help-39

[brave sluice]

seems right

[wheat remnant]

thanks! how do I know how convex the graph should be though

for a perfect sphere

[brave sluice]

i think it's an ellipse

[wheat remnant]

ou that would make sense

[brave sluice]

i think the height is pi times the distance from x to m

because halfway up the sphere the equator is 2pi * radius of the sphere

[wheat remnant]

what height are we talking about here

the highest point on the y axis?

[brave sluice]

yes

[wheat remnant]

oh yes true! Thanks so much, I'm happy with that :)

I'll close the channel now. have a nice day

.close

[pearl pond]

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[pearl pond]

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[rustic kernel]

been staring at this for the longest time. I'm so lost.

[rustic kernel]

So far, I think I use this?

the top one, Scalar by vector

from here, I have no idea how to proceed

I've always been dogshit with proofs

[sharp vigil]

write out a formula for w^T w in terms of w1,w2,etc

[pearl pond]

@rustic kernel Has your question been resolved?

[rustic kernel]

soo the summation of (w_1)^2 to (w_n)^2?

[sharp vigil]

yes

[rustic kernel]

I just use the summation symbol or is there another way to write that out?

[sharp vigil]

either the summation symbol or +...+

[rustic kernel]

how should I proceed?

[sharp vigil]

take the derivative wrt w1, w2, etc and put them in a vector

[rustic kernel]

so I just add x to everything right

should be all constants

I think I forgot how to derive, gimme a moment

I was thinking of integral

[pearl pond]

@rustic kernel Has your question been resolved?

[pearl pond]

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[bitter herald]

What does it even mean to have a "pole at infinity"?

[bitter herald]

that seems like nonsense

[blazing notch]

It would mean if you evaluate at a complex sequence z_k such that |z_k|->inf, then N(z_k)/D(z_k)->inf

Or if talking informally, a complex number with infinite magnitude, then X has a pole at that infinite magnitude number

[bitter herald]

Ok gotcha!

On another notes, I don't quite understand what they did here. Would you mind guiding me through it?

[blazing notch]

I'm sorry but u can't even read the x(n) that's written I can read the b^n and b^-n but not that parts next to them

[pearl pond]

@bitter herald Has your question been resolved?

[pearl pond]

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[craggy crypt]

I dont understand this at all can someone explain

[scarlet glade]

that looks like u substitution, do you know how to do u substitution?

[craggy crypt]

No

I dont know that

Not with the bounds of the integral

[dusk horizon]

It's actually just the chain rule

[craggy crypt]

I'm stuck I don't see it

[scarlet glade]

isn't it technically reverse chain rule or something?

[craggy crypt]

How are they forming that

[dusk horizon]

Pretend the upper bound is x instead of sinx. What would you do?

Nope, just normal chain rule

[craggy crypt]

I haven't integrated with variable bounds

[dusk horizon]

You're not integrating anything. You're differentiating

[craggy crypt]

Buf there's an integrate sign in there

[dusk horizon]

That's just part of the function. It's not asking you to integrate

[craggy crypt]

So what am I actually doing

Could you break it down please

[dusk horizon]

You didn't post the question, but the answer starts with "to differentiate the integral directly", so I'm assuming you're asked to find a derivative.

[craggy crypt]

The question is yea fund the derivative

Of that function

[dusk horizon]

In this case you're applying a rule, the fundamental theorem of calculus.

You should read that first

[craggy crypt]

I did

The first theorem is just finding the antiderivative right

[dusk horizon]

No, the part you want has no mention of anti derivative

[craggy crypt]

Oh

[dusk horizon]

The answer you posted tells you the part you need

[craggy crypt]

I'm looking at it

F(t) in this case is 14t^13

X is sin x

A is 1

[dusk horizon]

Okay. To start, pretend the upper bound is just x

[craggy crypt]

Okay

How do you differentiate an integral

[dusk horizon]

It tells you in the screenshot you posted

[craggy crypt]

I dont understand that

[dusk horizon]

Do you understand the individual symbols?

[craggy crypt]

Like I understand what everything is but I don't understand what it's telling me if that makes sense

[dusk horizon]

Try replacing f(t) with t^3, and a with 5

Then calculate both sides and see what happens.

[craggy crypt]

Okay let me try

I'm lost

[dusk horizon]

Do you know how to find the integral of t^3?

[craggy crypt]

Ysa it's 1/4 t^4

[dusk horizon]

Also, what is f(x) here?

[dusk horizon]

Now, do you know how to plug in bounds?

[craggy crypt]

X^3?

Let me write that one moment

[craggy crypt]

That was an educated guess what's the reasoning

[dusk horizon]

f(t) = t^3 means that anything you plug in gets cubed. f(2) is 8. f(3) is 27.

[craggy crypt]

Makes sense

[dusk horizon]

The d/dx is telling you to take a derivative, so do that

[craggy crypt]

Wow

So I need to apply that?

[dusk horizon]

Yes, though it's a bit more complicated with the sinx part, so it helps to understand the easier version first

[craggy crypt]

Yea thanks for that

Let me try brb

I'm a bit stuck

[dusk horizon]

!occupied

[pearl pond]

Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[dusk horizon]

The whole point is that you don't need to take any integrals. The right hand side tells you what will happen after you take an integral and a derivative

[craggy crypt]

Waif I wrote it wrong

[craggy crypt]

Yea

[dusk horizon]

You can shortcut the whole thing into one line

[craggy crypt]

True

Oh u don't even gotta differentiate really

Or integrate

Right?

[dusk horizon]

Right now that's correct

[craggy crypt]

I don't understand the 14x^13 on the left

For this function?

[dusk horizon]

Hold up. You gotta do the easy version first

[craggy crypt]

Okay

What's next

[dusk horizon]

And you don't need to take any integrals or derivatives. I only told you to do that so that you could see what was happening. The two were undoing each other.

[craggy crypt]

Yea okay

[dusk horizon]

You haven't done the first part yet, have you?

[craggy crypt]

What first part

[dusk horizon]

This

[craggy crypt]

Oh

[craggy crypt]

For this ?

Ah I see

[dusk horizon]

Yes

Okay, now, call the integral part g(x). Just to give it a new name so you don't get distracted by all the symbols

Oh, exponent is 13 but that doesn't really matter

[craggy crypt]

Yea sorry

[dusk horizon]

Good. Now you're being asked to find the derivative of g(sin x)

With no greater context, what is the derivative of g(sin x)?

[craggy crypt]

Im not sure, Is it cosx g(sinx)

[dusk horizon]

You're missing a prime

[craggy crypt]

The power reduces by one

[grim grove]

This

[dusk horizon]

<@&268886789983436800> this person has disrupted multiple channels, this one more than once.

[craggy crypt]

Can you explain

[dusk horizon]

Look up the chain rule

I thought you had a typo, but it seems you have a misunderstanding of it

[open blaze]

You are timed out for a day, learn to ask for help properly. Next offense will lead to escalated consequences

[dusk horizon]

The start and the end of that are correct, but the middle is not

But we only need the start and the end

[craggy crypt]

Wait do u apply the product rule

Why is the middle wrong

[dusk horizon]

There's no product in the original function.
So now in our question, what is g'(sinx)?

[craggy crypt]

Yea

[dusk horizon]

If you compare it to the first term, you must conclude that d/dx sinx =1

[craggy crypt]

What first term

[dusk horizon]

Just look at the top line

Something is equal to itself times something else

So that something else must be 1

But it clearly isn't

[craggy crypt]

Ah

Okay next

[dusk horizon]

Here

[dusk horizon]

You found g'(x) here

[craggy crypt]

14x^13?

So

g'(sinx) is 14 (sinx)^13 cos x?

That's it?

[dusk horizon]

Yup

[craggy crypt]

By the way how is the chain rule done correctly here

[dusk horizon]

You could just skip the middle expression, or you could write d/d(sinx) g(sinx)

[craggy crypt]

Oh

This?

Okay

[dusk horizon]

In books, you will see the letter u in place of sinx or whatever your inside function is

[craggy crypt]

Ah

Hey thanks a lot

Really

I appreciate it

[dusk horizon]

You're welcome

[craggy crypt]

One question we didn't really use the fundamental theorem of calculus really, we just did chain rule?

[dusk horizon]

We used it here.

In a sense, you could have taken the integral, plugged in the bounds, and then differentiated, because this function is pretty simple.

But you might also notice that the "take the integral and plug in the bounds" is actually the fundamental theorem of calculus.

[craggy crypt]

Yaa I did that for part one because that's the second theorem of calculus

[craggy crypt]

That's the second part of the theorem right?

[dusk horizon]

So we probably could have used either part for this question.

[craggy crypt]

I dont really understand the difference between part 1 and 2

[dusk horizon]

Yeah, it must be, since the first part is the derivative of the integral

[craggy crypt]

Yea

[dusk horizon]

The two parts are really saying the same thing, which is why they're parts of the same theorem

[craggy crypt]

Oh I see

[dusk horizon]

One part says that derivatives cancel integrals, and the other says that you can solve integrals with antiderivatives.

[craggy crypt]

Okay sweet

Let me note that

Right, thank you. I will now carry on doing questions. See you around and take care.

❤️

.close

[pearl pond]

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[sleek cobalt]

i need help my professor said that if anyone can solve this they can stay home tomorrow for class [f(2x) = 98x3.55 + 2x + 212]

[restive briar]

Solve it how

[pearl pond]

@sleek cobalt Has your question been resolved?

[hearty dew]

The way you write the equation is kinda confusing, if youre referring multiplication, you can put a dot or just use parentheses

[vast eagle]

f(2x) = 98(3.55) + 2x +212 ?
f(2x) = 98.155 + 2x + 212 ?
I didn't understand yet there is nothing to solve

[pearl pond]

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[robust nexus]

any way to do this without normal vectors and cross products?

[robust nexus]

just through linear combination is it possible?

cuz the direction vectors of the top plane should combine in a way to get the direction vectors of the bototm plane

wait how would I set up my matrix?

@cinder thistle

[cinder thistle]

so

it would be a rectnagle matrix

[robust nexus]

4x2 ?

or 2x4 ?

[cinder thistle]

2 colums 4 rows

[robust nexus]

could you please write out the matrix

so i can understand

i dont get what my b vector would be

[cinder thistle]

2 -3
1 1
-2 5
7 2

[robust nexus]

cuz im tryna equate it to two things

[cinder thistle]

M = this

and you want to show there is a A vector

so that

MA =

[robust nexus]

yep that part i got
i got lost on the b part

[cinder thistle]

3
-1
2
4

[robust nexus]

3 -1 2 4 i see

[cinder thistle]

and

-1
4
2
6

[robust nexus]

so id need two matrix ?

or just one

cuz in the rhs
we can only have one column ye?

[cinder thistle]

1 matrix

two eqns

[robust nexus]

oh so rhs has two columns ?

[cinder thistle]

no

[robust nexus]

in augmented matrix

[cinder thistle]

there will be 2 eqns

first is

2 , -3 | 3
1 , 1 | 1
-2 , 5 | -2
7 , 2 | 4

second is

2 , -3 | -1
1 , 1 | 4
-2 , 5 | 2
7 , 2 | 6

[robust nexus]

oh so doesnt this mean 2 augmented matrix?

[cinder thistle]

yes

but only 1 matrix

[robust nexus]

ohhhh 1 matrix but 2 augmented
trippy ngl

[cinder thistle]

hm

[robust nexus]

ye i got no solution

which means not parallel

i just dont get why we have two equations

have combination of the top direction vectors equal one of the bottom direction vectors

then write another aug matrix equating to other direction vector

i dont get the reasoning behind that...

[cinder thistle]

so

you want both vectors

to be linear combinations

so that is what you do

[pearl pond]

@robust nexus Has your question been resolved?

[robust nexus]

ye ill probs watch a couple more vids on linear independence and combinations

thank you though

appreaicate

[pearl pond]

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[urban sun]

i had trouble converting this to a perfect square

[urban sun]

I took 4 common from 4y^2-20y

Do i add 25 or 100

4(y^2-5y) +12x+67=0

to make a perfect square do i add 100 or 25

[warm osprey]

I don't think this is the right first step

[warm osprey]

I think 100

[urban sun]

first step is to make coefficient of y^2 equal to 1?

right

[warm osprey]

Yeah nvm I was wrong

[urban sun]

are u sure

[warm osprey]

So 25

[urban sun]

why 25

[warm osprey]

the coefficient of y afted dividing both sides by 4 is 5, 5 squared is 25

[urban sun]

yes but logically we need to add 25 in the term inside the bracket

so must be multiplied by 4 to

Oh wait

[warm osprey]

Divide both sides of the equation by 4 instead of factoring

[urban sun]

what to do with 67

[warm osprey]

divide it by 4

[urban sun]

ok

after taking lcm itd be 100

y^2-5y= -3x-(67/4)

add 25 on both sides

[warm osprey]

Then you get y^2 -5y + 3x - 67/4 = 0
Then y^2 -5y + 25/4 - 25/4 + 3x - 67/4 = 0 => (y - 5/2)^2 - 23 + 3x = 0 ....

[urban sun]

(y-5)^2= - ( 3x+67/4 -25)

So basically the same as adding 100

after taking common

[warm osprey]

This is not correct. Multiply the LHS out and you will get -10y (should be -5)

You need to add and subtract 25/4 instead of 25

[urban sun]

oh

so its same as adding 25

after taking common

[warm osprey]

I suppose

[urban sun]

ok

thnx

.close

[pearl pond]

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[astral pebble]

guys i have no idea how to start i js know i need to releate height to the radius

[edgy stone]

it's a sphere

try drawing lines to the center

a line from the surface of the sphere to the center will have length R

i see at least two important lines like that in that diagram

[astral pebble]

hmm r u saying that height of cone + radius of cone=2r of sphere?

[edgy stone]

i don't think that's true unfortunately

[astral pebble]

oh

[edgy stone]

big hint: how long are the black lines? how long is the purple line?

that should be all you need

[astral pebble]

,rotate

[astral pebble]

Hmm smth like this?

[pearl pond]

@astral pebble Has your question been resolved?

[pearl pond]

@astral pebble Has your question been resolved?

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[crimson cloak]

Open

[crimson cloak]

.close

[pearl pond]

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[lucid valley]

i need to solve exercises of this kind related to like the drawing thing "Determine for which values of k the point P(k-3,4-k) belongs to the second quadrant."

[k<3]

[lucid valley]

Cartesian plane

[regal herald]

you seem to have the result though

[lucid valley]

it may be a janky translation my book is in Italian and i dont speak Italian

[regal herald]

the words are all fine

[lucid valley]

but what does it want me to do 😭

[regal herald]

think about the restrictions on x and y

if its in the second quadrant is x positive or negative?

[lucid valley]

negative right?

wait no

positive

becouse it goes up

right?

or

Ohhh you mean the blue thing

yeah the answers are there somtimes

that dosent matter I gotta know how to do it

and im not sure how to do it

does that make sense?

[regal herald]

negative is right haha

so x<0
what about y

[lucid valley]

but it goes up in the second one no?

[regal herald]

up and down relate to y

left and right to x

second is top left quadrant

[lucid valley]

like this?

[regal herald]

yup

[lucid valley]

ty

[lucid valley]

how do i solve tough

[regal herald]

what restrictions are on x and y?

[open rivet]

huh? why not get a book in your language then?

[lucid valley]

im an immigrant

[open rivet]

ah, i see

apologies

[lucid valley]

no its fine dw

[open rivet]

are you not able to get some extra help then?

[lucid valley]

well they are helpful at school and it works with the rest of the subjects but

im very very bad at math

so

yk?

my teacher is nice but like i was bad at math in my home country i wont magically become great at it here

[open rivet]

fair enough

where were you in the question?

[lucid valley]

i didnt start doing it yet they were trying to explain it to me first

[open rivet]

okay, so lets go to the start

we want the point to be in quadrant II

that is the top-left quadrant

that means, x<0 and y>0, yes?

[lucid valley]

ye

sorry for the delayed response

[open rivet]

our x here, is k-3

and our y is 4-k

[lucid valley]

mhm

why k though

[open rivet]

thats our given point

look at the question

[lucid valley]

ahh

[lucid valley]

i see

[open rivet]

so, which two inequalities do we get?

[lucid valley]

k is bigger than 0?

[open rivet]

\begin{align*}x&<0\y&>0\x&=k-3\x&=4-k\end{align*}

[jolly parrot]

Bonk

[lucid valley]

OHHHHHHH

[open rivet]

then we fill in the x and y

[lucid valley]

by making a dot?

[open rivet]

no, into the inequalities i mean

i just noticed the second one is an x, mb

[lucid valley]

its ok does that change the stuff im supposed to do?

[open rivet]

no

so, what do inequalities do you get now?

[lucid valley]

im not quite sure what inequalities are in this case

i know its like 4>5

but i don't think its like that here right?

it means something else?

or is it like that

[open rivet]

do you see the arrows?

[lucid valley]

yeah

[lucid valley]

mhm

[regal herald]

they mean, if x<0 and x=k-3, then it must be that k-3<0, for example

[open rivet]

$\left.\begin{align*}
x<0 \
y>0 \
x=3-k\
y=4-k
\end{align*}\right\rbrace \implies \begin{cases}3-k<0\4-k>0/end{cases}$

[jolly parrot]

Bonk
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[open rivet]

fk

[lucid valley]

mr robot please dont do that to me its hard as is 😭

[open rivet]

$\left.\begin{matrix}x<0\y>0\x=k-3\y=4-k\end{matrix}\right}\implies\begin{cases}k-3<0\4-k>0\end{cases}$

yes!

do you see whats happening here?

[jolly parrot]

Bonk

[open rivet]

do you see what happened?

[lucid valley]

so k is 5?

[open rivet]

theyre inequalities

so it doesnt have to do with equal

[lucid valley]

x is a minus y is a +?

[lucid valley]

becouse -<0 and +>0

[open rivet]

look at the inequalities on the right

the k-3<0 and 4-k>0

try rewriting those into k ><?

[lucid valley]

0><0

[open rivet]

no....

[lucid valley]

im sorry im really bad at math

[lucid valley]

hey maybe you could show an example of a different exercise if you want?

[open rivet]

no its alright

[lucid valley]

maybe that will help me understand better?

[open rivet]

unless you are not following at all

[lucid valley]

im trying but im not sure what kind of inequality lm supposed to find

[open rivet]

an inequality is instead of an = (equality) there is a > or <

so, 3x+4=0 is an equality

but 3x+4<0 is an inequality

[lucid valley]

yes but that is already shown in the exercise so I'm not sure what im supposed to like

show

[lucid valley]

ohh

[open rivet]

we need to show for which k, the point is in the second quadrant

and a point is in the second quadrant, if x<0 and y>0

so from there we get, k-3<0 and 4-k>0

our k that we find, but satisfy both inequalities

[lucid valley]

i just understood that there are 2 ks💀

[open rivet]

its one k, its the same one

it just has to satisfy both of the inequalities

[lucid valley]

ohhh

wait isnt that impossible

how can that be if its simoultaniously less and more than 0

wait

K is 3?

4-3 is one and 3-3 is- nvm

wait

no

what if its 3.5

[open rivet]

$\begin{pmatrix}k-3<0\4-k>0\end{pmatrix}\iff\begin{pmatrix}k<3\4>k\end{pmatrix}\iff\begin{pmatrix}k<3\k<4\end{pmatrix}$

[lucid valley]

would that work

[jolly parrot]

Bonk

[open rivet]

do you follow this?

[lucid valley]

yes actually

so its 2?

no

its not 2 that wouldn't work in the first part

[open rivet]

it is 2

and its also 1

and also 0

and -1

and -2

and 1.4

and 2.99

[lucid valley]

I GOT IT? SORT OF?

OH MY GOSHHH

:0

[open rivet]

😄

[lucid valley]

YESSS

[open rivet]

its a range of values

[lucid valley]

THANK YOU

[lucid valley]

its that easy? its just that?

oh my goshhh

sorry i was overcomplicating it in my head or smth

[open rivet]

so looking at that last part, what is the range of k?

[lucid valley]

from - infinity to 2

right?

[open rivet]

so close

look at these

in words: k is less than 3, and k is less than 4

[lucid valley]

i meant minus infinity

like 2 , 1 , -1,-2,-3....

so its anything from 2 to any other number less than 2

[open rivet]

thats not the problem

[lucid valley]

oh-

[open rivet]

its any number less than 3

[lucid valley]

sorry

[lucid valley]

OHH

so starting from 2.99 to - infinity

[open rivet]

yesss

[lucid valley]

:0

AHHH

YAYYYYY

BONK THANK YOU 😭

[open rivet]

does this make sense?

[lucid valley]

yeah

[open rivet]

both of these need to be true

and for k<3 thats the case

[lucid valley]

do i need to do something else?

[open rivet]

https://www.geogebra.org/calculator/sddge6t5 here

take a look at this

play around with the slider

[open rivet]

move the graph a bit

the point is not visible

[lucid valley]

like this?

[open rivet]

go towards the origin

(0,0)

anyway

i have to go now

goodluck!

[lucid valley]

ty byee

oh look a P 😄

thank you

[pearl pond]

@lucid valley Has your question been resolved?

[lucid valley]

im not sure but your a robot

so you dont actually care

so eh

guys is it solved?

[wild token]

what is ur question

[lucid valley]

well im not sure if its that easy for the answer to just be like less than 3 but if thats it i think i understand it

lemme show you

[wild token]

show me the question

and where are you stuck

[lucid valley]

here

[lucid valley]

im not sure if I'm stuck or not i think it might have gotten solved?

thats my question did my question get solved or is this just a step?

[lucid valley]

if it did get solved then its ok i understand where the like 2.99 and - infinity came from

[lucid valley]

ykwim?

[wild token]

i mean i get taht u have a point P yes but ifeel like u're missing informations there ? what is the question even and what did they provide

which quadrant u talking about

[lucid valley]

2

[wild token]

yh but what is the equation

[lucid valley]

this is the question

[wild token]

can u pinpoint the second quadrant its probably a language barrier

[lucid valley]

the second quadrant is the one on the top left

[wild token]

u're working on space geometry?

then u have it solver

d

you just need k-3<0

<=> k<3

[lucid valley]

the cartisan plane

[lucid valley]

thank you thats what i needed to hear

thx bonk and everyone else that helped

[lucid valley]

you tooo ty

[wild token]

yw

[pearl pond]

@lucid valley Has your question been resolved?

[pearl pond]

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• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
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[floral sparrow]

Hi so the question I’m doing is question ex9L question 3c. And I’ve done this like 3 times now and I keep on getting it wrong can someone tell me what I did wrong?

[versed mica]

holy scribbles

[floral sparrow]

😓

Idk what I’m doing wrong

[random ermine]

,w integrate 6*cbrt(x) from 8 to 27

[versed mica]

$(27)^{4/3} \neq 74$

[jolly parrot]

Results provided by WolframAlpha

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Upgrade to WolframAlpha Pro!

Requested by mayervietorus

knief

[random ermine]

,w simplify 1044/4

[jolly parrot]

Results provided by WolframAlpha

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

Requested by mayervietorus

[floral sparrow]

Im just gonna re do it

[versed mica]

i told you your mistake

27^(4/3) =81

not 74

[floral sparrow]

idk then

How do I do then

It

[versed mica]

🤔

[limpid lily]

Well, first take the cube root of 27.

Then, take the result to the 4th power..

[random ermine]

how did you get 74

[floral sparrow]

Idk it just happened

Uh cube root of 27 is 3 right?

[limpid lily]

Yes.

[versed mica]

and 3^4 is

[random ermine]

74

[floral sparrow]

So then 3x3x3x3, so 3x3 =9 so then 9x3 is 27 and then 27 x 3 is 74

[limpid lily]

3^4 = 3 * 3 * 3 * 3 = 9 * 9

[floral sparrow]

Unless I’m tweaking

[limpid lily]

No, 27 * 3 isn't 74.

[floral sparrow]

I had like 2 hours of sleep 🙏🙏🙏

[versed mica]

27 x 3 = 74

🔥

[random ermine]

lol

[floral sparrow]

What is it gang

[versed mica]

🤔

[random ermine]

just use a calculator

[limpid lily]

Or do it on paper.

[versed mica]

try 20 * 3 + 7 * 3

[limpid lily]

Or that.

[versed mica]

or 30 * 3 - 3 * 3

[random ermine]

Or that.

[limpid lily]

If you forgot how to multiply on paper: https://www.wikihow.com/Do-Long-Multiplication.

[floral sparrow]

Oh wait

Why’d I think it’s to the power of 3

What the heck

Okay I’m tweaking

[versed mica]

even still

[floral sparrow]

It’s like 81

[versed mica]

no idea where 74 came from

[versed mica]

yep

[floral sparrow]

It just appeared

Idk how

[versed mica]

if you multiply an odd number by an odd number and get an even number a light should go off in your head

[random ermine]

me training this reflex every chance i get

[floral sparrow]

I think it’s the 2 hours of sleep getting to me

Anyways

Ty

[versed mica]

go to sleep

[floral sparrow]

no i must do this.

Maths before sleep

🙏

I’ll go to sleep earlier today…

Anyways

.close

[pearl pond]

Channel closed

Closed by @floral sparrow

Use .reopen if this was a mistake.

[versed mica]

what

that’s not my writing

i took a screenshot and circled their mistake with my phones pen

[pearl pond]

Available help channel!

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• After 15 minutes, feel free to ping <@&286206848099549185>.
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[boreal raptor]

Integration help
Why am I getting 2 different solutions?
Which method is correct

[boreal raptor]

Question at the top

Method 1 on the left

Method 2 on the right

[plush bramble]

both are correct

[boreal raptor]

I did method 1 but answer sheet says method 2
gpt also said method 1

[plush bramble]

just that C is different

[versed mica]

who said they were different

[boreal raptor]

oh constant..

i plugged into desmos

and they weren't the same

so i thought

huh.. getting 2 diff answers

[versed mica]

because you assumed they had the same C

[boreal raptor]

i forgot

yeah

i forget that with ln

that happens sometimes

which method should i use

[versed mica]

doesn’t matter

[boreal raptor]

both are acceptable?

thanks

.close

[pearl pond]

Channel closed

Closed by @boreal raptor

Use .reopen if this was a mistake.

[boreal raptor]

lol

[pearl pond]

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• After 15 minutes, feel free to ping <@&286206848099549185>.
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[supple roost]

$$
\lim_{x \to -\infty} F(x) = \lim_{x \to -\infty} (e^x + e^{2x \sin x})
$$

[jolly parrot]

Chrome

[supple roost]

1/e^infinity will be 0

[warm current]

Will it?

[supple roost]

e^(-inity) will be 0

[wet swallow]

That's e^0

[supple roost]

But i am not sure what should I do with e^(2xsinx)

[supple roost]

Hence 0

[wet swallow]

No

[supple roost]

1/e^infinity will be 0

Opps sorry

[wet swallow]

sinx oscillates between -1 and 1

[supple roost]

Yes sir

[warm current]

Intuitively, what do you think the limit is?

[supple roost]

I tried to think but not getting proper intuition

xsinx will be like increasing for x>0

And increasing x<0

x× [-1 1]

but negative sign so it will be max and minima both

e^max+, e^min-

DNE

Minima will be 0

And max is infinity

@warm current

[warm current]

Looks good

[supple roost]

Thanks

.close

[pearl pond]

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

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[plush bramble]

draw a triangle with angle sin^(-1) x

[celest heart]

alright!

what next tho, never had to use anything to draw

[warm current]

If you want a more mathematical proof, square both sides, and use cos²=1-sin²

[celest heart]

but that isnt allowed right cuz then i assume the proof is correct?

so i got to move from the left to the right

at least that is what I was told in lineair algebra maybe it is allowed in calc?

[warm current]

Okay, don't square both sides, square just $\cos(\arcsin x)$

[jolly parrot]

SWR

[severe quarry]

It's allowed to write equivalences, if this is equivalent to some simplified form and you know that simplified form is true, you can backtrack and follow that the first thing in your equivalence chain (the equation) is true

[celest heart]

Oh huh i was told i would get 0 points then

[severe quarry]

You can write an equivalence for squaring if you ensure both sides are positive

[celest heart]

Hmmm

imma do that first then but i want to use the triangle method after if that is possible

[severe quarry]

But you need to pay attention to what's an equivalence and what's not. Squaring is only an equivalence if both sides are positive

[celest heart]

Aaah alright thank you

[celest heart]

aaah okay ye the equivalence can sometimes still be tricky for me

[warm current]

Then yeah, you can do it without equivalence still. It just feels a little more obfuscated at times

[celest heart]

but how can i simplify (cos(1/sin(x))² ?

[warm current]

how did you get 1/sin?

oh

[celest heart]

ye they use arcsin and sin -1

[warm current]

$\sin^{-1}x\ne\frac{1}{\sin x}$

[jolly parrot]

SWR

[celest heart]

i suppose the sin -1 is 1/sin here then

[limpid lily]

No, it's the inverse function, like f^-1.

[celest heart]

but normally they write arcsin 💀

alright

[warm current]

$\sin^{-1}x$ is different from $(\sin x)^{-1}$. It's an unfortunate symbology

[jolly parrot]

SWR

[celest heart]

Pffff okay than how can i simplify cos(arcsin x )²

i know the normal rules with sin and cos but not with arcsin and stuff

[limpid lily]

Well, x is something you get after taking the sine of an angle. Let's say the angle is theta.

If sin(theta) = x, what is cos(theta)?

[celest heart]

pfff wait let me think about that

cos(arcsin x ?

but we already had that

[limpid lily]

To be more verbose:

arcsin(x) = theta
x = sin(theta)

Does that make sense so far?

[celest heart]

ye

[limpid lily]

OK, so cos(arcsin(x)) = cos(theta).

[wise gull]

I need help with this one pls

[limpid lily]

See the #❓how-to-get-help channel for how to get your very own help channel.

[celest heart]

Ooooooh

damn

ye that makes sense

[limpid lily]

So, if you have sin(theta), what is cos(theta)?

[celest heart]

idkkk

why do i not see this

[limpid lily]

OK, so you can use the Pythagorean identity.

sin^2(theta) + cos^2(theta) = 1.

[celest heart]

aaah okay

[limpid lily]

It's the main way of converting a sine to a cosine or vice versa.

[celest heart]

ooooh

so

sin²=1-cos²
sin(theta)=root(1-cos²(theta))

[limpid lily]

Right, but we want cos(theta) on the left.

[celest heart]

oh mb

just swap it

so cos(theta)=root(1-sin²(theta))

[limpid lily]

Right.

[celest heart]

right?

[limpid lily]

And what is sin(theta)?

[celest heart]

and sintheta is x

[limpid lily]

(from before)

Right.

[celest heart]

so sintheta ² is x²

DAMN

thank you very very much!

[limpid lily]

You're welcome.

[celest heart]

imma close it after i have taken a good note of it :)

[limpid lily]

OK.

[celest heart]

.close

[pearl pond]

Channel closed

Closed by @celest heart

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[celest heart]

.reopen

[pearl pond]

✅

[celest heart]

How could i have used the triangle method?

[limpid lily]

You can draw a right triangle with a hypotenuse of length 1.

Then, you can pick one of the non-right angles as the angle you're using.

Then, cos(theta) = adj/hyp = adj/1 = adj.

sin(theta) = opp/hyp = opp/1 = opp.

So, we have arcsin(x) = theta, so sin(theta) = x.

So, you write x as the opposite side.

Then you can use the Pythagorean theorem to find the adjacent side.

Then, cos(theta) = adj, so cos(arcsin(x)) = adj.

[celest heart]

aaah got it

thank you very much again

[limpid lily]

You're welcome.

[celest heart]

.close

[pearl pond]

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[pearl pond]

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

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[hidden imp]

i am an absolute beginner in algebra and i wanna see if my solve is correct

well i was guided a lot and i still think i messed up

[pearl pond]

@hidden imp Has your question been resolved?

[pearl pond]

@hidden imp Has your question been resolved?

[pearl pond]

@hidden imp Has your question been resolved?

[austere tinsel]

You're at least on the right track

Making a function f that maps elements of the sequence (a_n) to their place in the sequence n is definitely a great start, and then relating it to P(N) as a way of getting the carnality 2^N is fantastic

It kind of falls apart in the middle though, at least for me. I'm not sure what g or A is supposed to be here

@hidden imp If you're still down to work on this, do you wanna try explaining the purpose of g and A here? If you're not sure either though I'd be happy to help you patch your answer up!

[pearl pond]

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

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[rose sentinel]

Hello im looking for a good understanding of the DTFT. I get what it is but don’t understand its derivation or how it works intuitively. I know DFTs but I feel like there’s some gap between the two that I can’t grasp

[rose sentinel]

@last moth

[warm current]

!noping

[pearl pond]

Please do not ping individual helpers unprompted.

[rose sentinel]

He requested I pinged him in DMs

[last moth]

Can you describe your current understanding of DFTs and DTFTs?

[rose sentinel]

From my understanding:

Discrete Time Fourier Transforms are a discretized version of continuous fourier transforms in that they use samples taken at some sampling frequency rather than a continuous function of time. These samples are fed into a summation where the number of samples used and the index of the sample being summed are used to determine the angle of the vector in the complex plane that has a magnitude of x[n] After calculating all of these vectors for a given frequency, their sum will add to some value. This value will spike when the given frequency is close to or matches a frequency of a sinusoid approximated by the samples.

DFT is a discrete time to discrete frequency transform

DTFT is a discrete time to continuous frequency transform

From my understanding both utilize sampling, but the DTFT somehow gets a continuous frequency domain

Its formula is also different and I dont understand its derivation or much else beyond it

[last moth]

Do you remember the difference between Fourier series and Fourier transforms

That's very similar to the difference between DFTs and DTFTs

If you remember, Fourier series have a discrete frequency while Fourier transforms have a continuous frequency

That's because Fourier series act on a finite/periodic time interval whereas Fourier transforms work on an infinite line of time

A similar distinction holds for DFTs and DTFTs

DFTs are just Fourier series with sampled time bins, and DTFTs are just Fourier transforms with sampled time bins

Does that help @rose sentinel

[rose sentinel]

Yes

[rose sentinel]

What happens if I use the DFT to sweep continuous frequencies?

cant I just plug in any frequency I want

Or doesn't that remove its invertability or something if it doesnt match the same count

(I dont know much linear)

Also, I see what you're getting at, but how would you approach deriving the DTFT from the CTFT?

[last moth]

I'm not sure what you mean by plugging in. You apply the DFT to some finite vector of time-indexed values, and then you get out a finite vector of frequency-indexed values (of the same length). So the DFT is just a change-of-basis matrix.

Yes, for this operation to be invertible, the time-domain and frequency-domain signals need to be the same length. (They need to contain equivalent information.)

[rose sentinel]

i see

[last moth]

You can essentially just replace the integral over all time points with a sum over all time samples instead

The Wikipedia page has a section on this

[rose sentinel]

I saw that section but didnt really get it

[last moth]

The top formula is the Fourier transform, right?

[rose sentinel]

Right

[last moth]

The bottom formula just comes from replacing the integral with a sum, because now we have discrete time instead of continuous time

If we have discrete time samples spaced apart by T time (so ..., -2T, -T, 0, T, 2T, ...), then we just plug in t = nT

and we get the formula on the bottom

[rose sentinel]

why is Ts(nT) = s[n]?

[last moth]

s(nT) is just the value of s at the nth time sample

dt turns into T

[rose sentinel]

mhm

[last moth]

so you get T s(nT)

[rose sentinel]

right

[last moth]

and then e^(-i 2 pi f Tn) just comes from plugging in t = Tn

idk why they wrote nT in one place and Tn in the other, but same thing

[rose sentinel]

i get that but what is the significance of defining s[n] to be the sample value at nT times that time?

[last moth]

wdym nT times that time?

nT is the time that the nth sample takes place at

[rose sentinel]

I see but then why multiply that by T

[last moth]

T is dt

[rose sentinel]

yes, but why define an encompassing s[n] to include that

[last moth]

oh

[rose sentinel]

moreover why is T*s(nT) = s[n], I assume brackets signify a discrete function(?) and parenthesis imply continuous

[last moth]

yes, including the T in the s[n] here just makes the formula tidier so you get that the formula is just the dot product of s and e^(-i 2 pi f T n)

you could choose to normalize it a different way if you wanted, it doesn't really matter

then the formula would just be T * the dot product of s and e^(-i 2 pi f T n) instead

[rose sentinel]

wheres the dot product come in? I never really understood that part

[last moth]

oh, the dot product is just a sum

sum_n of s[n] * e^(-i 2 pi f Tn)

you can also write that as s · e^(-i 2 pi f T n)

same thing

[rose sentinel]

mhmm

what happens to the nT inside the s(nT) and why does multiplying it by T not only discretize it but also remove the interior T?

is it just pure convention?

I dont get it

[last moth]

if you think about s[n] as the magnitude of s at the nth time sample, scaled by how long one sample is, then it makes sense

the nth time sample occurs at t = nT

and one sample lasts for T time

so therefore we'd expect something like s[n] = s(nT) * T

we're abusing s here to mean two different things

on the left-hand side, s is a sequence of numbers, one value s[n] for each integer value of n; on the right-hand side, s is a function of continuous time

we've just decided to give them the same name for convenience

[rose sentinel]

Im still kind of lost

s[n] is the magnitude of s at the nth index, I get that. I dont get why it makes sense that n = nT

[last moth]

t = nT

n doesn't equal nT

[rose sentinel]

but the nth time sample occurs at t = nT so s(t) = s(nT), so why does multipling s(t)*T make it s[n]

[last moth]

perhaps a picture helps

you can think of the function s(t) as the red curve here

[rose sentinel]

mhm

[last moth]

then we discretize it by sampling at ..., -2T, -T, 0, T, 2T, ...

s[n] represents the area of the nth orange rectangle

[rose sentinel]

why the area?

why not just the value

s(nT)

[last moth]

you could do that too, you would just need to make the formula sum_n T s[n] e^(-i 2 pi f T n) instead for it to mathematically work out

it really doesn't matter so much since it's just a constant factor

[rose sentinel]

Okay i get the s[n] part

Im forgetting why area is important here though

[last moth]

you can just consider the value instead too

the area and value are proportional so it doesn't affect the theory at all, besides the constant factor of T somewhere in the formula

here they're just bundling (s(nT) dt) into s[n]

[rose sentinel]

right, but what is the meaning behind finding the area to begin with

is it just the inner product thing

where its 0 everywhere but equal frequencies

[last moth]

you could imagine that you're replacing the integral in the FT with a Riemann sum, which involves adding up areas of rectangles

[rose sentinel]

right

[last moth]

so s[n] = T s(nT) represents the width of the rectangle times the value of the signal

when you multiply that by e^(-i 2pi f Tn), you get the area of the rectangle in the Riemann sum

and then you can just take the sum

again you can just define s[n] = s(nT) if you want too, that's okay, there just needs to be a T somewhere in one of your formulas because it's the width of the rectangle in the Riemann sum

[rose sentinel]

wdym you get the area? I thought s[n] is the area

[last moth]

the height of the rectangle in the FT is the value of the signal s(t) times the exponential e^(-i 2pi f t)

so the area of the Riemann sum rectangle is T * (s(t) * e^(-i 2pi f t))

here the Wiki article is wrapipng T * s(t) into s[n] for convenience

and then s[n] e^(-i 2pi f nT) would be the area of the full rectangle for the Riemann sum

[rose sentinel]

whats the significance of that? I learned it off the back that its just two embedded integrals which are inner products <s(t),cos(nt)> and i<s(t),sin(nt)>

[last moth]

what's the significance of what

you understand why the formula for the DTFT needs to be sum_n T s(nT) e^(-i 2pi f T n) right

[rose sentinel]

i think so

i dont get why youre redefining the inside of it to be per say f[n] where the value is the area

[last moth]

I'm just saying it doesn't really matter what you define s[n] to be, up to a constant factor, as long as you're consistent

[rose sentinel]

ok

[last moth]

if you define it as s[n] = T s(nT), your formula will be nicer because there's no longer a T in it

One nice property is that the sum of s[n] will now be approximately equal to the integral of s(t), if you care about that

[rose sentinel]

so then when I have a list of sampling values T apart, I have to multiply them all by T too?

[last moth]

instead of the sum of s[n] * T being approximately equal to the integral of s(t)

but again it's up to you

[rose sentinel]

so s[n] isnt simply my sampling vector

its the vector time T

[last moth]

yes, the T gets accounted for somewhere

[rose sentinel]

okay i can get behind that

[last moth]

practically speaking you probably don't care about the exact frequency power values anyways, just the peak frequencies

[rose sentinel]

sure

[last moth]

so multiplying by T doesn't affect it one way or the other

[rose sentinel]

and so what makes that frequency domain continuous unlike the DFT?

[last moth]

you can plug in any value of f

this is because you have an infinite amount of time

your n ranges across all integer values

[rose sentinel]

but couldnt i sacrafice invertability in the DFT for a continuous frequency domain

[last moth]

I think by that you're talking about aliasing? like multiple different frequency-domain signals corresponding to the same time-domain signal

I'm not sure why you would want that

the point of a FT is that time-domain and frequency-domain signals represent equivalent information

if you can have multiple different frequency representations of the same signal, then what's the point

[rose sentinel]

fair

does the DTFT have aliasing too then?

[last moth]

fundamentally your frequency resolution is limited by how much time you record over

[last moth]

yeah, it has the same problem with the nyquist frequency since it also involves sampling

aliasing probably isn't the right word

[rose sentinel]

so the frequency domain repeats every multiple of the sampling f/2 right?

[last moth]

yeah

[rose sentinel]

like the DFT

[last moth]

exactly

[rose sentinel]

oka

alright

[last moth]

oh wikipedia has a picture

This is a table of what frequency spectra look like for each of the four FT techniques

Fourier transform, Fourier series
DTFT, DFT

as you can see the Fourier transform has continuous frequencies, and the DTFT also does, but it exhibits the nyquist rate, where you can see that at frequencies below -1/2T or above 1/2T, it starts repeating

[rose sentinel]

whats the FFT coming in there for?

[rose sentinel]

Hmm alright

[last moth]

sorry I was a bit imprecise here, the original frequency domain is from -F/2 to F/2, then it repeats again but backwards going from F/2 to 3F/2, then it repeats again but forwards from 3F/2 to 5F/2, etc

[rose sentinel]

alr

[last moth]

the FFT is just an algorithm to compute the DFT, you can ignore it in this picture

[rose sentinel]

sure, but why is it there

[last moth]

the FFT just computes frequencies spaced from 0 to F

[rose sentinel]

ohhh

[last moth]

which as you can see is enough to describe the whole frequency spectrum bc it just repeats periodically

[rose sentinel]

because N = F right?

[last moth]

so you can see here for the DFT, since it occurs over a finite time, its frequencies are discrete, and since it's discrete (sampled) time, its frequency spectrum is periodic

[rose sentinel]

alright

[last moth]

it's not exactly right that N = F since N is an index and F is a frequency, but the Nth frequency bin has a frequency of F; more precisely, the FFT computes the 0th, 1st, ..., (N-1)th frequency bin values

[rose sentinel]

wont the DTFT be off from the FT

[rose sentinel]

Well I mean the total sample count

I learned it as N = total and n is index

[last moth]

yeah N is the number of total samples

equivalently it's the total number of frequency bins

[rose sentinel]

right

[last moth]

yeah, you can see in the pic here, the DTFT contains less info than the FT because the DTFT contains a bunch of overlapping copies of the FT repeated over and over again

[rose sentinel]

well i mean more than that

because youre using riemann sums

if you let T go to zero i guess

[last moth]

no we're letting T stay fixed

[rose sentinel]

yeah but T is your sampling frequency right

[last moth]

but the effect of that is what I described about the frequency spectrum

it's your sampling period* yeah

[rose sentinel]

right

[last moth]

it's kinda magical

there's some nice interplay between the discrete and continuous versions of the Fourier transform

there's a relevant theorem called "Poisson summation" here but I won't get into it

[rose sentinel]

so for large values of T, our DTFT is innacurate right? atleast over non integer frequencies

[rose sentinel]

Mhmm

[last moth]

the DTFT would still be exactly a bunch of copies of the FT summed up

but as you increase T, the width of information in the DTFT gets smaller and smaller

if you increase T, the Nyquist sampling rate gets smaller and smaller

[rose sentinel]

And we retain more of the frequency spectrum right

[last moth]

less of the frequency spectrum

[rose sentinel]

Oh right yeah my bad

Other way

[last moth]

because you're smushing the copies of the frequency spectrum closer and closer together

[rose sentinel]

So but generally, if T is high, won’t it generally be inaccurate

Not because of the overlap necessarily

But just generally

because it only has bits and pieces of the signal

[last moth]

No, the overlap is the only distortion

it is still exactly equal to the FT, but a bunch of copies summed up