help-26

[agile harness]

cooked told you

you’re just integrating sin(2x)/2

[twin walrus]

wait why though

[agile harness]

because sin(2x)=2sinxcosx

thus

sin(2x)/2=sinxcosx

[crisp raptor]

physics trick: you can realize that you are integrating an odd function over an even Interval

[twin walrus]

is that a rule i should know?

[twin walrus]

whats an odd function?

[crisp raptor]

but its good actually finding the indefinite integral of this function

[agile harness]

a function such that f(-x)=-f(x)

you should know that sinx is odd

and cosx is even

an even function has the property: f(-x)=f(x)

[crisp raptor]

sin(-x)cos(-x) = -sin(x)cos(x)

[twin walrus]

and so how does it help me knowing if the function is even or odd?

[agile harness]

because of the property tobi mentioned

[crisp raptor]

symmetries in functions help a lot, whenever you want to evaluate an Integral, also in the complex plain which I jut realized a couple of days ago (even if you only want to find on Integral on the real axis)

[crisp raptor]

for example sin(x) from -pi to pi

the positive part cancels with the negative part here

[twin walrus]

I see

[crisp raptor]

so for odd functions

$\int_{-a}^a f(x) dx = 0$

[thorny flame]

tobi

[crisp raptor]

for all a

a being a real number

and for even functions

$\int_{-a}^a f(x) dx = 2 \int_{0}^a f(x) dx$

[thorny flame]

tobi

[twin walrus]

wait

so are you saying the problem states an odd function

and that the answer is just automatically 0?

[crisp raptor]

because it is integrated from -a to +a and a is pi here

yes

as an erxercise, maybe convice yourself by finding the indefinite integral of this function and plugging in the values

$\int_{-a}^a f(x) dx = F(a) - F(-a)$

[thorny flame]

tobi

[twin walrus]

wow it is

wait this is so helpful what

[crisp raptor]

this is true for every continues function

do you have an idea how to integrate

[twin walrus]

could you explain this one a bit more please

[crisp raptor]

$\int sin(x) cos(x) dx$

[thorny flame]

tobi

[crisp raptor]

sure

even functions satisfy the property that f(x) = f(-x)

[twin walrus]

I feel like I definitely learned this in calc 1 and wrote it down

but I cant find it

[crisp raptor]

let f(x) be an even function

just did the prove for this

on the first step I split the Integral in 2 parts
then I let v = -x in the first Integral, by that we have f(-v) in the Integral and we get a negative sign in the front from the dx
we know that f(-v) = f(v) because we assumed f to be an even function
we also know that we can split the upper and lower bounds with the cost of a negative sign

[twin walrus]

still here by the way, just trying to piece all of this together

[crisp raptor]

sure tell me if anything is unclear

or maybe an example of an even function, the cosine function, because we know that these marked areas have the same value, we just need to calculate the Integral from 0 to one of the lines to have enough information

[crisp raptor]

this is just proving this for any even function, but its the same idea

[twin walrus]

yeah a picture definitely helps

I get that now

since the two sides are equal

[crisp raptor]

its the same idea here, just that both sides cancel there

[twin walrus]

could you give me an example and let me determine if a function is odd or even?

I see that now thats so cool

[crisp raptor]

do you the exponential function

?

[twin walrus]

?

[crisp raptor]

e^x

[twin walrus]

odd?

[crisp raptor]

I mean do you know this function

[twin walrus]

like how would I go about determining if a function is odd or even

yes

[crisp raptor]

okay ill give you some examples

$sinh(x) := \frac{e^x - e^{-x}}{2}$

[thorny flame]

tobi

[crisp raptor]

we define the hyperbolic sine function like this

is it odd or even?

[twin walrus]

I have no clue

what is a hyperbolic sine function?

and what is the := ?

[crisp raptor]

":=" means defined by

so sinh(x) is just (e^x - e^(-x))/2

[twin walrus]

so is sinh(x) the same as sin(x)?

[crisp raptor]

no

[twin walrus]

whats the difference?

[crisp raptor]

sine and cosine give the coordinates of the unit circle

[twin walrus]

Yes

[crisp raptor]

sinh and cosh give the coordinates of a hyperbola

[twin walrus]

oh

[crisp raptor]

hyperbolic sine function

[twin walrus]

what class is that usually taught in?

[crisp raptor]

oh I dont know if you learn about this in school, im a physics major

I did not know about these functions in school

[twin walrus]

makes sense ive never seen it before

[crisp raptor]

but it does not matter if you only want to find out if a function is even or odd

you could check that with any function so I wanted to use something you probably have not seen before

[twin walrus]

ohhhh

so how would I go about determining if its odd or even?

just like a starting point

[crisp raptor]

when do we consider a function odd? when do we consider it even?

[twin walrus]

odd if f(-x)=-f(x) and even if f(x)=f(-x) like you said right?

[crisp raptor]

yeah! correct nice

so you have to check what you stated

$sinh(x) := \frac{e^x - e^{-x}}{2}$

[thorny flame]

tobi

[twin walrus]

so can I check by just making x whatever I want?

[crisp raptor]

if you want to check if sinh(x) is even you want to check if sinh(x) = sinh(-x) if you want to check if its odd check if sinh(-x) = -sinh(x)

[twin walrus]

so where do I start here Im confused...

[crisp raptor]

you want to check if you can write sinh(-x) as -sinh(x)

this has to be true if sinh(x) is an odd function

if a function is odd it has to be true that f(-x) = -f(x)

[twin walrus]

but how do I start checking that?

like I dont know how to start

[crisp raptor]

you know how sinh(x) is defined

[twin walrus]

yes

[crisp raptor]

so you can check what sinh(-x) is equal to

[twin walrus]

((e^-x)-(e^x))/2 ?

[crisp raptor]

yeah

now if sinh is an even function

this should be the same as sinh(x)

is this the case here?

or does sinh(-x) = sinh(x) ?

[twin walrus]

yes?

they look the same

[crisp raptor]

$sinh(-x) = \frac{e^{-x} - e^{x}}{2}$

[thorny flame]

tobi

[crisp raptor]

$sinh(x) := \frac{e^x - e^{-x}}{2}$

[thorny flame]

tobi

[twin walrus]

on second thought

no

the top is just flipped

right?

[crisp raptor]

yeah sinh(x) is not an even function

[twin walrus]

wait what

[crisp raptor]

maybe check if sinh(-x) = -sinh(x)

[twin walrus]

I thought we were checking if it was odd

[crisp raptor]

I just asked if sinh is odd or even

we already proved that it is not even

[twin walrus]

wait but I thought this rule was for checking odds

[crisp raptor]

yeah we showed it is not even, so lets check if it is odd

[twin walrus]

wait wait wait im kinda confused right now

how did we prove that it wasnt even?

[crisp raptor]

well you said the top is flipped right

so $\sinh(x) \neq \sinh(-x)$

[twin walrus]

yes I agree

but doesnt that prove that it isnt odd?

[thorny flame]

tobi

[twin walrus]

since those two have to be equal for it to be odd?

wait no

[crisp raptor]

most functions are not even and not odd!

[twin walrus]

im so confused right now 😭

what just happened

[crisp raptor]

if it works for you I can also try explaining this via voice chat

[twin walrus]

so this concept was only helpful in my case because I just so happened to have a function that was even or odd?

[twin walrus]

nono sorry

[crisp raptor]

yes, but this is often the case for elementary functions

[crisp raptor]

np

[twin walrus]

ok so let me get this straight

we proved that it wasnt even because f(x) !=f(-x)

how do we check if f(-x) == -f(x)?

[crisp raptor]

$\sinh(-x) = \frac{e^{-x} - e^{x}}{2} = - \frac{e^x - e^{-x}}{2} = -\sinh(x)$

[thorny flame]

tobi

[twin walrus]

doesnt that prove that its odd then?

[crisp raptor]

you said the top wasn't equal, but it only differs, by a -1

[twin walrus]

wait but it isnt

ahahahahahaha this is too much

but im getting it

slowly but surely

[crisp raptor]

do you see that the middle part is the same?

if you distribute the negative sign

or -1

[twin walrus]

no its not the same?

wait no it is

I do see that

it would be -e^x adding e^-x

which is the same as the left side

[crisp raptor]

nice and then we may recognize the definition of sinh(x)

so what you have to check for an odd function is just if f(-x) = - f(x) and for an even function if f(x) = f(-x)

maybe lets do an easier example

$f(x) = x$

[thorny flame]

tobi

[crisp raptor]

is this function odd or even or nothing of these 2?

[twin walrus]

neither right?

no

odd

sorry haha

[crisp raptor]

how would you prove this? lets say I disagree

[twin walrus]

because f(-x) = -x and -f(x) = -x

and that fulfills the requirement for an odd function

[crisp raptor]

yeah nice

what about $f(x) = x^3$

[thorny flame]

tobi

[twin walrus]

odd

[crisp raptor]

yeah nice

[twin walrus]

ok I understand now

[crisp raptor]

$f(x) = x^{101}$

[thorny flame]

tobi

[twin walrus]

odd?

[crisp raptor]

yeah odd power = odd function

even power = even function

[twin walrus]

yeah i see that

ok but going back to my original question

from way back before

of pi to -pi (sinxcosx) dx

[crisp raptor]

yeah

[twin walrus]

I use the sin2x =2sinx cosx rule

[crisp raptor]

(-pi to pi)*

[twin walrus]

oh sorry

yeah that

[crisp raptor]

you could yeah

[twin walrus]

and I get (sin(2x)/2)= sinxcosx

and I can just get rid of sinxcosx because they are the same

[crisp raptor]

yes works so far

[twin walrus]

I determine that the function is odd because there is only a sine right

and that proves that the answer is 0

or am I getting ahead of myself

[crisp raptor]

well yes it works like this, but you could have used the symmetry argument from the start, sine is odd, cosine is even and odd * even = odd

[twin walrus]

oh okay

didnt know it worked like that

just like negative*positive= negative?

[crisp raptor]

but now that you have already used this overpowered tool you can also find the Integral of the function

[crisp raptor]

thats exactly what odd * even is

or where it comes from

[twin walrus]

okay

[crisp raptor]

$\int \sin(x) \cos(x) dx = \int \frac{1}{2} \sin(2x)dx$

[thorny flame]

tobi

[twin walrus]

yes

thats what I have written down

[crisp raptor]

so first of all we can use the linearity of the Integral

with that I mean

$\int c f(x) dx = c \int f(x) dx$

[thorny flame]

tobi

[crisp raptor]

for a constant c

linearity is this and this property

[twin walrus]

oh I havent found F(x) yet right

[crisp raptor]

yeah thats what I was talking about

[twin walrus]

could I u sub the 2x

and put 1/2 on the outside of the integral

[crisp raptor]

yes

[twin walrus]

1/4 actually my bad

[crisp raptor]

you mean another 1/2 and with that 1/4 yes

then you are only left with

[twin walrus]

and am left with (-cos(2x))/4 +C

[crisp raptor]

yeah

[twin walrus]

and then i can just leave my answer as the written out version of F(a)-F(b)

which from this new thing I learned today I already know is 0

yeah?

[crisp raptor]

yeah plug in pi and -pi and the two terms cancel, because cosine is an even function

F(pi) - F(-pi) = F(pi) - F(pi) = 0

[twin walrus]

wow i love math when everything works out and nothing goes wrong

[crisp raptor]

saved mathematics once again, we did not prove 1=0 xD

[twin walrus]

this was a great learning experience with you

[crisp raptor]

yeah its cool

[twin walrus]

thank you so much for taking your time and being patient with me

Thank you!

[crisp raptor]

no problem, I wish you the best!

[twin walrus]

You too!

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[celest quest]

so

i have 0 clue

[surreal mural]

maybe try forming a basis of V that consists of a basis of ker T and try to do something with these vectors

[neon iron]

are we assuming those vector spaces finite dimensional?

[surreal mural]

I think they have to because otherwise the span can never fully be W or V

[neon iron]

yes, they have to. I am giving hint by posing more question

[surreal mural]

seemed more like an error in the question itself

[celest quest]

they are yeah

only finite dimentional vector spaces in this course, thats why they didnt specify

[surreal mural]

ah good

[celest quest]

that consists of a basis of ker T
what do you mean here?

[neon iron]

you don't need to consider the kernel

think about the basis of W, take the preimage, and work from there

[celest quest]

whats a preimage?

[neon iron]

inverse mapping if that makes sense

[celest quest]

uh not really sorry

[neon iron]

the preimage of a subset B of W is ${v\in V : Tv\in B}$

[thorny flame]

hahahahahahahaha from England

[celest quest]

oh so the vectors that map to the items in the subset

[neon iron]

yes, don't overthink it.

[celest quest]

im not sure what to do tho haha

say i take K = {k1,..,kn}

[neon iron]

Let ${w_1, \dots, w_m}\subset W$ be the basis of $W$

[thorny flame]

hahahahahahahaha from England

[celest quest]

but we dont know if T is invertible

what can i do with those?

[neon iron]

T is surjective

[celest quest]

yeah

oh

then there must be k1,..,km that map to those

[neon iron]

yes

you can say more actually

[celest quest]

what else

[neon iron]

read the assumption on T

[celest quest]

oh they span V

[neon iron]

yes

[celest quest]

does this prove they ker(T) = {0}?

[neon iron]

It seems that you only showed T(0)=0

[celest quest]

yeah but i started from the base of the image

hmm ig i guess i could by contrdiction

[neon iron]

To show ker(T)=0, you have to start with an arbitrary v, such that Tv = 0, and show v=0.

[celest quest]

assume ker is not 0

[celest quest]

oh yeah and that would work similarly

right?

[neon iron]

yes

[celest quest]

is this right?

[neon iron]

seems right to me

[celest quest]

yaay!

let me try b

$(A^tB)^t = B^tA$ right?

[thorny flame]

sibber

[neon iron]

yes

[celest quest]

alright im stuck again

im not sure how A being invertible helps

[thorny flame]

sibber

[celest quest]

i dont remember any other theorems

[celest quest]

=*

[neon iron]

just to be clear, we are assuming n > 1?

[celest quest]

yeah

well if its 1 then its trivial

[neon iron]

If n=1 then we can choose A = 1 and for all B we have B^t = B. So T(B)=2B which is injective...

that's why I asked

[celest quest]

yeah

oh we need to prove its not injective

[neon iron]

Ok, can you show $B\mapsto B^t + B$ is not injective

[thorny flame]

hahahahahahahaha from England

[celest quest]

oops brainfart

anyway

whats that arrow symbol?

[neon iron]

it means a mapping that takes B to B^t + B

[celest quest]

ah

[neon iron]

Alternatively, I can say let f(B)=B^t + B, show that f is not injective

[celest quest]

right

but that proves it for a specific A

[neon iron]

ignore the A for now

Just the mapping B \mapsto B^t + B

[celest quest]

ok

we can take any non symmetrical B, then both B and B^t would map to the same thing

[neon iron]

yes, using this you can show T is not injective

[celest quest]

right

now what

wait T?

how?

[neon iron]

Say we have two functions, $f, g: X\to X$. If $g$ is bijective and $f$ is not injective, what can you say about $f\circ g$?

[thorny flame]

hahahahahahahaha from England

[neon iron]

I should ask, is it injective?

[celest quest]

no

but that would require proof

[neon iron]

the proof should be simple isn't it

[celest quest]

now that i think about it yeah

wait

[neon iron]

Now, let $f:B\mapsto B^t + B$ and $g:B\mapsto A^tB$

[thorny flame]

hahahahahahahaha from England

[celest quest]

if we take 2 x's that map to the same thing in f,

and we give it to fg

oh wait no

we can take g-1(x)

and give those to fg and they would map to the same thing right?

yeah

[neon iron]

yes, and the preimages of the x's are distinct

[celest quest]

yeah

[neon iron]

that's all you need to conclude fg is not injective

[celest quest]

right

[celest quest]

but whats fg here

$(A^tB)^t + A^tB$

[thorny flame]

sibber

[celest quest]

rightt

so all thats left is to show that g is bijective

[neon iron]

or equivalently A^t is invertible

[celest quest]

why

oh would g(B) = A^t^-1B?

oh it is

great

thank you!

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[glacial cliff]

!status

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What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

@honest osprey Has your question been resolved?

[honest osprey]

1

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[fiery sigil]

How do I do this

[pearl fog]

does that mean the cylinder is on its side or smth?

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@fiery sigil Has your question been resolved?

[fiery sigil]

idk

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@fiery sigil Has your question been resolved?

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<@&286206848099549185>

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[minor compass]

how do you do this?

[minor compass]

i dont know where to begin

[opal vault]

what's the link between (x,y) and polar coordinates?

x = ?

y = ?

[minor compass]

i honestly dont know

oh sin and cos?

[opal vault]

well almost

you still need to tie it in with the radius r

[minor compass]

my teacher showed us Ay = |A|sin(theta)

and Ax = |A|cos(theta)

not sure if thats what were doing here though

[opal vault]

not sure what A is here

it's just a point right

instead of writing $A = (A_x, A_y)$, we just write $A = (x,y)$

[thorny flame]

rafilou2003

[minor compass]

ok

[opal vault]

and |A| is the radius/amplitude

known as r

[minor compass]

oh

[opal vault]

so y = rsin(theta)

[opal vault]

x = rcos(theta)

and the rest should be straight forward

[minor compass]

makes sense but if y =3x-2?

[opal vault]

yeah

well rewrite both sides of the equation

now that you know what x and y are

[minor compass]

x = -y-2) / 3

[opal vault]

nono don't manipulate the equation yet

[opal vault]

you know what x and y are equal to

[minor compass]

yee

[opal vault]

so just swap "x", 'y' by their values

[minor compass]

what do you mean

[opal vault]

well

if I said to you that y = 6

you would just write 6 = 3x-2

and solve for x

[minor compass]

yea

[opal vault]

so

[minor compass]

but what are the values here

[opal vault]

what if I told you

[minor compass]

there are nono

none

[opal vault]

that y is this value

[opal vault]

and x is this value

[minor compass]

yeah but its written weird

the euation is

equation

[opal vault]

wdym?

[minor compass]

y = 3x-2

like ?

[opal vault]

yeah, what's up

just

plug the values in

[minor compass]

idk what ur talking abt

ther are n values

[opal vault]

this is the value of y

y = rsin(theta)

[minor compass]

3x-2

[opal vault]

the value of y is "rsin(theta)"

[minor compass]

ok but right now y = 3x-2

where am i supposed to put 3x-2

[opal vault]

uh huh

[opal vault]

nowhere, just keep it

[minor compass]

ok so y = rsin(theta) 3x - 2

[opal vault]

no

you REPLACE y by its value

[minor compass]

i know but thats what ur basically aying to dp

[opal vault]

rsin(theta) = 3x-2

[minor compass]

oh

ok

[opal vault]

and now

you also have the value of x

[minor compass]

i keep forgetting were doing algebra

[opal vault]

so you can replace x by its value

[minor compass]

oh

o leme try

so

rsin(theta) = 3rcos(theta) -2

so the answer is b?

wait nevermind idk how to solve

[minor compass]

ok now what

[opal vault]

alright so

you see that the answer you want

is of the form

[minor compass]

wait im right it is answer n

[opal vault]

"r = ..."

[minor compass]

b

yea

[opal vault]

yeye

so you want to solve for r

[minor compass]

so you birng cos to the left side

[opal vault]

we put everything in r on the left side

[minor compass]

OH IM SMART

[opal vault]

rsin(theta) - 3rcos(theta) = -2

ye

[minor compass]

lol

[opal vault]

and then you factor r

and you divide

[minor compass]

ok 1 more

ill open a new one thne i go sleep

.close

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[snow mantle]

Im trying to figure this out but the solution doesnt make any sense to me. Where have they pulled the 35 from in the "61.4-35=26.4157"? My notes to follow in the next post

[snow mantle]

When I work out the angles, I get 181 degrees

[neon iron]

what do you have to find?

[snow mantle]

THe heading he was on walking from vertex C back to A

[neon iron]

okay so u gotta find x?

[snow mantle]

And I have that at 333.6

but I dont understand teh solutions theyre giving me

WHere does the 35 come from?

And what am I doing wrong to get 181 degrees in the triangle?

[neon iron]

im in a class rn, ill need to see the question properly

hence to find the angle made by the bearing you subtract 360 degrees from it

[snow mantle]

thanks

[neon iron]

wait no

[snow mantle]

Im slightly concerned the set up is bad because when I add up the interior angles I get 181

[topaz sinew]

@snow mantle Has your question been resolved?

[topaz sinew]

@snow mantle Has your question been resolved?

[neon iron]

it is

but

ive never come across such word problems

[snow mantle]

It is set up incorrectly?

Or it is to something else?

[neon iron]

ofc it is if youre getting 181

there is an error

i just cant see through it

[snow mantle]

Yeah but Im not sure thats my fault or theirs

Sin(75)/5.5=Sin(C)/5=Sin(A)/4 gives me the angles, when I run through sin-1 of
A = 61.4, B = 75, and C = 43.6 which when added up gives me 181.

[solar wagon]

does this make sense for the 35

parallel lines angles thing

if youre getting 181 you mightve just rounded at an earlier stage

if you use a more accurate value than 5.5 for calculating the bearing you get a different answer

[snow mantle]

I think my notes are the same as yours up until the blue writing. I dont think Im rounding anywhere.

[solar wagon]

youre rounding for calculating 61.4

since 5.5 isnt the exact answer

instead of 5.5 if you used the expression sqrt(41-40*cos(75)) the angles will add up to 180

[snow mantle]

Ahhhh, yeah let me try that. let me finish up what I moved on to

and then Ill rework this

[snow mantle]

Ok, yes on the rounding

[snow mantle]

Though the rounding they give, is the same as mine

[solar wagon]

yeah and that will result in you getting angles which sum to 181 degrees

theres nothing wrong with that

since the question never mentions triangles, that was never a constraint

[snow mantle]

This is a trig mini module I have to get done this summer

And the 35 makes sense too now looking at it but I cant articulate it

the 35 is like working the other way around from 0/360, where as Im working fro the 3rd vertex, and so the 35 seems to appear out of nowhere?

[solar wagon]

yeah its just because u drew it differently

[snow mantle]

Ok

Ill have to stare at the 35 though for a while to see hwere its coming from

O

its because the 215-180=35

?

[solar wagon]

does it not make sense from this?

[snow mantle]

I see the 75+70+35 = 180

[solar wagon]

the 35 is the bearing of that black line

i initially just added it because figured itd be useful later

[snow mantle]

Yeah, ill just have to look at it myself for a bit, but I sort of get it.

Thanks

[solar wagon]

np, yeah it should click if u just ponder on it a bit/let it simmer in ur head for a while

returning to it later or smthn maybe

[topaz sinew]

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[wintry urchin]

I just learnt about row vectors and row space
This whole time I've been able to easily interpret results from linear algebra by thikning about the column vectors and their span (the column space)
I am now wondering what the row vectors represent
What I mean by this, is like I interpret column vectors as where ihat, jhat, ... land after some transformation, and I am able to understand this transformation by considering the column space.
With row vectors though I don't know how to begin to interpret what it means.

[topaz sinew]

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@wintry urchin Has your question been resolved?

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[neon iron]

question b

[frail reef]

Take the augmented matrix and row reduce it

You should then see how the different values of alpha relate to the situation of how many solutions there are

[topaz sinew]

@neon iron Has your question been resolved?

[neon iron]

What now?:

[frail reef]

try to reduce it fully (ie you are able to divide by anything including alpha)

for now

and try and get down to the 4x4 identity matrix in the first 4 columns

[neon iron]

so make everything alpha?

[frail reef]

as in fully row reduce it (allowing for divisibility by say alpha, ie having a 1/alpha somewhere)

you should get a 4x4 identity matrix and then a bunch of terms involving alpha in the 5th column

[topaz sinew]

@neon iron Has your question been resolved?

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[mellow root]

How can I express sin3x + cos1.5x with a single trig identity? I need to prove that this function is periodic.

[mellow root]

I tried expressing it as:

2sin1.5xcos1.5x + cos1.5x = 2cos1.5x(sin(1.5x) + 1/2)

[opal vault]

wdym with a single trig identity?

proving the function is periodic doesn't need much rewriting I believe

[neon iron]

The sum of two "similar" periodic functions is periodic, this is a general fact.

[mellow root]

I mean, if I can express it as sin(some angle) or cos(some angle) this would immediately prove that it's periodic
I've done similar problems like this before

I need to prove it @neon iron (this sounds so mean lmao I'm sorry)

[opal vault]

yeah but you won't be able to this time

does this look like cos(some angle) to you?

[mellow root]

umm no

[opal vault]

alright

well it's periodic

[mellow root]

so what do we do

[opal vault]

just know that you're adding two periodic functions together

[mellow root]

Yeah but the exercise asks me to prove it somehow.

[opal vault]

and they have VERY similar periods

[mellow root]

they do

[opal vault]

we just have to find the period then

[mellow root]

yeah

how?

[opal vault]

alright

what's the period of the sin3x part

[mellow root]

2pi/3

[opal vault]

alright

so everytime you add 2pi/3 to x

[mellow root]

the other one 4pi/3

[opal vault]

you get back to where you started

[mellow root]

yeah

[opal vault]

and here if you move by 4pi/3

you get back to where you started on that term

maybe

you can find a multiple of both

[mellow root]

yeah
so it 'contains' the period of the former

[opal vault]

that BOTH make them go back to where they started

[opal vault]

exactly

[mellow root]

so the period is ... 4pi/3?

[opal vault]

yep

at least 4pi/3 is one of the periods

to check this, just compute f(x + 4pi/3)

if you get f(x)

then you win

[mellow root]

ok
can we pls try another one now
like for example this one:

sin6.7x + tg8.9x

[opal vault]

alright

so the previous example was kinda easier

[mellow root]

so for the first one the period is 20pi/67

[opal vault]

since the period of one englobed the period of the second

[mellow root]

yeah

for tg we get a period of 10pi/89

this is where the problems start for us ... lol

[opal vault]

ok so

now we need to find a period

that englobes BOTH of those

meaning a multiple of 20pi/67

that is also a multiple of 10pi/89

[mellow root]

yeah

[opal vault]

well how about something like

20pi

[mellow root]

I saw online people finding the LCM but how do they do that? we're literally dealing with fractions

[opal vault]

exactly, if there is a denominator, just pick the gcd for the denominator

since 67 and 89 are coprime

the denominator of the period we're looking for

is gonna be 1

then

we just pick the LCM

of the numerators

20 and 10

it's 20

so 20pi

it's easy to check that 20pi is a multiple of 20pi/67

and it's also a multiple of 10pi/89

[mellow root]

ok let's test it with the graph ig

[opal vault]

let's... not

the pattern repeats after a huge amount of time

and the graph is unreadable because tangent flies off to infinity every so often

[mellow root]

this graph will come to me in my nightmares ...
forget about what I suggested before

[opal vault]

anyways if you go about a general way

say you have $f(x)$ of periodicity $\frac {p_1}{q_1}\pi$

[thorny flame]

rafilou2003

[opal vault]

and $g(x)$ of periodicity $\frac {p_2}{q_2}\pi$

[thorny flame]

rafilou2003

[opal vault]

then their sum or product is gonna be periodic

[mellow root]

ok
so the lcm of the numerators divided by the gcm of denominators?

[opal vault]

exactly

this is ALWAYS gonna be a period

*pi

[mellow root]

ok
I will make a note so that I don't forget
and the same approach goes here right? sin1.2x + sin3.4x + sin4.5x
yeah makes sense

[opal vault]

definitely you can add any finite amount of periodic functions

BUT the period of each

always has to look like

[neon iron]

Let me add one more thing, the function is gonna be periodic, but in general, you cannot find the smallest positive period.

[opal vault]

$\frac pq \pi$

[thorny flame]

rafilou2003

[neon iron]

and the LCM method is not reliable

[mellow root]

oh so this way we don't find the smallest period ...

that's bad bad

[opal vault]

we don't always get on the smallest period

[mellow root]

is there any other approach?

I completely forgot about it

[opal vault]

not any other that's reliable 100% of the time

like a small example like

[mellow root]

I see, so unless there is some obvious pattern we can't really tell

[opal vault]

(sin(2x) + sin(10x)) - sin(2x)

like the first function has period pi

[neon iron]

or simpler sin(x) + (-sin(x))

[opal vault]

the second funciton also has period pi

[opal vault]

let's not deal with period "0"

[opal vault]

so both functions have period pi and yet their sum is of a smaller period

[mellow root]

I just saw once how a guy used a similar approach

it was in stack overflow, unfortunately cannot find it rn((

but he used it to express everything with one identity only

that immediately proves the periodicity

[opal vault]

yeayea that works sometimes

[mellow root]

can we do it with our examples?

oh no we can't (cuz look here the angles are the same)

[opal vault]

I would argue that if your goal is periodicity, it's way easier done with what we did

here's a theorem for you:

if f(x) and g(x) are periodic of periods T1 and T2

and if T1/T2 is rational

then their sum, product, etc... is periodic

[mellow root]

their ratio is not?

[opal vault]

their ratio is periodic

[mellow root]

ok

It's just kind of sad that we still won't be able to find the smallest period
but at least we can prove that it's periodic

[opal vault]

if the function is periodic and non constant

and you find a period T

then the smallest period looks like T/n

[mellow root]

yup

[opal vault]

with some positive integer n

usually it doesn't take long to find

[mellow root]

does it have to be a factor of the numerator?

[opal vault]

wdym

[mellow root]

of T in our case

[opal vault]

we don't know what T looks like

[mellow root]

nvm
I mean, it could be any number

yeah, but I think it would be pretty tough for us to find n through testing

it could be any positive number, right?

or did I get it wrong

[mellow root]

so, for example when we found 20pi as our period

How would you go about finding the n for it?
Or is there a way to figure out whether the period we currently have is the smallest or not?

To use it as some guidance ig

[opal vault]

it's actually pretty complicated

[mellow root]

@opal vault I asked too many questions, sorry

[opal vault]

like there is no actual easy method to finding the smallest period

[mellow root]

yeah
I asked GPT and it suggested proof by contradiction, but again I think it is not as intuitive for less obvious cases

eg proof that the period we have is the smallest or not

this is for the very first example I gave you

sin3x + cos3/2x

I can't find anything online about this proof

I've learned a lot, thank you guys! @opal vault @neon iron
If you do not have anything to add, I can close the channel)

.close

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[mellow root]

I'm back ...
Literally the next problem asks me to find the main period of the following function:

(2^sinx - 2^(-sinx))/cosx

[mellow root]

The answer is supposed to be Pi
which means I should express it as either tan or cot?

<@&286206848099549185>

[neon iron]

write it as sinh (the hyperbolic sin) might make the expression simpler

[mellow root]

ok

[neon iron]

checking pi is a period is not difficult, again, the difficult part is to check that it is the main period.

[mellow root]

how do we get rid of that 2 ...

I'm pretty sure they expect me to simplify this to tan or sth

Maybe somehow find the period of 2^sinx?

[neon iron]

I'm not sure if they really want you to prove formally or just plot the graph and "spot" it or make an educated guess. Proving tan(x) has period pi is quite a challenge already (c.f. https://math.stackexchange.com/questions/941803)

[mellow root]

I think we don't need to proof the period of Tan specifically)) just express it as tan is enough ig

so like if I somehow make this look sth like tan(angle) or cot(angle), I think we would be done here

[neon iron]

you can't express that function as tan

[mellow root]

yeah ...
then how can we find its smallest period

@neon iron
all of these functions have a *smallest period of PI

we need to prove it

maybe there is some pattern?

these and the one I mentioned above

[neon iron]

what examples do they give you, and how do they prove it?

[mellow root]

this problem is marked as hard, the book doesn't have any examples like these
ig they want me to figure it out on my own/with a teacher

[tender solstice]

I was asked about periods for functions, I'd start by taking the derivative and then looking at the critical points.

When I find the smallest repeating critical point gap, i'd go in and measure that part of the function to find its smallest period.

Sorry again if i'm being anti-helpful.

No idea about proving it but it seems to be a good starting point if you just need to justify it without proving.

[topaz sinew]

@mellow root Has your question been resolved?

[mellow root]

Yeah, we can def try derivatives and observe the function's behavior)) but that chapter is not covered here yet lol, so I assume they want us to use some other approach

sorry for the late reply I was away from my laptop:/

[topaz sinew]

@mellow root Has your question been resolved?

[mellow root]

I will skip this problem for now, if you guys come up with a solution, please let me know 🤎

.close

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[peak edge]

where do I go from here

[peak edge]

I dont think Im understanding this

or did I do something wrong?

I'm not sure what to do with the 8x+20 on top

[tender solstice]

another u sub, which is (x^1/2 - 2) ( x^1/2 +2) maybe? (WRONG, oops)

Not sure that will work or not, sorry just a guess

[vernal matrix]

4 * (2x + 5)

[peak edge]

I see it now

So the answer is B?

Its C

yes yes

thx guys

.close

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[ivory sorrel]

Prove that if $0 \leq x <y$ then $x^n < y^n \forall n \in \N$

[thorny flame]

veni, vidi, veni

[ivory sorrel]

I DON"T want to use induction here

if possible

[gleaming reef]

it's free if you consider this tbh

👍

[ivory sorrel]

I did use that when I did this yesterday, yes

[gleaming reef]

yes bc( y-x) must be positive

[ivory sorrel]

but don't I have to prove this?

[gleaming reef]

it's trivial

expand it out

[ivory sorrel]

Got it

[gleaming reef]

cool cool

if you are done @ivory sorrel

you can close the channel

[ivory sorrel]

I actually have another question, I suspect I can use the same method here too

[gleaming reef]

let's hear it

I gotta cook soon

[ivory sorrel]

Prove that if x<y , and n is odd x^n<y^n

[gleaming reef]

but if you can send the q soon

[gleaming reef]

same thing

[ivory sorrel]

yeah, I know

[gleaming reef]

but like hmm

consider parity

[ivory sorrel]

but instead like $x^2<y^2$

[thorny flame]

veni, vidi, veni

[ivory sorrel]

that's trivial to prove

[gleaming reef]

it's really easy to see that each of these terms

will be positive

[ivory sorrel]

I get that

[gleaming reef]

n-1 is even

n-2 is odd but y is odd

[ivory sorrel]

thanks

[gleaming reef]

so -1 * -1

ye

keep being a chef @ivory sorrel

[ivory sorrel]

now $x^3<x^2y$

[thorny flame]

veni, vidi, veni

[ivory sorrel]

and $x^2y<y^3$

[thorny flame]

veni, vidi, veni

[ivory sorrel]

hmm

[gleaming reef]

gotta give the full problem

chef

[ivory sorrel]

This is the same problem

[gleaming reef]

just restate the problem

it's useful for you too

[frail reef]

you could just factor this

assuming again your info is x < y

[ivory sorrel]

Prove that if x<y , and n is odd x^n<y^n

[gleaming reef]

x^2y - x^3 = x^2 ( y-x)

@ivory sorrel

[ivory sorrel]

yes

[gleaming reef]

you can guarantee that's greater than 0

assuming x != 0

[ivory sorrel]

I mean what I'm actually trying to prove is that x^3<y^3

[gleaming reef]

what

didn't you prove the general case

for x^n < y^n

where n is odd

[ivory sorrel]

i have, yeah

but I was thinking of trying induction now

[gleaming reef]

so doesn't this come automatically?

okay

[ivory sorrel]

so I have x^n<y^n as my inductive step

so $x^{n+1} < y^n x$

[thorny flame]

Veni, vidi, perii

[ivory sorrel]

and $x^ny<y^{n+1}$

[thorny flame]

Veni, vidi, perii

[ivory sorrel]

ooh

$\frac{x^{n+1}}{y^n}<\frac{y^{n+1}}{x^n}$

[thorny flame]

Veni, vidi, perii

[ivory sorrel]

so $y^{2n+1}>x^{2n+1}$

[thorny flame]

Veni, vidi, perii

[ivory sorrel]

and here 2n+1 is odd \forall n\in N

now , if $x^n=y^n$ to prove $x=y$, if n is odd

[thorny flame]

Veni, vidi, perii

[ivory sorrel]

I mean one way is just $x^n-y^n=0$

[thorny flame]

Veni, vidi, perii

[ivory sorrel]

factorise

and that would give x=y?

is that enough

[gleaming reef]

well

can we show that this can never be 0

[ivory sorrel]

not really

[gleaming reef]

uh

yes we can

[ivory sorrel]

x,y <0 is possible

[gleaming reef]

every term is positive

or wait

[ivory sorrel]

let x=-1, y=2

[gleaming reef]

x can be positive and y could be negative

ye

[ivory sorrel]

Well, this feels mechanical

I think I'll try another method?

$x =\frac{y^n}{x^{n-1}}$

[thorny flame]

Veni, vidi, perii

[ivory sorrel]

and $y = \frac{x^n}{y^{n-1}}$

[thorny flame]

Veni, vidi, perii

[ivory sorrel]

I guess I could re-write this as

$y = \frac{(\frac{y^n}{x^{n-1}})^n}{y^{n-1}}$

[thorny flame]

Veni, vidi, perii

[ivory sorrel]

uuh

[gleaming reef]

chef

what's going on

[ivory sorrel]

I'm trying a different method

[gleaming reef]

can I go cook some food

[ivory sorrel]

$y^n = \frac{y^{n^2}}{x^{n^2-n}}$

[gleaming reef]

and come back to veni viid peri

[thorny flame]

Veni, vidi, perii

[ivory sorrel]

sure, why do you have to ask me

I mean I now kind of have to prove the same thing

[frail reef]

are you trying to prove this?

[ivory sorrel]

yes

I looked at MSE

looks like the same method applies

[frail reef]

cant you just use a contrapositive argument?

[ivory sorrel]

I don't think I've done that yet

[frail reef]

$A \implies B \iff \neg B \implies \neg A$

[thorny flame]

eugene_krabs_has_cake

[frail reef]

you seen that before?

[ivory sorrel]

oh right

yeah

ok, so if $x=y \to. x^n=y^n$

[thorny flame]

Veni, vidi, perii

[frail reef]

$x \neq y \implies x^n \neq y^n$

[thorny flame]

eugene_krabs_has_cake

[frail reef]

for odd n

[ivory sorrel]

I mean I'd use the same argument here

[frail reef]

well havent you already proven it?

you prove if x < y then x^n < y^n for odd n

[ivory sorrel]

yeah, I proved that here

[frail reef]

ok then youre done

[ivory sorrel]

Oh

wow

thanks

.close

[topaz sinew]

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[ivory sorrel]

Thanks a lot both of you

[frail reef]

(assuming that proof is correct)

[ivory sorrel]

👍

[topaz sinew]

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[turbid pebble]

do I have to integrate 1/4u but also if I do how do I do that

[loud oasis]

you integrate 1/(4u) the same way you would integrate 1/(4x)

[turbid pebble]

is that 1^(-4x+1)/4x+1

or 1^(-4x+1) / -4x+1 with both being negatives

[loud oasis]

no

[turbid pebble]

oh ok

[loud oasis]

do you know what the integral of 1/x = x^{-1} is?

[turbid pebble]

no

[loud oasis]

do you know the derivative of ln(x)?

[turbid pebble]

oh 1/x

[loud oasis]

yes. so following that, what is the integral of 1/x ?

[turbid pebble]

ln(x)

[brave fox]

,w integral of 1/x

[thorny flame]

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Requested by cyclic_tree

[turbid pebble]

okay i see

[brave fox]

huh. I was sure it had absolute value?

[thorny flame]

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[loud oasis]

to be extra general we usually write ln(|x|), since that has a larger domain

[turbid pebble]

so would the integral of 1/4x be log(4x)?

oh cus you separate the 4

sorry yeah

okay thank you

sorry so like since i'm doing u substitution it is (1/4*lnu) and I just plug in my u and + c

.close

[topaz sinew]

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[topaz sinew]

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[mystic siren]

(A)i,j=|i-j|
A is nxn.

calculate detA.

any can walk me through solving it?

[mystic siren]

chatgpt seems to have incorrect solutions

also, was on my test yesterday

and there's no solutions

and conflicting answers

[cursive patrol]

what kind of course is this from?

[topaz sinew]

@mystic siren Has your question been resolved?

[mystic siren]

linear Algebra 1

[cursive patrol]

do you need to prove your answer?

[mystic siren]

yes of course

[cursive patrol]

what have you covered so far?

[mystic siren]

all the det() operations you can do, linear transformations and more

[topaz sinew]

@mystic siren Has your question been resolved?

[topaz sinew]

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[minor tapir]

Pls help

[cedar wagon]

Where are the asymptotes for tan (x) ?

[minor tapir]

-pi/6 and pi/6

[cedar wagon]

No just for tan(x) not the function

[minor tapir]

odd multiple of pi/2?

like pi/2 , 3pi/2 , 5pi/2 etc

[cedar wagon]

Indeed, so the number a, would be the solution of a*pi/6 = pi/2

[minor tapir]

Ah ok thanks

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[outer portal]

I found a potential way to answer this question

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[ivory sorrel]

Prove that $|\frac{1}{x}| = \frac{1}{|x|}$

[thorny flame]

Veni, vidi, perii

[ivory sorrel]

so to start I considered $(|x|^{-1})^{-1}= |x|$

[thorny flame]

Veni, vidi, perii

[ivory sorrel]

so from that it follows that $|x|^{-1} = \frac{1}{|x|}$