help-27

[thorn birch]

3

[torn lagoon]

yes

you got it!

that should be the answer

[thorn birch]

just 3?😭

[torn lagoon]

yes

[thorn birch]

thank uu

i got an 100%

[torn lagoon]

np! if u want some good resources for practice, check out khan academy or organic chemistry tutor on youtube

[thorn birch]

Thank u so much

[torn lagoon]

anytime!

[thorn birch]

.close

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[arctic temple]

Is the smallest possible N that works is 4?

[calm patio]

is this C

im kinda confused

wait no

[arctic temple]

#❓how-to-get-help

[sick fulcrum]

I love that book

[calm patio]

what?

[sick fulcrum]

understanding analysis

yes, 4 seems to work

[calm patio]

oh wait

mb

[wicked turtle]

the 4th sequence element is -1/4

is that in the interval (-1/4, 1/4)?

[calm patio]

i dont see it

[arctic temple]

5th term won’t work as we require an open epsilion neighbourhood right?

[sick fulcrum]

oh sorry I forgot this book uses >= N

[wicked turtle]

i only inferred that based on them saying N=3 for (-1/2,1/2)

[sick fulcrum]

yep

[arctic temple]

I can handle the 4th term: 1/4 in interval (-1/3,1/3)

[wicked turtle]

but no one is talking about the nbhd (-1/3,1/3)

they'are asking about epsilon = 1/4

[arctic temple]

oh wait, they are asking the N which works for epislion=1/4

[wicked turtle]

yep

[arctic temple]

ie after which term in goes into the (-1/4,1/4) and never leaves

[wicked turtle]

correct

[arctic temple]

i see, that’s from N=5

I didn’t see it earlier, thanks

I appreciate that

[wicked turtle]

sure, nw

[arctic temple]

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[chrome sandal]

can anyone explain how to do this?

[chrome sandal]

theres 2 more js like this but diff

[restive river]

,rotate

[restive river]

Which parts do you not understand?

[chrome sandal]

everything

i mean some parts

where do i start

[restive river]

Always start with the parenthesis first

[chrome sandal]

the (5 - 1.5)?

after that do divide it by 5/7?

oh wait do i do (-2)^3 = -8 then add 1/4 = 0.25 THEN i divide it but 5/7 THEN (5 - 1.5) = 3.5 then 3.5 x whatever 0.25/ 5/7 equals

that just popped in my head

[brittle burrow]

1/4 put next to a bracket without any sign is implied multiplication

[restive river]

uuuuh

Yeah

It's $\frac{1}{4} * (5-1.5)$

[woven radish]

What should I do: Physics

[restive river]

not $\frac{1}{4} + (5-1.5)$

[woven radish]

What should I do: Physics

[chrome sandal]

holy shit

[restive river]

Also

Division is just multiplication of the reciprocal

[chrome sandal]

wait whats the diff

1/4 x (5 - 1.5)

?

[restive river]

yeah, the difference is one is multiplication and one is addiction

[chrome sandal]

ah

also

blurry for some reason

how to do this

<@&286206848099549185>

[loud furnace]

which part are you stuck on / don't understand?

[chrome sandal]

top part

and after you do (3.4 - (3^2 - 1.8))

what does that random 4 do

[loud furnace]

top part would be -0.75 (from -3 / 4) x 16
after (3.4 - (3^2 - 1.8)), multiply it by 4 (from the outside)

it works the same as like 4(3) where you multiply it, just written different so its easier to read :3

[chrome sandal]

so when i get the answer for this i multiply it by 4?

[loud furnace]

correct

[chrome sandal]

wait can you explain the top part

more

mb if im asking to much

[loud furnace]

nuh uh all good dw

[loud furnace]

you get -0.75 from dividing (-3 / 4) and then you multiply 16 x -0.75

which would be the same as taking 3/4 of 16, but make it negative

[chrome sandal]

ohhhh

that would equal -12

then -12 over

[chrome sandal]

this x 4

[loud furnace]

-12 / 4(your solution for [(3.4 - (3^2 - 1.8))])

[chrome sandal]

yo

tysm

[loud furnace]

👍

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[sweet veldt]

Right or wrong proof

[dapper tiger]

what's E ?

oh even ig

so you're saying that if x² is odd then x²-1 is even

but the problem is that x²-1 could be even without being divisible by 8

you don't know if your k/4 is really in Z

wrong proof

several right proofs are possible, and the direct one of the contrapositive is quite nice

x²-1 = (x-1)(x+1)
so if x is odd, both x-1 and x+1 are even, and one of them must be divisible by 4 since they are "consecutive evens"
so if one is divisible by 2 and the other by 4, their product is divisible by 8

[devout snow]

@sweet veldt Has your question been resolved?

[sweet veldt]

ah i see, i took the algebraic manipulation too far lmao

for ur proof, i don't get this "so if x is odd, both x-1 and x+1 are even"

oh wait

any odd +- 1 is even

damn that's a crazy proof, so simple

thanks a lot

i have this other question:

its quite long, but I think it's straightforward, could you check if it's correct please @dapper tiger

[dapper tiger]

m²-n² = (m-n)(m+n)
two numbers have same parity iff their sum/difference is even (*)
it immediately follows that if m and n have same parity
m-n and m+n are even, and so is m²-n²
reciprocally, if m²-n² is even, m-n or m+n is, so by (*) m and n have same parity

so we've proven m²-n² even iff m and n same parity
which is the same as m²-n² odd iff m and n different parity

another way would be to say that m and m² have same parity
same for n and n²
so m²-n² is odd iff m-n is odd, ie iff m and n have different parity by (*)

[sweet veldt]

so this is the contrapositive for the original statement

[dapper tiger]

it's the equivalence of the negations

but there are at least 10 ways of proving it close to what I did, the main point is not what I did but more like the general idea

[sweet veldt]

do u know if my method is correct tho (i don't wanna rewrite ths again 😭)

[dapper tiger]

I'll check

[sweet veldt]

i should probably think more about the different possibilites and rewordings of the statement before just jumping into a direct proof the next time

though sometimes my mind is just blank and not creative, so i found that just starting to write stuff helps to get the flow going, even if it's less "efficient"

[dapper tiger]

for P => Q

you are right

but case 2 is quite useless

you can just say m = 2k and n = 2l+1 without loss of generality

since case 2 is just relabelling m as n and vice versa

you can always name the odd one n

however, for -P => -Q, both cases are necessary, since in one both are odd and in the other both are even

[sweet veldt]

what does generality mean

Like if i just straight up erased Case 2 for P => Q, is there some sentence i can substitute so that it basically summarizes my intentions

[dapper tiger]

"without loss of generality, let's assume m = ..."

the case 2 is just a renaming of m and n

you can always pick m to be the even one such that you're always in case 1

[sweet veldt]

damn ok that helps so much

gonna have to do some more iff proofs so that will cut some time

tysm for the help

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[marsh locust]

Could someone help explain to me how to solve this using special product patterns?

[dapper tiger]

solving a number doesn't mean anything

[stone stump]

I assume they want you to notice that its the same as (30+1)^2. which is of the form (a+b)^2

[marsh locust]

Where does the +1 come from tho

[brittle burrow]

31=30+1

fr

[marsh locust]

How would I solve it from there

[brittle burrow]

you can expand it

[stone stump]

do you know what (a+b)^2 is equal to?

[brittle burrow]

with special product patterns

[marsh locust]

ight i got it fam thanks

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[shell gyro]

Can someone help me prove that n points can always be intersected by log2(n) lines?

[karmic abyss]

Is that actually true? Your third sketch has $8$ points and $4$ lines, but $4 > \log_2 8$

[woven radish]

@karmic abyss

[shell gyro]

wait no you're right. i cant think rn.

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[karmic abyss]

To be fair, those points actually could be intersected by three lines

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[timber cairn]

A toy train rolls around a horizontal 1.0-m-diameter track. The coefficient of rolling friction is 0.10.
How long does it take the train to stop if it's released with an angular speed of 20 rpm?

[timber cairn]

I've tried solving this using some basic angular kinematics equations, but I don't think I can get a proper answer without the mass of the train... I think I need that to get the angular acceleration

[brittle burrow]

friction =-0.1N=-0.1mg, so acceleration=-0.1g?

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[restive river]

can someone explain how we decide to plug in q2 or q1 for the field plz

[obsidian bridge]

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[restive river]

thanks

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@restive river Has your question been resolved?

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[little lynx]

add the product

[little lynx]

h

hi

i need help on another question

Add the product of (-16) and (-9) to the quotient of (-132) by 6

do i like divide 132 and 6?

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[tame matrix]

yes

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[restive river]

anyone able to explain this

[devout snow]

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[strong willow]

well what is that you do not understand here specifically ?

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[frosty mantle]

@quasi flame

[frosty mantle]

c = 4000*(1−δx,0) + 8000*n

sorry

[quasi flame]

What is that symbol

Go the left of x

[frosty mantle]

its the symbol for the kronecker delta

[quasi flame]

I have no idea what that is lol

[frosty mantle]

setting it to be
delta(n, 0) will make it always return 0 for anything other than 0 and 1 for 0

so then you can do 1 - delta(n,0) which will make it return 0 for 0 and 1 for anything else

[frosty mantle]

so 0 here = 0

[quasi flame]

I sort of understand does that symbol always represent that specifically

[frosty mantle]

anything other than zero will be essentially 4000 + 8000n

[frosty mantle]

nah

its just the lowercase greek letter delta

[quasi flame]

So you would need to show that image and c = 4000*(1−δx,0) + 8000*n

For it to make sense

?

[frosty mantle]

in that notation

delta(x,0)

it should make sense

im not aware of another prominent function that has the specific notation

[quasi flame]

Ok I guess

Thanks

[frosty mantle]

specifically it should be written as

$c = 4000 * (1 - \delta _{n, 0}) + 8000 * n$

[woven radish]

accialto

[quasi flame]

Thanks👍

[frosty mantle]

mhm

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[torpid ferry]

k(x) = e^x + 3 for the domain {x=-2,-1,0,1,2}

i) represent the function on a mapping diagram
ii) state whether the function is one to one or many to one

[torpid ferry]

I can probably do part ii however part i is what I am confused on

How am I supposed to find the value of e?

Unless its talking about eulers number but I don’t think it is

[brittle burrow]

why wouldn't it be talking about e

[pseudo basin]

yeah this looks like e should be euler's number

you can and should use a calculator to calculate all five values of k

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[sturdy sage]

anyone know how I can simplify this more?

[sturdy sage]

the answer is one of these three

[wheat pawn]

you can simplify the last line with the distributive properties

[sturdy sage]

That's true. Thank you

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[restive river]

Hi

[restive river]

The first one is done by the teacher

The second one is mine

I want to know what did I do wrong?

[dark tundra]

Because 2x+pi doesn’t go to 0

[pseudo basin]

tan(2x+pi)/(2x+pi) does not approach 1

[wheat pawn]

check the tan(2x+pi) by itself when x->0

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[restive river]

.

[restive river]

Doesn't tan( x) / x aproch 0 ?

[pseudo basin]

as x goes to what?

[restive river]

To 0

I mean

[pseudo basin]

then no, tan(x)/x would go to 1

[restive river]

Aproch

1

Yes

[pseudo basin]

but tan(x+2pi)/(x+2pi) as x goes to 0 doesn't approach 1

[restive river]

Can't we consider (x+2pi) as X

For example tan(5x)/5x will approach 1

If we consider 5x as X

[dark tundra]

Yes, but you have to change the value X approaches

[pseudo basin]

but if x goes to 0 then x+2pi will go to 2pi and not to 0

[restive river]

Why isn't it the sames as tan(5x)/5x as x approach 0, here we wouldn't change anything

[dark tundra]

This only works because as x->0, 5x->0 also

[pseudo basin]

if x goes to 0 then 5x goes to 5*0

and 5*0=0

[restive river]

Oh

Because we have addition ?

[dark tundra]

But as x approaches 0, x+2pi approaches…

[restive river]

And it only works with multiplication

?

[dark tundra]

I would just check what value the new thing approaches regardless

Just to be safe

For example, x->0 implies x/5 ->0, but it doesn’t imply 1/x ->0

So I don’t like ascribing hard rules

[pseudo basin]

^

[restive river]

If I had for example 2x.pie instead of 2x + pie , it would have worked corect?

[stable herald]

Yeah, as the term 2x(pi) collectively approaches 0, still

[pseudo basin]

pi not pie

but also again like this is just basic substitutions in limits

it's not a special case

[restive river]

Yeah I get it now, thanks guys

.Close

[dark tundra]

lowercase

[restive river]

Oh, ok

.close

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[acoustic mirage]

hello couldd someone explain to me the concept of limits with the little arrow under "lim" like x->x0 it goes like that in my lectures

[acoustic mirage]

plis

sorry for the french parts.... 😿

[frosty cradle]

the arrow means "approaches"

[acoustic mirage]

yeah that i somewhat understood but theres something about intervals hold im just translating the part

The "tends towards" pressure means that for every number ε (>0) as small as one wants, there exists an interval ]x0-a , x0+a[ such that for all x in this interval (x # x0 ) , x belongs to the interval ]f(x0) - epsylon, f(x0) + epsylon [

what does that mean

[sand dove]

The 'part with intervals' should mean "no matter how close we want f(x) to be to L (with precision epsilon), we can find some neighborhood of x0 where we always have that precision"

[acoustic mirage]

then theres a graphic representation

so epsylon's like precision ? ive never encountered it before

i have no idea whats it used for

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[vestal notch]

What's the difference between c and d

Like I get that d is when population decreases every 10 years

In what case would it have been c

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@vestal notch Has your question been resolved?

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[ivory bluff]

How do i prove that if $p\equiv 1 \pmod 4$ where p is a prime, then a solution to $x^2\equiv -1 \pmod p $ exists

[woven radish]

bigpufik

[stone stump]

do you know that an element c is a square if and only if c^((p-1)/2)=1 mod p?

[ivory bluff]

didnt know that actually no

so a number n is only a square if for all prime p $n^{\frac{p-1}{2}}\equiv 1 \pmod p$?

[woven radish]

bigpufik

[stone stump]

I meant *c is a square mod p if that wasnt clear btw

[stone stump]

yes

well, odd primes

[ivory bluff]

Yeah ofcourse

this reminds me of qudratics reciprocity

[ivory bluff]

you mean that n is co prime with p, well yeah

[stone stump]

I mean that c is a quadratic residue mod p

[ivory bluff]

Like you mean that c is a part of Zp and a possible residue of n^2 mod p?

I dont seem to fully understand what you mean

[stone stump]

there is a solution to x^2=c mod p

[ivory bluff]

So if c is a solution to x^2=c mod p, then we can go the other way and say that x is only a square if c^p-1/2 = 1 mod p?

[stone stump]

I meant that there is a solution x iff c^((p-1)/2)=1 mod p

exactly what you are supposed to show for c=-1

[ivory bluff]

Ok, and how do i prove this?

I mean this theorem, the problem i asked becomes trivial

[stone stump]

one of the directions is easy if you know fermat

for the other, remember how many solutions a polynomial equation can have

[ivory bluff]

I see, what is this theorem called btw?

[stone stump]

not sure if it has a name

there is probably also a more elementary way to show it specifically for -1

but I don't know it.

google probably does

[ivory bluff]

my text book hints to use wilsons theorem

[stone stump]

hmm

[ivory bluff]

but i dont see how i can use $(4k)!\equiv -1 \pmod p$

[woven radish]

bigpufik

[ivory bluff]

factorials are never squares

[stone stump]

the factors from the factorial pair up nicely actually

that is much easier

remember p-k=-k

[ivory bluff]

I dont immeadietly see it

[stone stump]

try it out for some small primes

[ivory bluff]

I think i still dont see what you aim to do with this part

[stone stump]

write down 12!

[ivory bluff]

Yeah

[stone stump]

but replace the last few numbers all in that way

[ivory bluff]

you mean 13-1, 13-2?

etc.

[stone stump]

yes

[ivory bluff]

so i get 4k! on the other side?

[stone stump]

no

you have 1,2,3,4,5,6,-6,-5,-4,-3,-2,-1

now play match up

[ivory bluff]

OHHH you are so right ill get a square

and that means they haev to have the same residue

thanks!

.close

[devout snow]

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[steel halo]

what does this mean exactly?

[carmine pumice]

k(x) = 0 at x = -1

[dry oxide]

k(x)*

[carmine pumice]

my bad yeah

[steel halo]

that's weird, doesn't that just mean an x intersection?

[unreal eagle]

any point where k(x) = 0, which is y = 0, which means x intercept

also called a root of the function

[steel halo]

so my course is just deliberately trying to confuse me by introducing terminology never before used before the mock exam

lmfao

[unreal eagle]

if you're familiar with functions, root shouldn't be a new term

[steel halo]

i've never heard it used that way before

and also they never used that term, they always said x-intersections

for all exercises

[carmine pumice]

has a root, has a solution, has an x intercept, has a zero, all mean the same

[steel halo]

well, at least a godsend to learn this today and not tomorrow on my exam lol

thank you

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[elfin jewel]

I’m struggling to figure out where to start with A and B. Can anyone explain to me where to start?

[elfin jewel]

,rotate

[unreal eagle]

limit definition of derivative for A

[elfin jewel]

?

[unreal eagle]

(f(x+h) - f(x))/h

[elfin jewel]

What do I ply in for H

Plug

Help

[devout snow]

@elfin jewel Has your question been resolved?

[elfin jewel]

No

I need help

Badly

<@&286206848099549185>

[tight bane]

@elfin jewel 🤷 nice question

But i dont know

[elfin jewel]

I’m screwed aren’t I

[elfin jewel]

.close

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[indigo turtle]

need help w this task

[devout snow]

Please don't occupy multiple help channels.

[proud perch]

let's bust out the limits

all these limits are as x goes to inf

lim f/g = 0
lim h/1 = inf
we want to find lim f/(g(h))

[devout snow]

@indigo turtle Has your question been resolved?

[indigo turtle]

Yes did that

I Found a counterexample with

f = x

g = x^2

h = log(x)

But I think it’s not a valid one cuz h has to be N -> N

[indigo turtle]

Can I somehow prove that when f/g is already 0 then f/g(h) MUST also be 0 since g(h) is a function N -> N and not 1

So h at least has to be x

[indigo turtle]

Can I just insert h/1 in f/g(h)?

And say f/g(infinity) = 0 or smth

Idk LMAO this is so confusing

<@&286206848099549185>

[proud perch]

h does not have to be at least x

h could be log(x), loglog(x), ...

any arbitrarily slow function greater than 1

[indigo turtle]

No cuz

h has to be N -> N

And that doesn’t hold for log functions

[proud perch]

good point!

you could still do floor(log(x))

[indigo turtle]

I didn’t have that in my course yet

[proud perch]

it doesn't have to be a one-to-one function does it?

[indigo turtle]

What is 1 to 1 sorry

[proud perch]

like does each natural have to be mapped to a unique natural

[indigo turtle]

Oh no i don’t think so

[indigo turtle]

I mean yea I could say smth like floor or log(x) only for x element N

But I think we have to prove or disprove it differently

[indigo turtle]

Since we didn’t have that yet

[proud perch]

i think you might be on the right track. just assert that h must be at least x

[indigo turtle]

Okay thanks

[indigo turtle]

Wait when I say h has to be at least x

Then can’t I insert into f/g(h)

So that I have f(x)/g(x)

Which would be the original thingy which was 0

[proud perch]

yep exactly

which i think is what is intended

but at the same time i don't think it's factually correct

because we have no way of knowing if h is at least as large as x

also look at:

[indigo turtle]

Yes true

[proud perch]

f = n
g = n^2
h = sqrt(n)

then f/g(h) = n/n = 1

[indigo turtle]

Thanks a lot tho u helped me get on the track for this one at least 😭

[proud perch]

but i guess we run into the same issue for N to N thing. which is strange

[indigo turtle]

Yea cuz sqrt is also not defined for all naturals

Guess this task is just made like that

So I can’t use sqrt log etc etc

[devout snow]

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[stone verge]

I need help with the function table and graphing

[stone verge]

Oh ok, thank you

I think i get how to do it now

[woven radish]

Şêro

[stone verge]

Alright

Oh okay, thank you!

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@stone verge Has your question been resolved?

[stone verge]

Yes

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[lament cradle]

what are the U and upside down U notations?

[lusty sapphire]

They're all sets, so you can use set equality to try to prove.

Then yes

[frigid estuary]

they are big union and intersection, respectively

.close

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[brittle burrow]

sen is sine right

you use trig identities, like sin^2+cos^2=1, and double angle

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[wicked hedge]

how do I do this

[wicked hedge]

choice B right?

[devout snow]

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[open tusk]

How can i get a girlfriend

[brittle burrow]

https://www.youtube.com/watch?v=ya_D9IwB3-s
might be helpful
take some advice from the big guys

do you have an actual question or not

[open tusk]

The onion

💀

Just close it

[brittle burrow]

you do it

[main gull]

It's your channel, you have to close it

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[viral rapids]

does $\abs{x} ^n=\abs{x^n}$

[woven radish]

Jash

[brittle burrow]

$\abs{-\frac15}^{\frac12}\neq\abs{\left(-\frac15\right)^{\frac12}}$

[woven radish]

chlamydia

[brittle burrow]

but mostly yes

[viral rapids]

does | | of a complex number mean the magnitude

[brittle burrow]

come to think of it yeah

[viral rapids]

,w |isqrt(1/5)|

[woven radish]

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[viral rapids]

wait you get the same thing

[devout snow]

@viral rapids Has your question been resolved?

[devout snow]

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[barren garden]

why are the bottom 2 wrong?

[barren garden]

(I substituted 24.9 and 25.1 into the approximation equation to get those 2 numbers)

[restive river]

READ THE INSTRUCTIONS

[barren garden]

so I just add 4 0's or what

dont think that's it

.close

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[timid mantle]

Hello, I'm trying to work my way through Strogatz's lectures on Nonlinear Dynamics, and am just digging into this one in particular https://www.youtube.com/watch?v=eZmzmQW-fAA
I've gotten to minute 6, in which he's setting up the governing equations for the forces acting on a bead threaded onto a rotating hoop.
As he's writing the equation for centripetal force he says "that would have strength mass times velocity squared divided by the distance of the axis" but then he proceeds to write on the board m * rho * omega^2 and says "I'll leave it to you to check thats the same as what I just said"
I'm struggling to understand this - it seems to me the equation should be (mass*omega^2)/rho. I don't want to get lost this early in his lecture, could someone help explain my misunderstanding?

[devout snow]

@timid mantle Has your question been resolved?

[devout snow]

@timid mantle Has your question been resolved?

[timid mantle]

I've figured it out - when he refers to velocity in his quote he's referring to the velocity of the bead (rho*omega) not angular velocity (omega) . So centrifugal force = mass * velocity / radius = mass * rho ^2 * omega^2 / rho = mass * rho * omega ^2

[devout snow]

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[restive river]

do i round up or down on a date? 💀

idk if i should put 600 or 601 days

[karmic abyss]

What's the context?

[restive river]

simple interest

[winter patrol]

can you show the original question

[devout snow]

@restive river Has your question been resolved?

[restive river]

sure

[devout snow]

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[viral rapids]

if you have like $\dv{x} (\ln{x})\biggr|_{x=-\frac{1}{2}}$

[woven radish]

Jash

[viral rapids]

and you get -2

does that represent the slope in the complex plane

[devout snow]

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[balmy crater]

How is that the range ?! 😱😱

[winter patrol]

how are you getting y<-9

[balmy crater]

I thought that the range was just the bit from completing the square

[winter patrol]

be more specific

[ocean beacon]

Here domain is x<3

[balmy crater]

Oh

So what do I do

[devout snow]

@balmy crater Has your question been resolved?

[devout snow]

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[wide bluff]

what does neighborhood mean ??

it means at some region around the point excluding the point ?

[wicked turtle]

typically on the real line, "neighborhood of x" means "open interval containing x"

possibly it means more specifically "open interval centered at x"

depending on context/author

[wide bluff]

can the open interval contain x as (x,a) ?

or (a.x)

like boundary

[wicked turtle]

no neither of those intervals contains x

[wide bluff]

what about [x,a)

[wicked turtle]

that's not an open interval

the point is that there's some epsilon > 0 such that the series converges on the interval (x - epsilon, x + epsilon)

[wide bluff]

i dont get what is analytic, like the series converges thats it or does it have to also be equal to the function

[wicked turtle]

the latter

it has to converge and be equal to the function

[wicked turtle]

there are examples of functions whose taylor series converge but not to the function

[wide bluff]

but here the lower equation

the series does not equal to the function

it says its analytic

[wicked turtle]

"analytic at x=1"

not necessarily for all x

[wide bluff]

its not analytic at x = 1

according to the definition

[wicked turtle]

why not?

[wide bluff]

cuz like 1/(1+x) only converges for mod x less than 1

not at x = 1

so how is it analytic at x=1

[wicked turtle]

that just says its taylor series centered at x=0 doesn't converge at x=1

for "analytic at x=1" you would want to center the series at x=1

[wide bluff]

how do we know its radius of convergence centerered at x = 1

do i create the series and use ratio test

and also taylor's theorem

is there no shortcut ?

[wicked turtle]

maybe easiest to just use the fact that the ratio of two analytic functions is analytic at any point where the denominator is nonzero

otherwise you'd have to compute the taylor series which seems grungy

,w taylor series for 1/(1+x) centered at x=1

[woven radish]

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[wide bluff]

really ? can i prove this for polynomial case atleast

cuz this might be the key to link the analytic definition of singular points to the polynomial specific case

[wicked turtle]

this MSE answer points to a PDF that has a proof

https://math.stackexchange.com/a/2140412

[wide bluff]

this is the polynomial specific case

i just dont see how does this links to analytic general definition

[wicked turtle]

well all polynomials are analytic

[wide bluff]

what

[wicked turtle]

so the ratio of two polynomials is analytic at any point where the denominator is nonzero

[wide bluff]

so polynomials arent analytic at denominator zero point ?

i mean their ratios

[wicked turtle]

ratio of polynomials

right

[wide bluff]

what makes their ratio not analytic at denominator zero

[wicked turtle]

the function isn't even defined there

[wide bluff]

but the series still might converge to something right

[wicked turtle]

sure

but recall the definition of analytic

series must converge to the function

[wide bluff]

yeah

what if the series and function both diverge to infinity

then also no ?

[wicked turtle]

also no

[wide bluff]

i wish i could find some good notes on this, i have been searching books, and most DE books do a bad job to dive deep into series solutions

it feels like they add it for formality

[wicked turtle]

the pdf linked above looks decent

[woven radish]

Spooderman

[wicked turtle]

all complex analysis books will treat this, but real analytic is more subtle than complex analytic, so you want to look in a real analysis book

[wide bluff]

ok

[wide bluff]

for non polynomial like sinx/x we can replace sinx by its maclaurin and it becomes a polynomial so its analytic everywhere except denominator zero right ??

[woven radish]

Spooderman

[wide bluff]

is this gpt 4

[livid geyser]

this is too correct to be chat gpt

[pseudo basin]

this really looks like chatGPT.

[brittle burrow]

#help-23 message
opinion on this?

#help-15 message

[wide bluff]

bard

so like are all infinite polynomial series analytic if they converge ??

if we take series of ln(1+x) , it converges for all x so its analytic at all x ??

i mean if the polynomial series isnt a series of a function like log

but it itself is considered a function the infinite series itself

then if it converges for every x then its analytic ??

.close

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[devout snow]

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[brittle burrow]

what have you done

[urban hornet]

I tried substituting xsinx+cosx=t but that doesn't help much

<@&286206848099549185>

[restive river]

I'm trying

[waxen harness]

I am done. I did it with the same substitution you made.

you see, take that substitution and call it f(x). Then f'(x) = x cosx and f''(x) = -x sin x + cos x .

[restive river]

what did you take f(x)

[waxen harness]

f(x) = x sinx + cosx

[restive river]

how did you get f'(x) = xcosx

[waxen harness]

Just differentiate it. You will get the same.

[restive river]

how

tell me the steps

it should be xcosx + sinx -sinx

not xcosx

[waxen harness]

Well, use product rule for x sinx.

So , it will be sinx + x cosx

And then the derivative of cosx will be -sin x

[restive river]

fuck sorry I made a super silly mistake

I need sleep

[waxen harness]

Its ok.

[devout snow]

@urban hornet Has your question been resolved?

[urban hornet]

What did you do after this

[waxen harness]

So I did that I wrote the expression in terms of f(x) f' and f''.

Then separated and applied integration by parts twice.

A little long.

[urban hornet]

Ok

.close

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[restive river]

Wtf is going on at the bottom here?

[restive river]

for b, the book gives y=sqrt(e){x+1/2} but I just don't see how that can possibly be right

Tried writing it using sqrt(e) and e^1/2 but I dont see it

My god

nevermind now ive posted it I realise what ive done, typical

fml

.close

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[broken robin]

Hello

[broken robin]

For the following rational function, sketch its graphs, identify its domain and range and the asymptotes.

I was wondering would y = 0?

[timid whale]

Yes

[broken robin]

Sorry if it's a easy question, but it's my first time encountering x as a numerator in graphing?

or x in the numerator would simply represent 1?

[long pasture]

any guidance/example from your teacher about this Q?

[broken robin]

Nothing with x as the numerator

But here's the closest one i can find in our lecture notes

[long pasture]

that's good, can you show the page please

[restive river]

well do you know that denominator ≠ 0

[long pasture]

i see

ah

so, quadratics only

[broken robin]

Yeahhhh, ik i suck at math haha

[long pasture]

no worries

[long pasture]

everyone has their times of not knowing much

let's start

the easiest start would be checking the denominator ≠ 0

as delta G said

[long pasture]

this is because, if denominator=0, it will be undefined

[broken robin]

woudnt the horizontal asymptote be =0?

[restive river]

I should've revised conic sections

[long pasture]

you can try check by doing
lim x to +infinity and lim x to -infinity

[broken robin]

What would the value of x be?

The numerator x i mean

[long pasture]

well, x value can be in all domain

you mean you wanna find which x value is valid for f(x)?

which i assume f(x)=x/(2x²-7x+6)

[broken robin]

Yes

[long pasture]

well, for domain, i think it's hard to explain

you'll just have to check is there's any x that f(x) is not valid

[broken robin]

Wait, so the horizontal asymptotes is y=0

[long pasture]

i.e. f(x) is undefined in this case

[long pasture]

yes

[broken robin]

So the remaining task would find the vertical ones

[long pasture]

yep

[broken robin]

Then I would be able to solve the rest?

[long pasture]

yep

just be careful if it's positive or negative

and the range is a bit weird too

[broken robin]

Am i in the right direction?

[long pasture]

yea, x cannot be 3/2 or 2

[broken robin]

so My VA would be x=3/2, 2 then HA would be y=0?

For the domain, I just set the denominator to zero then completely ignore x(numerator)?

[long pasture]

that's right for this case, denominator to zero => those specific x's not in domain

[long pasture]

good

[broken robin]

Nice, I'll proceed to finding the range

Sorry, i was having a hard time solving the ranges so i proceeded in graphing in first

Did i graph it correctly?

[long pasture]

lemme check
oh for x in (3/2,2)
it should be negative instead

to find the range, e.g. finding some local max and min, you'll have to use differentiation

[broken robin]

I'm stalling in finding the range

[long pasture]

do you know how to do differentiation?

[broken robin]

I think not 😓

[long pasture]

i see, it's okay

well, then you'll just have to sketch the graph by checking the negative and positive ranges for f(x) for different ranges of x

e.g.
(2,-inf) is decreasing from +infinity and positive something like that

[broken robin]

Would the range be 0 since y=0?

0-infiniti

[long pasture]

well, it's not that simple

I'm really unsure why your teacher/prof would have you doing this type of question, if you have only done quadratics

this would be insanely hard if so

[broken robin]

we have 4 sems per year lmao

and he's usually late 15 min, so yeah

[long pasture]

there's no use even if i told you the answer... so i bet just skip the range for this one?

after that you can try plugging in some numbers into the functions, and try plotting it out

it will be really narrow

and increase/decrease very quickly if you use normal 1:1 scale

[broken robin]

Yeah I'll try what your suggesting

I guess this is it

Thanks for the help 😄

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[ivory bluff]

let $S={x| x\in Z_n^*, \gcd{x+1, n}=1}$ show that $\prod_{x \in S} x \equiv 1 \pmod n$ if n is an odd positive integer.

[minor bough]

hello

[woven radish]

bigpufik

[ivory bluff]

I mean I think you have to just show that the multiplacitive inverse of x is a part of S

[minor bough]

let $S={x| x\in Zn^*, \gcd{x+1, n}=1}$ show that $\prod{x \in S} x \equiv 1 \pmod n$ if n is an odd positive integer.

[woven radish]

jkcwakcjw

[ivory bluff]

hi

[minor bough]

Yes then you have to demonstrate that it is congruent to 1 modulo n

[ivory bluff]

thats obvious tho no?

[minor bough]

yes, that is all you need to do

[ivory bluff]

but I dont see how that shows that its a part of S

[minor bough]

Yeah

This was my original idea:

Identify the elements of S and their properties.
Utilize properties of S to rearrange the product
Show that this product is congruent to 1 modulo n

[ivory bluff]

I think Id like to finish this on my own, but thank you for your time

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[restive river]

I have some question regarding this

[stone stump]

if any of them are zero

can you give a few examples of vectors in that set?

[restive river]

i meant

if x_1 is equal to 0

will x_2 also be equal to 0

[stone stump]

why should it. they are independent

[restive river]

yeah that i what i thought

but then i dont see how this will be true

since then it could be that when you add them together it will equal a number above 0

or below i suppose

[stone stump]

yes and it isnt true

[restive river]

I am a bit new to this

could this line of reasoning work

[stone stump]

this line of reasoning works, yes. the execution is a bit flawed

[devout snow]

@restive river Has your question been resolved?

[restive river]

can you explain to me why?

if you could try to be less vague and evasive in your responses it would help

[stone stump]

check again which vectors you actually defined

your second vector is (10,10,10)

[restive river]

y_1 could just be 0

and it will still hold

[stone stump]

yes

[restive river]

Will this hold for every vector of that sort

x*y=0 will be false

3x^2y^17y^3=0

will be false?

[stone stump]

do you mean that a set satisfying that condition will not be a vector space?

yes

[restive river]

Yes i do

[stone stump]

unless it is only one variable

eg just x=0

[restive river]

Yes, so in a more general case:
(c for constant)
ck^n * ky^n * k*z^n
where both n and k, can be positive, negative and 0

[stone stump]

yes

[restive river]

from the set [2-infinity)

i terms of how many terms you have

if you have two or more than two terms that only are times together, it wont hold

[devout snow]

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[wheat mica]

how do I solve this 2x^2 +8x < 0

[wheat mica]

I want to use the square root formula -b +- square root ( b^2 - 4 * a * c) / 2 * a

But I don't have a value for C and I don't have a calculator

so what value do I set for c?

I tried 0 and 1

[hybrid snow]

2x^2 + 8x = 2x^2 + 8x + 0

What would c be

[wheat mica]

I first did that with 0

but then I got a result that doesn't make sense to me

[hybrid snow]

Show me your work

[wheat mica]

one moment

I'll draw it

[hybrid snow]

Just send a picture of your work

[hybrid snow]

That's right

What is 0 divided by 2

[wheat mica]

I've been told that you can't divide a number by zero

it causes an error

[hybrid snow]

That's not the same as 2/0

0/2 means you're dividing 0 by 2

Not 2 divided by 0

[wheat mica]

oh I didn't know that

so dividing 0 by a number is ok but not any other number by 0

[hybrid snow]

Dividing 0 by any number that isn't 0 is equal to 0

[wheat mica]

so the final solution is -8 < x < 0

actually I forgot I need to divide by 4 and not 2

so its -4 < x < 0

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[smoky quest]

how do i work out simultaneous equations graphically?

[hybrid snow]

Find where they intersect

[smoky quest]

ik but idk how to find out how they intersect

[hybrid snow]

You just graph the equations and you can legit just see where they intersect

Like

The grid lines and shit that are on the graph should tell you where they cross

[smoky quest]

....

[hybrid snow]

Unless you need to do it algebraically

Show me the problem

[smoky quest]

smth like

y = 2x + 4
y = x(squared) - 3

[hybrid snow]

Okay yeah so show me your graph of those equations

Like either sketch it or graph it on desmos

You can estimate where they intersect

Like just give some intuitive estimation of the intersection points

[restive river]

or you can equate both y values and you can get the points

[smoky quest]

brb

[hybrid snow]

They wanna do it graphically

[restive river]

mb

[devout snow]

@smoky quest Has your question been resolved?

[devout snow]

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[quartz shell]

Can anyone help me prove this

[devout snow]

@quartz shell Has your question been resolved?

[devout snow]

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[wheat talon]

how do i find the inverse function of this?

[wheat talon]

i dont know where to go from x = (2-y^3)^5

[hearty yoke]

x^1/5= 2-y^3
y^3=2-x^1/5
y=(2-x^1/5)^1/3

[wheat talon]

what?

[hearty yoke]

you raise both sides to the power of 1/5

and then get y^3

[wheat talon]

ohh

[hearty yoke]

and then raise both side to the power of 1/3

to get y

[wheat talon]

thanks a lot

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[dusty stone]

Can someone explain where to go from here? I don’t understand how the vector being parallel to the plane helps or where it comes in

[dusty stone]

If you have any ideas please ping me

[dense jay]

if a vector is parallel to a plane, then the vector is perpendicular to the normal vector of said plane

@dusty stone

which means the dot product of them is 0

[dusty stone]

so (1i-2j).(ai+bj+ck) = 0?

If its parallel dont you just take the planes coordinates as the vector?

Isn't there something like that?

[dense jay]

(1 -2 0).(a b c)=0

the planes coordinates?

[dusty stone]

so 1a -2b = 0 so a = 2b

[dusty stone]

I cant remember exactly but isnt there a way to turn a planes coordinates into the perp vector at that point

Or is that just the vector?

are the coordinates of a plane the same as the vector from 0 to the plane?

[dense jay]

you dont have the coordinates of a point on the plane, even if you did it wouldnt be much help to know a single point, the normal vector of the plane is all you needed

[dusty stone]

Oh ok I think I get it now

I'll give it a try thanks bro

[hearty yoke]

isnt the cosine of angle with +ve z axis = c/|v|

|v| =1 so c=1/2

[dusty stone]

Wait what?

Where did you get that from

[dense jay]

you could have a vector (parallel to) on the +z axis eg (0 0 5)
do the dot product to get 5c=5 |v| cos 60

|v|=1 hence c=cos60

[hearty yoke]

if a vector is ai + bj + ck then angle between vector and +ve z axis is c/|v|

[dusty stone]

if it where between vector and the x axis would it be a/|v|?

[hearty yoke]

yes

similarly angle between vector and y axis is b/|v|

[dusty stone]

i see

I didn't know that

Hey so

I’m confused on why the z axis is k

It’s when x and y are both equal to 0

But doesn’t that mean k can be anything

Mark scheme

[hearty yoke]

conventionally unit vector i represents direction of x axis and j for y and k for z

[devout snow]

@dusty stone Has your question been resolved?

[dusty stone]

Yeah but they’re using 1k as the z coordinate, saying that the z coordinate 1 is the only place in the positive z axis, but it could be any point that is belong z =0 right?

[hearty yoke]

it doesnt have to be 1k, you can take it as anything like nk because you are also putting its magnitude in the denominator

we dont care about the magnitude, jjust the direction because angle will be same for any vector in the k direction

you will divide by n instead of 1 for vector nk

@dusty stone

[dusty stone]

Oh that makes a lot more sense

Thanks bro

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[mint orbit]

Sketch a graph of a function with the following attrivutes: f(x) is concave upwards everywhere and local max and one local min.

[mint orbit]

Is this sketch fine. I was basically wondering when showing maximum and minimum points do I have to constrain the function to a certain domain where we then determine maxima and minima? Or can we base the behavior of the arrows for x e r for all values of x. (-infinity, positive infinity)?

[restive river]

That is not a local maximum