawny hollow]

But when you divide that factor away you lose that solution

[last canyon]

ahhhhh ok

[tawny hollow]

So divide can crudely be though of as removing solutions

[short pilot]

Cz sometimes when u put any operation solutions vanish

[tawny hollow]

While multiplying can crudely be thought of as adding solutions

[last canyon]

got it

[last canyon]

whats this about then

[tawny hollow]

Well in this instance, I noticed that whenever cos 2x = 0, it actually didn’t matter since it gave no solutions

I.e sin and cos can’t be simultaneously zero

So in that case it was fine to divide, but we don’t know that beforehand

So we have to check

[last canyon]

ok

ill pretend i dont know that yet then lol

so now what do we do

[tawny hollow]

Well if we go back, what did you have left after taking away the common factor?

[last canyon]

so thi sis when its not 0

when the common factor isnt zero and we can divide

[tawny hollow]

Sorry what?

Oh right yeah

Precisely, we’ve accounted for when it is zero

Which in this case also turns out be solutions

So now we can exclude these

And divide by that common factor

[last canyon]

so we have 2cos(2x)=sqrt6(sinx+cosx)

[tawny hollow]

Mhm

[last canyon]

then cosx-sinx= sqrt(3/2)

[tawny hollow]

Hm, how ?

[last canyon]

^

ive checked it on wolfram alpha

[tawny hollow]

I’m playing dumb here because there’s an important step you’ve missed in this, and I would like to see your reasoning

Yeah so while your solution may be correct, in general there’s an important step which is the same one I’ve mentioned again and again

[last canyon]

becuse sinx+cosx

the thing we end up dividing by and timesing by does not equal zero

[tawny hollow]

Right that is correct, but before this

What do you do?

You’ve multiplied by something correct?

[last canyon]

times by cosx-sinx

ah

need to check this dsnt equal 0?

[tawny hollow]

Yup!

[last canyon]

okie dokie

[tawny hollow]

Because if not we could be introducing solutions

[last canyon]

ok

so

when we check this

these solutions are ones we exclude

becuse they are added

[tawny hollow]

Yes almost!

[last canyon]

almost? what am i missing lol

[tawny hollow]

It could be the case that the equation before actually has a solution like this

[last canyon]

ah ok

[tawny hollow]

So if we remove it, we remove an actual solution too

[last canyon]

so what do we do?

[tawny hollow]

So we check when cos x - sin x = 0

And then see if our equation right before 2cos(2x) = sqrt(6)(sin x + cos x) satisfies any of these

If not we exclude them like you said

An example to illustrate this is say the equation x = 0

Certainly is x = 0 a solution

Multiplying by 0 we have 0 = 0

But all x satisfy this equation

[last canyon]

i see

[tawny hollow]

So if we remove all x, then we also remove the correct solution x = 0

[last canyon]

so from this i get x=pi/4+npi

as u said before tan(x)=tan(x+pi)

[tawny hollow]

Yup!

Of which pi/4 and 5pi/4 are in between 0 and 2pi per our restriction

[last canyon]

yup

[tawny hollow]

Now we’ll have to (sadly) check these with the equation before

Incase they happen to be actual solutions

[tawny hollow]

Notice the problem we have by ending up like this

Then we have to check all real x

[last canyon]

pi/4 dsnt work

[tawny hollow]

Which is not really possible

[tawny hollow]

Correct!

[last canyon]

5pi/4 dsnt work either

so both excluded

[tawny hollow]

Nice!

[last canyon]

ok so now we know to exclude these solutions

[tawny hollow]

Correct

And I might hear you say, that all of this work is really painful and I agree. In some instances you can solve the problems slightly differently to avoid doing stuff like this

[last canyon]

that sounds like my lecturer lol

[tawny hollow]

And remember how we ended up like this? It’s because we either divide or multiply by potentially zero stuff

Lmao

[last canyon]

nah i get it dw good to know

and its not too difficult it just knowing to do it

[tawny hollow]

Yup!

[last canyon]

ok so now do we just continue

[tawny hollow]

Yeah

And as you mentioned cos x + sin x is non zero so we can confidently divide that away like you already did

Then you have a final equation to solve

Oh I noticed a redundancy here, was quite confused what happened

[last canyon]

hm?

[tawny hollow]

So when we had 2cos(2x) = sqrt(6)(sin x + cos x)

[last canyon]

yup

[tawny hollow]

You don’t have to multiply by anything

[last canyon]

hm?

[tawny hollow]

Just notice that cos(2x) = cos^2x - sin^2x = (cos x - sin x)(sin x + cos x)

So you can divide sin x + cos x here directly

[last canyon]

ah

[tawny hollow]

So sorry about the work you just did above

[last canyon]

ahahah nws

i didnt notice either

[tawny hollow]

Eh, that’s how we learn stuff!

In any case we have a final equation left and then we’re done

If we haven’t missed anything else

[last canyon]

ok

so

im struggling with cosx-sinx=sqrt3/2

[tawny hollow]

Hm, a useful trick is to know how to rewrite a sum acos x + bsin x

Have you seen it before?

[last canyon]

ah

i feel i definetly have but have indeed forgotten it

in fact i definetly have a while back in secondry school lol

[tawny hollow]

Mhm all is good! I half remember it only, but we should be able to derive it again without much effort

[last canyon]

cool

[tawny hollow]

I’ll directly do it on the sum we had, so assume we can write cos x - sin x = Asin(x + B)

[last canyon]

yup

[tawny hollow]

Where A and B are to be determined

By using addition formula for sin we have in the RHS that

Asin(x + B) = A(sinx cosB + cosx sin B)

Would you agree?

[last canyon]

can we use this, this is what i remember

[tawny hollow]

Yeah!

Yeah we could just directly apply it here I suppose if you’d like

[last canyon]

sorry ahaha this is what i have in my head didnt mean to interrupt

[tawny hollow]

So A = sqrt 2

[last canyon]

yup

a=π/4

+npi

[tawny hollow]

Yeah B in this case seems to be 3pi/4

[last canyon]

hm yeah

ur rigth

[tawny hollow]

Yeah alright cool!

[last canyon]

wait why sorry

[tawny hollow]

Now you can rewrite cos x - sin x as sqrt(2)sin(x + 3pi/4)

[last canyon]

as

ah

cos u are writing it in sinx form

[tawny hollow]

I’m terrible with the formula but I just solved the system cos B = -1/sqrt2 and sin B = 1/sqrt2

[last canyon]

hm wait

[tawny hollow]

And B = 3pi/4 seems to be a valid solution

Hm, what is confusing you?

[last canyon]

could i write it as kcos(x-pi/4)

[tawny hollow]

If it’s +pi/4 yes

Since cos(x - pi/2) = sin x

Sorry -pi/2

Typo

[last canyon]

but using this i got α=pi/4

ahhhhhhh

forgot my minus 1

nvm know what i did

[tawny hollow]

It might not be taking into account the quadrants

I think that formula only works for the 1st and 4th quadrant iirc

In any case, let’s use cos

Then we can rewrite cos x - sinx as sqrt(2)cos(x+pi/4)

[last canyon]

wait so

cosx-sinx

a=1

b=-1

k=sqrt2

[tawny hollow]

Yes

And what’s arctan(-1)?

[last canyon]

tanα=b/a

3pi/4

[tawny hollow]

Or -pi/4

[last canyon]

true

[tawny hollow]

So that’s why it becomes +pi/4

Because you have to be careful since this formula only
Works for the 1st and 4th quadrants iirc

[last canyon]

ok

got it

so sqrt2cos(x+pi/4)

[tawny hollow]

Ye so just replace that in your equation from before

And it should be easier to solve

[last canyon]

got it

x+pi/4=pi/6

?

[tawny hollow]

We have two scenarios here

[last canyon]

hm

[tawny hollow]

x+pi/4 = pi/6 + 2pi n

[last canyon]

ah

[tawny hollow]

Or x + pi/4 = -pi/6 + 2pi k

Since cos is even

[last canyon]

where did the k come from

whats k

[tawny hollow]

Some integer possibly different from n

Just to make it clearer I suppose

[last canyon]

ok so could n not just be k? sorry if that sounds dumb i just dont understand this part

[tawny hollow]

Yeah it could, just thought I’d use a different letter, since I sometimes think using the same letter might be confusing when they’re (possibly) not even the same

[last canyon]

hm ok

would k then be negative? since cos(-x)=cosx

[tawny hollow]

Yeah that could be possible, remember k or n are just whole numbers, so it doesn’t really matter how you write it down

As long as you find the right n and k, such that x is in between 0 and 2pi

[last canyon]

if n is any integer tho, does it not account for k

[tawny hollow]

Nope that was a falsehood I always had, I thought that since we wrote n all the time I could use the same n, for the other cases, but there are (rare?) instances where n and k are most definitely different integers

[last canyon]

ah ok i dont think ive been taught that yet

i think for what ive been taught stick to n for now maybe

but good to know

so x=-pi/12 +2pin

[tawny hollow]

It’s just a dumb artifact of laziness and it causes confusion as soon as we have more cases to consider, so good to use different letters imo

[tawny hollow]

Yeah, but remember n from this solution might be different the other one you had

So we can’t just use the same n we found

[last canyon]

ok but where did k come from

[tawny hollow]

it’s just the same as 2pi *n, aka some amount of whole turns

[last canyon]

ok

ahhhhhhh

no i get it now

bevuse it could be a different amount of terms but equal the same

[tawny hollow]

So k here is signifying that it might require a different amount of turns, because if we use the same letter n again, we might falsely think we can just use the same amount of turns again to avoid checking for what n it’s valid

I hope I’m not confusing you further, just think k here is some letter to mean any whole number

[last canyon]

yeah i get it now

[tawny hollow]

So if you decide to use n, just remember we’ll have to check for which n it’s valid again even if you’ve already done it for the other case you had

[tawny hollow]

Alright cool!

[last canyon]

so for solutions we have all the ones when we looked at the common factor before

and now these two we just found?

[tawny hollow]

Wait which two was it?

Not sure if that was clear

Yeah I think this is a perfect example actually when k and n is different

But I do have to admit it might be obvious in this instance to see why, so it’s not really surprising

[last canyon]

why?

@tawny hollow

i rlly have to go now so sorry if i do not respond but i rlly apprecaite all you help

[tawny hollow]

since cos(-x) = cos(x), so if x = -pi/12 + 2pi is a solution with n = 1, then so should -x, but then we have a negative solution, so we need to tweak n

yeah no worries!

[last canyon]

ok cool

thanks

[vocal sleet]

@last canyon Has your question been resolved?

[vocal sleet]

Channel closed

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[vocal sleet]

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• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
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• Be polite and have a nice day!

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[quasi kayak]

Pretty self explanatory I need to solve this integral.

idk where to start.

[woeful stream]

what's tg? tangent?

[quasi kayak]

yep

tangent

[woeful stream]

maybe factor out the 1+tan^2

[quasi kayak]

I did that

[woeful stream]

and use some trigonometric identities

[quasi kayak]

give me a second @woeful stream

if I substitute cos x = t it wouldn't be right I suppose?

maybe the whole sin x / cos x?

[woeful stream]

what identity do you know involving $1+(\tan{x})^2$

[twin meteor]

Element118

[quasi kayak]

good question?

oh wait it's that cos x + sin x

thinghy

= 1

[woeful stream]

hint 2: what is the derivative of $\tan x$

[twin meteor]

Element118

[quasi kayak]

uh..

so basically I write tan as sin x / cos x

and do the derivate this way?

[woeful stream]

you can [find the derivative]/[differentiate] sin x/cos x

[quasi kayak]

2/ cos^2 (x)

right?

chain rule?

oh nvm

is that thinghy

when you have f'(x) * f(x) = f(x)^2/2? @woeful stream right?

[woeful stream]

you seem to have an extra factor of 2

[quasi kayak]

it's 1/cos^2 x?

[woeful stream]

yeah, not sure what error you made, check again

[quasi kayak]

saw it thx

[woeful stream]

$\frac{d}{dx}\frac{\sin x}{\cos x}=\frac{\cos x\cos x-\sin x(-\sin x)}{(\cos x)^2}$

[twin meteor]

Element118

[quasi kayak]

yea it was a mistake on my part

[woeful stream]

do you see how to solve the integral now

[quasi kayak]

I see the first part because it is f'(x) * f(x)

but the second part is f'(x) * f^2021(x)

[woeful stream]

integration by substitution?

[quasi kayak]

I know this equals to tg^2 x /2

because it's the formula

thinghy

but the second is idk lmao

OH WAIT

I SUBSTITUTE

TG(X) = T

AND THE DERIVATE IS DT

right?

THXXX

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[frail dew]

I really do not understand how this works still...

If I let u = a constant, how is that a helix?

I can't even tell if u is the red or blue here.

I would imagine red since blue would be the 'angle' provided by v, or how much of a circle it completes.

but then how is the red a helix?

[blissful sentinel]

This is not a very helpful thing to say

(In the future, don't delete Moderator pings so that other moderators can see that the ping was addressed btw)

[frail dew]

oh, whoops, sorry

[blissful sentinel]

no worries

[blissful sentinel]

u here is the distance from the z-axis

so if we keep u constant and only move along the v grid curves, we'll get the blue one

Which is a helix

[blissful sentinel]

And if you keep v constant and only vary u, you'll get the red line

I think you're confusing keeping a variable constant vs changing a variable.

[frail dew]

ohh thats how it works

yeah

thanks!!

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[blissful sentinel]

It's a little tricky

You're welcome!

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[faint frigate]

hi

[urban edge]

was the hint i gave you of any help?

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@faint frigate Has your question been resolved?

[faint frigate]

i wasnt able to tget any further

@urban edge

[urban edge]

what happens if you keep putting in more 12s to the set {6,12,12}

[faint frigate]

the mean of smaller set approaches mean of all integers?

[urban edge]

yes

[faint frigate]

but dont there have to be three distinct numbers

[urban edge]

what would be the best choose of three diatinct integers if you want a small difference?

[faint frigate]

1, 2, 3?

idk

[urban edge]

yeah three consecutive intgers

[faint frigate]

right

and i just keep repeating one?

[urban edge]

it might be easiest to think about -1,0,1

then keep adding 0s until the two means are close enough

[faint frigate]

keep repeating the middle term?

[urban edge]

yes

that way the smaller mean is -1/k and the bigger mean is 1/k

[faint frigate]

how would i write out a solution though

and how do we know this is the smallest value for x

[faint frigate]

just to confirm this is just to make thinking easier right

[urban edge]

yes

[faint frigate]

because all the integers are +ve

[urban edge]

adding 1 to all terms does nothing

i mean difference of means stays the same

[faint frigate]

thats true

[urban edge]

you can always put the middle integer as 0 and then you want to minimize the difference between the negative terms and positive terms

[faint frigate]

?

because solving like this we get k = 2027

[urban edge]

remember the difference between these two are less than 1/2025

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[rocky widget]

how do i multiply this

[rocky widget]

matrixs

[pale perch]

im not even sure what im looking at

is this what you were given?

[rocky widget]

had to spell "math is cool"^

then mutliply it by (a) 2 1 4
3 2 5
1 0 2

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<@&286206848099549185>

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[ivory violet]

for partial fracitons, why when making x=0 we dont include the A and x in the equaiton i circled?

[ivory violet]

pls @ if any1 replies

[lucid bane]

A times 0 is what :3

@ivory violet

[ivory violet]

uhhh

0...

interesting

ty haha

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[shut zenith]

Any help

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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.

[shut zenith]

<@&286206848099549185>

Please i need for a project non school related

<@&286206848099549185>

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<@&286206848099549185>

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[shut zenith]

<@&286206848099549185> why wont you help

[sage rampart]

dude make the image bigger pls

[mint gate]

Do u want step by step?

a)
F=m.v
V=f/m (just put the values)

b)
K=1/2v^2 (just put the values)

c)
n = K/E

d)
n = 2v/v+ve (this ve is the result of the answer A)

e)
n0= ni . np

ni is the result of the answer C
np is the result of letter D

f) I = v/g
Remember the v is the answer of A

g)
S= 1/i

Just remember this is a Math discord channel

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[finite hatch]

for this i got n= 20 or n=11

how do i know whcih is right

[short pilot]

Check then

[short pilot]

Put n=11 and n=20

Check

11/2(2*15+(11-1)1)

11/2(30+10)