help-0

[alpine sable]

do this

[alpine sable]

OHH

like

every

thing needs to bbbe

yes

BUT

power of 4

okay

wait gimme a sec

I WANT TO BE VERY CLEAR

from what i wrote, dont ever, ever do [
\p{a+b}^n = a^n + b^n
]

[alpine sable]

it is not the same situation and i want you to fully accept that the above is wrong

wdym?

[alpine sable]

like this

if the qn asked

for indices

would my answer be correct

^

[alpine sable]

wait why so

im confused on why thats wrong

[gray isle]

still don't get what you mean by asked for indices

[alpine sable]

if the qn asked for indices, the expression is bracket

so

my answer was based off the law of indices

would it be correct

if

the qn

asked for

indices

and not

simplifying completely

[gray isle]

its wrong regardless of what its asking

[alpine sable]

damn 😔

[gray isle]

what specific law were you applying?

[alpine sable]

honestly i dont know im confused by my own answer

[gray isle]

refer to what alex posted

[alpine sable]

caan you explainn why this is wrong

and

what like

situation

i should write like that

OH WAIT

I KNOW

NOW

the crab method?

[alpine sable]

two brackets

(a+b) (a+b)

then

a^2 + ab + ab + b^2 = a^2 +2ab + b^2

[lone heart]

@alpine sable Has your question been resolved?

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[charred dock]

Cos2x = sin x

[charred dock]

How do i solve this

Test in 20 minutes also

It looks like it should be easy but idk what identities I could use

[gray isle]

you can apply double angle identity here, to express this as a quadratic equation in sin(x)

[charred dock]

Cos^2x-sin^2x

So, cos ^2x - sin^2x = sinx

Then what?

[gray isle]

the double angle identity for cos has 3 forms

[charred dock]

Oh no

You're right

Rip

.close

[lone heart]

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[trail crow]

hey

[trail crow]

i need help

nvm im done

.close

[lone heart]

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[lone heart]

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[alpine sable]

\newcommand{\Mod}[1]{\op{mod} #1}
if $a,b\in\Z_{\ge0}$, then
[
(2^a - 1)\Mod{2^b-1} = 2^{a\Mod b}-1
]

[alpine sable]

how do i prove this?

[marsh rapids]

X^n - 1 = (X^m-1)(X^(n-m) + X^(n-2m) + ... + X^(n-qm)) + X^(n-qm)
Where n = qm+r is the euclidian division
Right ?

[alpine sable]

yes

oh i see

i think i see where u r going with this

[marsh rapids]

By long polynomial division

[marsh rapids]

Inspired from the proof that (X^n-1) ^ (X^m-1) = X(n ^ m) - 1

[alpine sable]

ok i gotchu thank

.close

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[serene vault]

wiw

[serene vault]

can someone help me through this

and help me find the pattern

i know when n = 3 the output is 11

then n =2 the output is 6?

but wth is the point

n=4 is 18

[hallow vessel]

what is this code for lmao

[serene vault]

just lookn

do you have any idea?

it appears to increase by every additonal odd number as n increases

[hallow vessel]

its the fibonacci sequence

do you know it?

[serene vault]

i do not

[hallow vessel]

okay wait

its not the fibonacci

but its a recursive sequence

[serene vault]

?

[hallow vessel]

one sec

here you go

when n = 5, is the output 29?

[serene vault]

nope

ouput is 27

i forgot what this method was called in math

[hallow vessel]

hmm

[serene vault]

but when you take the difference

of two sets

so like n=5 output minus n =4's

then take that and subtract it by n=4s output by n =3s

[hallow vessel]

yea i think thats called a recursive sequence

[serene vault]

u know what im talkn ab?

its not recursive

its called something else

[hallow vessel]

ah i dont know the name then ahhaa

but it seems recursive to me since you are using the previous results to get a current result

[serene vault]

i wonder what it can be used for tho

[hallow vessel]

no clue lol

i wish i knew

whats the algorithm called?

[serene vault]

i wasnt given a name for it

[hallow vessel]

where did you get it?

[serene vault]

just asked what big O would be

[hallow vessel]

in what context

ahh right

yea sorry im not too sure what it could be used for

id suggest trying to look for it on stack exchange

[serene vault]

what formula could i derive from it?

[hallow vessel]

like, what the equation for it is?

[serene vault]

yea

in a way where i dont have to go crazy writing down all the steps on paper

is there a formula i can just use instead

[hallow vessel]

okay let me think a little

okay

the formula is

for n in Z+, output = n^2 + 2

[serene vault]

how did you get that?

[hallow vessel]

i just worked it out on paper by looking at the first 3 vaues

values

check it

do n = 11

answer should be 123

[serene vault]

its right

[hallow vessel]

okay great!

[serene vault]

thats very intuitive

nice

thank you

[hallow vessel]

no worries at all

you actually couldve found it yourself

so

what did you notice

from the pattern

[serene vault]

each iteration it would add an additional odd number

from n=2 to n =3 it would be +5

[hallow vessel]

yep

[serene vault]

from n =3 to n=4 +7

[hallow vessel]

then

[serene vault]

n=4 to n =5 +9

[hallow vessel]

u compare n to the output

do you understand the significance of noticing that?

[serene vault]

n=3 11 n =4 18

[serene vault]

not really

[hallow vessel]

okay so

what kind of increase is this?

theres a word to describe it

[serene vault]

ohh

umm

exponential

[hallow vessel]

nope ahaha

[serene vault]

rip

[hallow vessel]

i mean

for

+3

+5

+7

[serene vault]

linear

[hallow vessel]

yep exactly

so what this tells me

is that if i make an equation

its going to be quite straightforward

nothing too fancy

[serene vault]

if the increase is linear then how the heck is the formula exponential

[hallow vessel]

so the formula isnt actually exponentiall

polynomials arent exponential

if it was $2^x$, then thats exponential, but $x^2$ is not exponential

[ocean seal]

shavet

[serene vault]

is that cause the exponent is fixed?

[hallow vessel]

yes exactly

[serene vault]

ohh ok

[hallow vessel]

so the main thing to notice is that

what actually is linear

something very specific is linear

and then i think it will click in

[serene vault]

yea it does

if i were to think of big theta in how long this thing runs

i could just do n-1?

[hallow vessel]

so lets see

[serene vault]

since i starts at 1

[hallow vessel]

can you send the code again?

[hallow vessel]

did you find big O?

[serene vault]

isnt big O just n?

[hallow vessel]

yes it is

so every time the code loops

is there anything it can cut out?

[tall topaz]

What language is this

[serene vault]

pseduo

[serene vault]

hmmm

id guess it can cut out i?

[hallow vessel]

i dont actually think there is anything that it can cut out

so i think

i could be wrong

but i think theta(n) is correct

from my understanding

big theta is the average case

or the tight bound

also u never answered ahaha

what actually is linear here

what is increasing at a linear rate

[serene vault]

i

i is increasing linear

[hallow vessel]

not quite what i was looking for

[serene vault]

lmao

[hallow vessel]

the thing you pointed out earlier

odd numbers

is that linear?

what are those odd numbers

[serene vault]

x is linear

y isnt tho?

[hallow vessel]

okay

so

+3, +5, +7, ...

what is that?

those numbers

what are they representing

where do i find them

[serene vault]

the growth rate

by taking the differnce of each increase in n

[hallow vessel]

yes, the rate of change

okay

so

rate of change, what maths does that relate to?

[serene vault]

slope?

[hallow vessel]

calculus

have you done calculus?

[serene vault]

ahh this is where im fucked

nahh never took calc

[hallow vessel]

thats okay ahah

so in calculus

if you have a polynomial

you can find that

the derivative of x^2, is linear

so we have

n^2 + 2

the derivative of n^2 + 2 is 2n

and look at our change?

3, 5, 7, 9, ...

what is the difference?

2!

so we can find that our rate of change is linear

[serene vault]

waiiitttt

[hallow vessel]

and that means the output will be quadratic growth

are you in high school? or university

[serene vault]

bruv im in my masters lol

ahahahha

i got alot of catching up to do

so this is basic calculus?

[hallow vessel]

yep ahha

[hallow vessel]

ohhh i see

id say

try and learn basic basic calculus

you dont even need calculus for this

but it helps to answer your question about why is the growth linear

but the final output is quadratic

i would suggest

try and learn mathematical induction

its really good to help spot these patterns

https://www.math.waikato.ac.nz/~hawthorn/MATH102/InductionProblems.pdf

try some of these

you can always ping me if u need some hellp :)

[serene vault]

tysm

going to try these out over the school break

looks like good practice

thank you!

.close

[lone heart]

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[slim chasm]

hey just curious if i've done this right

**n^3 +3n^2 + 2n

Let x = n^2

x^2 + 3x + 2**

[placid zinc]

So let's look at n³

n³
= (n)(n²)
= nx

But not x²

[slim chasm]

so how would i do this?

the question is to prove that n is a positive integer divisible by 6

[ionic jewel]

you mean prove that expression is divisible by 6 given that n is a positive integer?

[slim chasm]

yeah oops

[ionic jewel]

is this supposed to be induction?

[slim chasm]

no clue

[lone heart]

@slim chasm Has your question been resolved?

[rancid ice]

Would factorising it into n(n+1)(n+2) change anything?

[lone heart]

@slim chasm Has your question been resolved?

[rancid ice]

n^3+3n^2+2n = n(n+1)(n+2)
For any three consecutive integers, minimum one is divisible by 3 and one is even (divisible by 2)
So if n(n+1)(n+2) is divisible by 2 and 3 n^3+3n^2+2n is divisible by 6
Therefore n^3 + 3n^2 + 2n is divisible by 6 given that n is a positive integer

[lone heart]

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[swift saddle]

How cross product 3d

[storm ridge]

What

[lone heart]

@swift saddle Has your question been resolved?

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[swift saddle]

.reopen

[lone heart]

✅

[tacit arch]

https://www.khanacademy.org/math/linear-algebra/vectors-and-spaces/dot-cross-products/v/linear-algebra-cross-product-introduction

[swift saddle]

Bad

[fallen verge]

Wdym bad

[lone heart]

@swift saddle Has your question been resolved?

[swift saddle]

Bad video

[lone heart]

@swift saddle Has your question been resolved?

[lone heart]

@swift saddle Has your question been resolved?

[tough lintel]

you make people not want to help anyone else with your kind of ungrateful attitude

[swift saddle]

.close

[lone heart]

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[kind patio]

what happens

[kind patio]

to the n on the top

[tall topaz]

you divide the top and bottom by n^3 thats what happened

[kind patio]

i get that part

but once its at n/1-9/n^2

where des the n go

[tall topaz]

They just calculate the limit of the denominator and show that it's finite and non-zero so a_n diverges to infty

[kind patio]

does only the denominator matter?

[tall topaz]

it commputes

,, \df{\lt{n}{\infty} n}{\lt{n}{\infty} 1 - 9/n^2}

[ocean seal]

Pure

[lone heart]

@kind patio Has your question been resolved?

[kind patio]

is that one of the tests you can do for this?

ohh wait

so I had gotten infinity/1 is that the same thing they got

[lone heart]

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[calm anchor]

,rotate

[calm anchor]

,rotate

[calm anchor]

Can this be right?

It doesnt feel right

[limpid turret]

$a^{bc}$ is not normally equal to $(a^b)^c$ in the complex field.

[ocean seal]

SWR

[calm anchor]

Oh

[limpid turret]

wait

[calm anchor]

But it is normally?

[limpid turret]

one sec

[calm anchor]

Oki

[limpid turret]

lemme think abt this actually

(dusts off complex analysis book)

Yeah I was mistaken. This (mostly) holds. If x is an integer, it will be correct. Otherwise, you need to consider principal branches

For example, if x=2, then both expressions give you 1.

And if x=1/2, then you get sqrt(-1), and both expressions give you i as expected

But if you do x=1/3, then things get a little weird. Cube root of -1 is -1 in the space of reals, but in complex field, you need the principal root of -1

Which is $\frac{-1+i\sqrt{3}}{2}$, I believe

[ocean seal]

SWR

[limpid turret]

And that is, indeed, equal to $e^{\frac{i\pi}{3}}$.

[ocean seal]

SWR

[limpid turret]

So yeah, it holds if you stick to the right branch

[calm anchor]

Wow

What an explanation

Thank you

What are you studying/have studied?

[limpid turret]

I graduated a while ago

I got a degree in physics with a minor in math.

I would've double majored but college was expensive. 🫰

[calm anchor]

Cool!

Im doing a base year and still trying to figure out what im going to do

Physics is on the top of my list

So far

Thanks again!

.close

[lone heart]

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[limpid turret]

Physics is great, if you have the concepts down. CS is good money. Math major is hard and you just gotta be built for it.

[calm anchor]

.reopen

[lone heart]

✅

[calm anchor]

I have heard physics is the hardest

Next to math

[limpid turret]

imo math is harder

But it depends on your talents

I got straight As in physics and struggled to get Bs in math

[calm anchor]

I have never done any coding

I wrote hello world in python

So i feel like picking a cs program

Is risky

I dont know if ill like it or hate it

But i find physics interesting

[limpid turret]

If you want to learn physics, some CS knowledge will help. It's nice to know how to run any kind of simulation or calculation.

[calm anchor]

Should i do it on my spare time

Or will they (most likley) teach me there

[limpid turret]

I think most university physics tracks will require or suggest one CS course.

Are you already in college?

[calm anchor]

Im at chalmers

[limpid turret]

idk what that is

[calm anchor]

A university in sweden

[limpid turret]

ah

[calm anchor]

But im in a base year

[limpid turret]

Talk to a physics professor, ask what they think

[calm anchor]

So next year ill pick a real program

[limpid turret]

Get an idea of what kind of career you want to do in physics (if that's the track you want)

Even as a first year, it's totally fine to talk to your professors and ask what they think of the profession, gauge if you would be a good fit for it.

Imo, in this age of technology, some basic programming knowledge is a must for physics or math degrees

[calm anchor]

What do you work as now?

[limpid turret]

I do software engineering now. But I worked in astronomy for 5 years before that.

[calm anchor]

Oh cool!

How does the work ”work”

Ive never really seen a day of life for an engineer

How do you know what to do in a day

[limpid turret]

Engineer work is turning a big project into 1,000 little projects and getting them all done as soon as possible

[calm anchor]

Do you do 9-5 days or is it that a project must be finished before this date

[limpid turret]

As an example, say I wanted to build Windows 12. Break it down into its many features: interface, programs, accounts, cloud storage, kernel system, etc.
Then break each of those into sub features. e.g. interface would include graphic design, user interaction logic, theme design, so on. Those keep getting broken down until they're singular tasks that's reasonable for one person to do.
Management structures teams to work on each bit, then it all slowly comes together. You do your tasks, know what your team is doing, and provide support where needed.

I set my own hours, but have deadlines

[calm anchor]

That sound pretty fun

[limpid turret]

It can be

[calm anchor]

Seeing the project going forward and being on a team making it go faster forward

Thats def something i need

I took a gap year, working in a hotel reception

[limpid turret]

It is, but you start to realize just how big some projects are.

You likely end up working on such a small thing you would've never even thought existed before

[calm anchor]

And it was killing me to have every days work reset as if it never happen

[limpid turret]

Yeah I can see how that can be brutal

I worked retail for 4 years, so I know how draining it gets

[calm anchor]

Yep..

[limpid turret]

anyway, take some time to think about what you like doing. Ask professors, look for clubs, immerse yourself in new things and see what you enjoy.

[calm anchor]

Ty for the guidance

And nice pfp

[limpid turret]

Not my dog, personally, but I love it

[calm anchor]

Same hehe

[limpid turret]

Nice to see a fellow good boi enjoyer

[lone heart]

@calm anchor Has your question been resolved?

[lone heart]

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[serene vault]

what method would i use to solve this?

[tacit arch]

Trial and error

[serene vault]

huh?

this was the whole problem

im doing (a)

the first picture is what i broke it down to

[tacit arch]

https://www.merriam-webster.com/dictionary/trial and error

n eventually outgrows log(n)

So does n/50

[serene vault]

there must be a more thurough way

[tacit arch]

So just trial and error for it

[serene vault]

so just plugging in random numbers for n?

[tacit arch]

Do smarter trial and error

[serene vault]

ahaha

i could just use demos to see the intersect

but that leads to me learning nothing

[tacit arch]

https://en.m.wikipedia.org/wiki/Lambert_W_function
If you want to learn something

[serene vault]

bruh

i feel like thats something i should focus on after the semester is over

looks like a rabit hole

[tacit arch]

,w solve log(n) = n/50

[ocean seal]

Results provided by WolframAlpha

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

Requested by rie.mann

[tacit arch]

Then do this

[serene vault]

can i just solve the inequality?

[tacit arch]

What happens when you try

[serene vault]

cant bring that exponent of n anywhere

just makes it more confusing

do you think this question is asking for a range?

also

it appears carlos has a faster algo after 1.01 but it declines and becomes slower than alans somewhere around the 400 mark

.close

[lone heart]

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[idle quest]

not sure how this is wrong

[charred jewel]

change of base formula, you cant just change the log from base 10 to base e like that

[idle quest]

ohhh okay, I got it now thank you

.close

[lone heart]

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[dense glacier]

why cant u pull out the 54?

[ocean whale]

You can

What makes you say you can't?

[dense glacier]

When I did it that way I got a different answer

[ocean whale]

Show your work then

[dense glacier]

The integral I got was wrong 54arctan(6x) + C, the real integral is 9arctan(6x) + C, i dont have a camera rn

[ocean whale]

What were the steps that you did to get that

You can use paint if you're on windows

[dense glacier]

I got my u = 6x and my a being 1, I got plugged it into 1/a times arctan(u/a) + C, and then multiplied by the 54

[ocean whale]

You applied the u sub process wrong, yes u = 6x is correct but you need to find the derivative of u as well

To be in terms of u

[dense glacier]

Oh I see that makees a lot of sense I would've multiplied by du/6

thank u brother

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[wraith flame]

how do i fix this piece wise situation?

[wraith flame]

<@&286206848099549185>

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[serene moon]

can any1 help me

idk

its not on the standard chart of sin values for angles

and idk how to use the addition/subtraction formulas

none of those combinations add up to sin = pi/12

[ebon sparrow]

unit circle?

[serene moon]

yes

?

[ebon sparrow]

use that

[serene moon]

how

[ebon sparrow]

,tex .unit circle

[ocean seal]

Akira (fumo)

[serene moon]

it's not on the unit circle

pi/12

[ebon sparrow]

[serene moon]

its gotta be a combination

but i cant figure out which vectors subtract to pi/12

[stark crater]

perhaps it would be easier for u to see if u converted that to degrees

[vague coral]

pi/12 = pi/3 - pi/4

[serene moon]

how did u figure that out

[vague coral]

sin(a+b) = sin(a)cos(b) + sin(b)cos(a)

[serene moon]

thanks btw

[vague coral]

guessing

[serene moon]

oh theres no algebraic formula or anything i feel like im missing out on some intution or sth

ok thanks

[serene moon]

Thanks

right it would

👍

[stark crater]

You can factorize 12 into 4*3 and that combo becomes a bit more obvious

since when you add you need to find common denominator which involves multiplying

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[oblique mango]

But according to the MCQ
13 is not one of the answers
and the correct answer is given to be 10

What am I doing wrong?

[lone heart]

@oblique mango Has your question been resolved?

[oblique mango]

<@&286206848099549185>

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@oblique mango Has your question been resolved?

[drifting beacon]

You need to use chain rule here

$h’(x)=f(x)(-2g’(1-2x)) + g(1-2x)f’(x)$

[ocean seal]

Afi

[oblique mango]

Makes sense!
Thank you

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[unborn shell]

How do I visualize this one?

or rather just the part w the conditional probability definition

the intersection of x >=3 is not X = all values of x since x can be 0, 1, 2,

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[unborn shell]

<@&286206848099549185>

<@&286206848099549185>

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[rain warren]

Teacher says x = 85, I say x = 95. Can someone please check which is correct

[autumn rampart]

95 thats just a subtraction fail

[rain warren]

Ok thx

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[surreal sky]

how do I change the dx to like dt or whatever it is suppose to be?

[coral thorn]

are you talking about substitution?

[cyan seal]

$$t = \ln{x}$$
$$dt = \frac{1}{x}dx$$
$$dx = xdt$$

[ocean seal]

Rub05

[mellow grail]

if u assume ln x = t, then dt/dx = 1/x => dx = x dt

[surreal sky]

yes

so dx = xdt?

or would I now apply the substitution and get e^tdt

so like this

[cyan seal]

Correct

[surreal sky]

yay thanks

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[open pumice]

help me on this please

[strange meadow]

polar coordinates

[open pumice]

yes

i know but

θ

the region of θ

[golden canyon]

The region of integration is half a disk - another disk, because x <= 0

so pi/2 <= theta <= 3pi/2

[open pumice]

but why its 3pi /2

its ok for pi /2

[golden canyon]

if theta = 3pi/2, then cos(theta) = 0, so x = 0

[open pumice]

when i draw the disk

y <= 0 or y >=0 ?

because it mentionned that x <=0 but not y

[golden canyon]

both, just x <= 0

[open pumice]

so i should i work on that space

[golden canyon]

yeah

[open pumice]

r is between 1 adn 2

[golden canyon]

yes

[open pumice]

and θ is between pi/2 and 3pi /2

ok thx buddy

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[timid jolt]

How would you do q18 without expanding using binomial theorem?

[stuck cape]

Which ?

[timid jolt]

question 18

[stuck cape]

Do you know about "n"th root of unity

[timid jolt]

yes

wait but how is that useful

[stuck cape]

You can find sin5θ , sin3θ

[timid jolt]

this is what the solution says but i don’t get it

[vale crag]

$$\cos\theta = \frac{e^{i\theta} + e^{-i\theta}}{2}$$ $$\sin\theta = \frac{e^{i\theta} - e^{-i\theta}}{2i}$$

[ocean seal]

aPlatypus

[vale crag]

the z in their solution being e^itheta essentially

@timid jolt

[timid jolt]

i’m still quite lost

could you show me the next step?

[vale crag]

well then $$\sin^5\theta = \left(\frac{e^{i\theta} - e^{-i\theta}}{2i}\right)^5 = \frac{1}{32i}\left(e^{i\theta} - e^{-i\theta}\right)^5$$

[ocean seal]

aPlatypus

[vale crag]

if you really wanna see the (z - z^-1)^5

[timid jolt]

thank you i understand 🙂

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[heavy turtle]

Could someone please explain the solution?

[heavy turtle]

I'm a bit confused as to how x^2 + y^2 + z^2 = 2

[lone heart]

@heavy turtle Has your question been resolved?

[heavy turtle]

<@&286206848099549185>

[dire tiger]

x^2 + y^2 + z^2 = 2 is because magnitude of vector a is equal to vector c so magnitude of vector a is sqrt(2) and magnitude of vector c is sqrt( x^2 + y^2 + z^2 )

[heavy turtle]

how is the magnitude of vector a sqrt(2)

[dire tiger]

Because it's i+j and it's magnitude is sqrt(1^2+1^2)

[heavy turtle]

OH

[dire tiger]

Magnitude of a vector xi +yj is sqrt(x^2+y^2)

[heavy turtle]

I completely forgot about that part ahaha

Thank you!

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[teal holly]

Hi, Im not sure why the derivative of dy/dz with respect to x is that. Does it have to do with the chain rule or something?

[teal holly]

<@&286206848099549185>

[slate jolt]

i mean what are the definitions of y z and x relative to each other?

also under certain circumstances you can differentiate in any order you want

i think thats just that + chain rule

[teal holly]

this is the original question, and they do that to reach the final form

[slate jolt]

but is this verified here?

[teal holly]

is what they did basically differentiate that with respect to z? and then put dz/dx? i dont understand why they add that at the end?

[slate jolt]

they swap d/dx and d/dz

because probably they assume y is C^2

or its stated somewhere

then they use chain rule

on dy/dx

[slate jolt]

or not really because its weird here x and z are not just the variables of y

they assume y is regular enough

and x and z

probably

[teal holly]

what do you mean regular enough? This is high school stuff so they dont do much more than this on second order DE

[slate jolt]

im writing all of this because this swapping is not true in general

and you would have to justify it properly potentially

[teal holly]

is this what's going on perhaps?

[slate jolt]

uh?

whzts happening here?

[teal holly]

the dz cancel right?

so it's fine writing in that way?

[slate jolt]

noooononono

[teal holly]

it's the only way i could think to reach that result

[slate jolt]

chain rule isnt "the dz cancel"

whats happening is

d/dx(dy/dz) = d^2y/dxdz = d^2y/dzdx = d/dz(dy/dx)

chain rule

=d/dz( dy/dz * dz/dx)

= d^2y/dz^2 dz/dx

[teal holly]

ok i think i understand

for that question do you know if theres another method that doesnt involve this?

[slate jolt]

not really

thats how i would do it

but i would probably add some assumptions about y

[teal holly]

Ok, which is that it is regular you said? I dont think i need to say that in the question but why is do i need to say that?

[slate jolt]

but probably its ok since x is very regular when defined in function of z

dont worrt

its university lvl math

but ask your teacher

"do i have the right to swap derivatives"?

of different variables

and see what they answer

[teal holly]

Ok, will do. Thanks for the help!

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[graceful jungle]

Ok so my maths teacher gave us the graphs of the derivative of a function and then asked us to graph the function, including the values of any turning points/intercepts. This is the graph of the derivative.

[graceful jungle]

I know the equation for this is y=m(-x^3+3x-2)

and so the equation for the actual function is

so i was solving for the intercepts and got to

[storm ridge]

What about a +c term?

[graceful jungle]

and im unsure how to solve x^3-6x+8 = 0

[graceful jungle]

we haven't really learnt about those so i figured we weren't meant to count that since idk what to do with that

ig it would have a +c

[storm ridge]

Oh

[graceful jungle]

nah i want to to do this properly

but we don't know integration

[storm ridge]

Wait so the +c term doesn't matter here is it?

[graceful jungle]

well let's say it does

[storm ridge]

Hmm

[graceful jungle]

wait actually just leave it out i asked my teacher

[storm ridge]

Oh ok alr

[graceful jungle]

but im stuck on how to solve this without cubic formula

[storm ridge]

Ok so we know when the derivative is 0, from the graph

Those will be the turning points for the actual function

[graceful jungle]

yeah i haven't done that bit yet

i was tryna find the intercepts first

[storm ridge]

Ok well if the try to find the turnin points of x^3 - 6x +8, what do we get?

[graceful jungle]

so the derivative of that is 3x^2 - 6

and you set that to 0

[storm ridge]

Right

Yes

[graceful jungle]

so x = +- root(2) right

sry idk how to format with latex

[storm ridge]

Correct

No it's alr

So now how do we know which point gives the maxima of the function and which is the minima?

[graceful jungle]

you look at second derivative

[storm ridge]

Yes

[graceful jungle]

which is 6x

so the maxima is x = -root(2)

sorry maxima

[storm ridge]

Yeah

So minima is sqrt2

I mean at x= sqrt2

[graceful jungle]

yeah

[storm ridge]

So how about we plug in x=2 in that cubic equation

Sqrt 2 sorry

[graceful jungle]

oh um

so x^3 = sqrt(8) = 2sqrt(2)

[storm ridge]

Sqrt 8*

[graceful jungle]

6x = 6sqrt(2)

[graceful jungle]

oh yeah mb sry

[storm ridge]

Np

[graceful jungle]

so you have 2sqrt(2) - 6sqrt(2) + 8?

[storm ridge]

Yes

[graceful jungle]

so 8 - 4sqrt(2)

[storm ridge]

But that's the minimum of the entire function

And it's greater than 0

[graceful jungle]

wait what sorry

wait lemme go back through this

[storm ridge]

Sure

[graceful jungle]

so we had x^3 - 6x + 8 = 0

we found the turning points at x = +-sqrt(2)

[storm ridge]

Yeah

And then the minimum

At x=sqrt2

[graceful jungle]

yes

[storm ridge]

And the minimum was this

Which is greater than 0...

What does it mean?

[graceful jungle]

wait so which one is the minimum

oh wait im stupid

the minimum (x,y) = (sqrt(2), 8 - 4sqrt(2)) right

[storm ridge]

Yes

[graceful jungle]

but idk what it means

[storm ridge]

But the y coordinate is the minimum value of the function

[graceful jungle]

oh it only has one solution

[storm ridge]

Look, if the minimum is greater than 0, can the function ever be equal to 0?

[graceful jungle]

^^ this right

at the intercept

[storm ridge]

Nah wait my logic is flawed

Coz it's a cubic so it has atleast one real root

Even if the minimum is greater than 0

It could be the local minimum

Mb sorry

[graceful jungle]

nw

so how do you solve that cubic without cubic formula? is it possible

[storm ridge]

You could use this, but cmon now, do you really want to?

There should be another way

[graceful jungle]

i mean i have the answer to that

but its like i wanna find a nice way

[storm ridge]

Yes exactly

[graceful jungle]

<@&286206848099549185>?

[storm ridge]

Are you allowed to use the cubic formula tho?

[graceful jungle]

i mean yeah

i might just use it tbh

dont got time to be doing all of this

[storm ridge]

Yeah coz I graphed it and the x coordinate is in a decimal

[graceful jungle]

yeah it is

[storm ridge]

Hmm

[graceful jungle]

and idk what that is as a surd

[storm ridge]

Well if you're allowed then it should be alr ig

[graceful jungle]

ok then ig

[storm ridge]

Yeah

[graceful jungle]

thanks

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[storm ridge]

Np

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[tranquil nexus]

I need to calculate for which t the Matrix is invertable

i need to use Laplace to calculate the determinant

[foggy stirrup]

we know that a matrix is invertable if det(A) ≠ 0

[tranquil nexus]

yes

[foggy stirrup]

so using laplace we simplify the determinant of A so we can find for what values of t det(A) ≠ 0

[tranquil nexus]

i calculated that and i get $24t^2 + 10t + 7$

[ocean seal]

キマイラ

[tranquil nexus]

i did = 0

but i get some weird numbers which cant be right

[golden canyon]

no, it's correct. I put it in wolfram

[tranquil nexus]

yea but how do i know for which value t the matrix is invertable?

thats the point i dont understand

[foggy stirrup]

it would be every value thats not the t's we got here

because when there is any other value than that then det(A) ≠ 0 which means it is invertible

[lone heart]

@tranquil nexus Has your question been resolved?

[tranquil nexus]

can you explain again what you exactly mean?

i dont understand it fully

[foggy stirrup]

ok so when det(A) = 0, A is not invertible

[tranquil nexus]

yes

[foggy stirrup]

when t is the solution to $24t^2 + 10t + 7=0$ det(A) = 0

[ocean seal]

Laito

[foggy stirrup]

so for every other value of t, A is invertible

[tranquil nexus]

ohh okay i understand

do you think that would be enough for the problem

[foggy stirrup]

yeah, you found all the t for which A is invertable

is that all the problem asked?

[tranquil nexus]

yes thank you

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[lone heart]

@gusty ocean Has your question been resolved?

[lone heart]

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[dusk jackal]

its an exponential graph use the intercepts

or any real visible points

[odd drift]

(-1,-3)?

[dusk jackal]

they didnt give much info so i guess you just have to assume

[dusk jackal]

yes and (0,1)

also it tends to 3 as x->infinity so thats you c value in ab^x+c

[odd drift]

how do I write the eq

Do i follow abx+c?

[dusk jackal]

yes

also its b^x coz x is an exponent

[odd drift]

is the a and b -1 and -3?

[odd drift]

ohh ok ok

[dusk jackal]

since we know c=3

y=ab^x+3 then sub in (0,1)

1=ab^0+3

1=a.1+3

1=a+3

a=-2

now use the other point to get b and you done

[odd drift]

how did it become a.1?

[dusk jackal]

any number raised to the power of 0 is 1

so thats how i got rid of the b

[odd drift]

Oh okl

Is b 1?

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[ebon sparrow]

How did you get that

[lone heart]

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[surreal sky]

what to do now?

[fickle blade]

Use partial fractions possibly

[keen pasture]

Or add and subtract 1 in the numerator to simplify first

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[tulip oak]

I tried using squeeze theorem, similar to the proof for lim x->0 sinx/x, but that lead nowhere. Now I'm looking back to the limit laws to see if my statement for the reciprocal is correct. Can anyone help out?

The limit in the box is the original question I need to evaluate for.

[tulip oak]

The question is a Trigonometric limit

Also, I don't know what L'hopitals rule is yet, so can someone help solve it without that knowledge?

[slate jolt]

you know the lim of sin(x)/x?

[tulip oak]

Equals 1

Yes

[slate jolt]

yes

[tulip oak]

As x-> 0

[slate jolt]

ok so what is the limit of (f(x))^2 if lim of f(x) is l

and l is a real number

[tulip oak]

L^2 I'd hope

Based on limit laws

[slate jolt]

yes but why?

whats the limit law

you are reffering to?

[tulip oak]

The function is squared, so assuming it's not discontinuous like you mentioned L is real, then you'd know that the limit would also be squared

One sec let me find it

[slate jolt]

yes

its about continuity

if g is continuous at point lim(f(x)) -->a = l

then lim (g(f(x)) when x-->a

is equal to

g( lim(f(x)) = g(l)

[slate jolt]

yeah

but the general rule is that

of continuity

[tulip oak]

Mhm?

[slate jolt]

re read what i said

just before

[slate jolt]

here

[tulip oak]

Why are you creating a composite function?

[slate jolt]

i'm not creating anything

its just a property

about limits

and function composition

[tulip oak]

Ok I think I understand that

What about it?

[slate jolt]

so now apply that to g(x) = x²

and f(x) = sin(x)/x

since g is continuous at 1

which is the lim of f

at x -->0

then lim (g(f(x)) when x-->0

is just g(1)

= 1² = 1

now we have the same logic

but h(x)= 1/x

[tulip oak]

Nono that's fine I understand that property, but the issue is the question is asking for its reciprocal,

f(x) = x^2 / sin^2x

Not sin^2x/x^2

[slate jolt]

yeah

thats the secondd step

[tulip oak]

Ah

[slate jolt]

is h continuous at lim( g(f(x)) x->0?

[tulip oak]

No

[slate jolt]

why?

[tulip oak]

Asymptote

[slate jolt]

whats that

?

[tulip oak]

F(x) is x right?

[slate jolt]

lim (sin²x/x²) in other words

f(x) = sin(x)/x

[tulip oak]

Oh

And g(x) is that squared

Hmm

[slate jolt]

we already know that limit

[tulip oak]

This will be 1

[slate jolt]

yeah ok

so is h continuous at 1?

[tulip oak]

Yes I think so

[slate jolt]

yes

so?

apply the result

[tulip oak]

I don't understand

H(x) is 1/x, or in otherwords the reciprocal of what we just did

Hang on let me get my laptop

[slate jolt]

yes but the limit is 1

so the reciprocal of 1

is well

1

[tulip oak]

So let me organize what you just did, one sec

Is this what you're saying @slate jolt ?

[slate jolt]

almost

theres no lim on the numerator

[tulip oak]

Would it be incorrect to say it though?

I see what you mean, the function is already 1 / whatever

[slate jolt]

i mean its true but doesnt help with understanding

[tulip oak]

Its just extra yea

[slate jolt]

i could say 1= e/lim(1+1/n)^n

its true but doesnt help you to grasp what 1 is

[tulip oak]

😱 oh ma gawd

[slate jolt]

orwhat you should do

[tulip oak]

lol got it

Okay, well if that's the summary, then I think I understand it now

[slate jolt]

the theorem is about composition

and continuity

[tulip oak]

Could you also say that is a continuation of limit laws? Which build on to that, the composition and contiunity principles?