help-13

[proven summit]

I thought you were a different person for a sec because of the pfp change 😭

[plain granite]

nah its just me

i need to mentally manifest

[crimson sedge]

can you tell me what this recursion means in english?

[plain granite]

i would have put berdly but i dont wanna say something like “oh my sweet gamer player 2…” or something

[plain granite]

hmmm

[proven summit]

okay you can think of it like this

[plain granite]

then it goes to x^-1 / 1 ?

[latent bloom]

How about we do this:

$x^k = x^k \times \frac xx$

[wraith dagger]

VulcanOne

[plain granite]

x^k would go on top right

[latent bloom]

Yeah if we keep multiplying it by x/x

The x at the denominator will keep going with it as well

[plain granite]

yeah

i believe though

ok

[plain granite]

wait no i dont show me an example or does the x on the bottom stay the same

[latent bloom]

Wait you're still confused about the z^-1?

[plain granite]

yes somehow

[latent bloom]

Alright

Let's start with 8

8 is 2^3

And let's keep dividing it by 2

When we divide 8 by 2, we get 4 = 2^2

[plain granite]

yeah

[latent bloom]

After that, we get 2 = 2^1

[plain granite]

yep

and then 1

[latent bloom]

After that, we get 2/2 = 1 = 2^0

After that, we get 1/2

Do you know what happened here?

[plain granite]

0.5!!!!

[latent bloom]

Yeah

We divided 1 by 2

And if you remember, 1 = 2^0

[plain granite]

yeah!!

[latent bloom]

And when you keep dividing, the exponent kept decreasing by 1

[plain granite]

yes

[latent bloom]

So that means when we divide 2^0 by 2, we gotta decrease the power of 2^0 by 1

0-1 = ???

[plain granite]

-1

[latent bloom]

Thus, 1/2 = 2^-1

[plain granite]

YEAH

[latent bloom]

That means that 1/z = z^-1

[plain granite]

how did i not see

[latent bloom]

1/number = number^-1

[latent bloom]

Dw we can't see a lot of things till we go through them from a different perspective

[plain granite]

oh ok ok

[latent bloom]

Alright does that answer all your questions?

[plain granite]

hm we’ll see

[latent bloom]

If you got more questions, feel free to post them

[plain granite]

do i just put 7b?

[latent bloom]

Mhm

[plain granite]

ok ok

[latent bloom]

You're getting good at this

[plain granite]

is this correct

[latent bloom]

Yep yep

Nice

[plain granite]

ok ok!

this too?

[latent bloom]

Yeppers

[plain granite]

oh ok!

12-14?

[latent bloom]

Hmm

12 is almost correct

You meant to write 6^-1 right?

@plain granite

[plain granite]

hm

ok ok

6^-1 c^5 ?

[latent bloom]

Ye

13 is correct

14 is correct

[plain granite]

alr alr

[crude brook]

You need to do 16-18 too or you good?

[plain granite]

if i need it

i’ll just say it

[crude brook]

Aight

[latent bloom]

The pfp is doing its effect lmao

[plain granite]

heh… glad to see its working

[cedar kiln]

@plain granite Has your question been resolved?

[plain granite]

i can open another one when i need it

[cedar kiln]

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[drowsy path]

@wheat spruce

[wheat spruce]

Okie

[drowsy path]

u can send ur question here

[wheat spruce]

Ty

TySm

[drowsy path]

No problem man

[drowsy path]

,rccw

[drowsy path]

20. i or ii?

[wheat spruce]

Both

[drowsy path]

given,
3 tan A = 4

Have u done

Any work?

assuming u know

$\sin \theta = \frac{1}{\csc \theta},
\cos \theta = \frac{1}{\sec \theta},
 \tan \theta = \frac{1}{\cot \theta}$

[wraith dagger]

dexa.cld

[drowsy path]

here we can find the value of tan A

[wheat spruce]

Okie tysm

[drowsy path]

tan A = 4/3

[wheat spruce]

Ok tysm

[drowsy path]

Ahh?

[wheat spruce]

Tysm means thank you so much

[drowsy path]

i mean i know 😭🙏

sin A / cos A = 4/3

find cos and sin

Then use relation

[wheat spruce]

Okie

Okie

Okie

[drowsy path]

And plug in values

Proved

[wheat spruce]

Okie

Tysm

[drowsy path]

np

2nd also kinda same

just put value

proved

[wheat spruce]

Okie

Tysm

[drowsy path]

np

[cedar kiln]

@drowsy path Has your question been resolved?

[cedar kiln]

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[spring forum]

I've done A^(-1)(3; 1) by letting (3; 1)=A(x; y) and then solving for x and y. I got x=11/13 and y=6/13. But I'm not sure how to do the one for B^(-1), since the coordinates of the endpoints of (BA)i and (BA)j aren't integers

[spring forum]

cat bit

[hollow trail]

there are some points on the BA grid that do appear to be on integers

[spring forum]

ooh :o that actually helps, tysm

I'm going to try doing it and see if I have any more problems

I think I got it, tysm cloud

.close

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[civic sage]

what do i do to find B

<@&286206848099549185>

[safe cloud]

So a rule ab parallelograms is that those diagonals meet at 90 degrees

U can find the gradient of AE

Negative reciprocal for gradient of BE

Use E coords for line equation

Equate to line AB and find intersection

Sorry I’m not in a situation to work through it rn but I hope this helps

[safe cloud]

MB RHOMBUS

Wait is it a rhombus in the question tho

[civic sage]

no

[safe cloud]

R u certain? All rhombuses are parallelograms but not all parallelograms are rhombuses

[civic sage]

yeah im aware. it says parallelogram

[safe cloud]

No I know but it could still be a rhombus, bear with me as I check this

[civic sage]

doesn't seem righ

t

[safe cloud]

Okay yeah no clear way to see

Gonna need to crack out the pen and paper, will be back shortly

@civic sage OKAY I am back

Sooo

What I did

I found the line AC

Using points A and E

I then found the change in x between A and E

And bc the magnitude of vector AE and EC are the same

I added this change in x onto the value of x at E

To find the value of x at C

Plugged this into my AC line equation

And found coords of C

I then used the fact that BC is parallel to AD to determine it would have a gradient of 1/4

And using the coords of C, was able to get a line equation for BC

Equated this to the given equation for AB to find x at B

Then used one of them to get the y coord

And that’s that

Posting solution with spoiler

[civic sage]

thanks

[safe cloud]

No problem

[civic sage]

.close

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[limber dawn]

the second step

[wraith dagger]

brandon

[limber dawn]

in the first step, you have -(2n + 1) on the top

then you expand it

in the second step

second line i mean

-(2n + 1) = -2n - 1

nice

np

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[sturdy escarp]

these are all pretty chill for the most part

managed to go 29-31

stuck on 32-34

32 I wrote as 10x* series k=0 to inf -x^k

$10x \sum_{k=0}^{\infty} (-x)^k$

[wraith dagger]

Paul04

[sturdy escarp]

pretty sure that's right

[lyric plank]

a common trick with producing power series from rationals is to take integrals, derivatives, or perform partial fractions/divide things out

[sturdy escarp]

hmmm

interesting

[lyric plank]

i should say common tricks*

you just have to undo what you did afterwards

but its pretty easy to perform integrals and derivatives on power series after you produce them

[sturdy escarp]

okay I think I kinda get what ur saying, can we try these problems?

[lyric plank]

mhm

[sturdy escarp]

alright, how did I do here

[dusk goblet]

this is good, just bring the x inside

and the 10 for that matter

[sturdy escarp]

so $ \sum_{k=0}^{\infty} 10x*(-x)^k$

[dusk goblet]

use exponent rules

[sturdy escarp]

$\sum_{k=0}^{\infty} 10x*(-x)^k$

[wraith dagger]

Paul04

[dusk goblet]

x^a * x^b = x^(a + b)

[sturdy escarp]

(-1)^k * 10x * x^k

[dusk goblet]

note that (-x)^k = (-1)^k * x^k

[sturdy escarp]

okay yeah

[dusk goblet]

simplify further

[sturdy escarp]

erm

[dusk goblet]

^

[sturdy escarp]

(-1)^k * 10 * x^(k+1)

[dusk goblet]

yes

and the radius of convergence?

[sturdy escarp]

-1

?

[dusk goblet]

🤔

[sturdy escarp]

oh

[dusk goblet]

how can a radius be negative

[sturdy escarp]

oh lol right

oopsie

[lyric plank]

ur totally right with the lower bound though

[sturdy escarp]

okay wait so I gotta do ratio test and allat?

[dusk goblet]

eh

[sturdy escarp]

okay then R=1

but like how

[dusk goblet]

you already have it

[sturdy escarp]

my instinct is ratio test

[dusk goblet]

so what’s the interval of convergence

[sturdy escarp]

-1,1

needa do convergence test to figure out closed or open

[lyric plank]

this is correct btw, we just sort of have a shortcut bc we know the interval for 1/(1-x)

[sturdy escarp]

okay makes sense

[lyric plank]

1/(1+x) could be a little diff bc now we're alternating

its the same here

but it could be diff in general so it is good to check

[sturdy escarp]

okay

so we get

(-1,1)

now what

[dusk goblet]

you’re done

[sturdy escarp]

oh

[lyric plank]

u did itt

[sturdy escarp]

that's wild

okay cool

[lyric plank]

33 and 34 are where the derivative and integral tricks become useful

[sturdy escarp]

yeah I was gonna say

perhaps a derivative for 34

[lyric plank]

mhm

[sturdy escarp]

for 33 idk let's try ig

what I first ended up doing was

expanding

[dusk goblet]

no

[sturdy escarp]

1/1-(100x^2)

but ig that's wrong

[dusk goblet]

try writing it as the derivative of some function

[lyric plank]

for 33, that looks close to a geo series right

[sturdy escarp]

yes

[lyric plank]

it would be convenient if that exponent was gone

[sturdy escarp]

okay

[lyric plank]

how could we get rid of the exponent

[sturdy escarp]

take sqrt?

[dusk goblet]

no

[lyric plank]

thats a good thought

but we ideally want an operation that's easy to perform on power series

[dusk goblet]

as a hint, what’s the derivative of 1/x?

[sturdy escarp]

ln|x|

[dusk goblet]

no

[sturdy escarp]

oh

[lyric plank]

like integrals and derivatives

[sturdy escarp]

-1x^-2

soz

[sturdy escarp]

okay

[dusk goblet]

do you see how that’s useful here

[sturdy escarp]

lemme see

oh

-2(1-10x)^-3 * -10?

[dusk goblet]

no

[lyric plank]

thats a little worse for us

[sturdy escarp]

breh

[dusk goblet]

integrate

[lyric plank]

we wanna somehow reduce the exponent

[sturdy escarp]

oh

lol right

[lyric plank]

how could we go from -2 to -1

knief kinda gave the answers

[sturdy escarp]

I get it

it's

(1-10x)^-2 --> -(1-10x)^-1 ?

[lyric plank]

at the very least, an integral will give us the exponent we want

we just need to be careful with coefs

[dusk goblet]

what about the 10?

u = 1 - 10x

[sturdy escarp]

🤦‍♂️

1/[10(1-10x)] + C

[dusk goblet]

it cancels with the - from the -10

[lyric plank]

ooh yeah

i can do integrals

yay

anyway thats perfect

and now u can do a power series

[dusk goblet]

find a power series for this (also forget the + C)

[lyric plank]

okay but specifically we should understand why we dont really need the +C

it's because we'll eventually take a derivative to undo the integral

that constant will matter if we do the reverse operations

which you'll see in 34

[sturdy escarp]

1/10 * series k=0 to inf -10^k = -(1)^k * 10^k ?

[dusk goblet]

@tm_29495 why did you do an @everyone ping a few weeks ago

[sturdy escarp]

oh

[lyric plank]

i think ur missing an x

[sturdy escarp]

wait

[proper palm]

what?

[dusk goblet]

in that other server

[sturdy escarp]

1/10 * series k=0 to inf -10^k = -(1)^k * 10x^k ?

[dusk goblet]

you pinged everyone then left

🤣

[proper palm]

which one ?

[sturdy escarp]

|10x| < 1 . 1/10=R

[proper palm]

ahhhh

🫣

[lyric plank]

make sure (10x) is in parentheses but that looks good

[sturdy escarp]

awesome

okay but

we didn't reverse anything here

did we?

[lyric plank]

not yet

we still have to

bc this power series is of the integral of our function

we want the power series of the original function

[lyric plank]

so dont worry about IOC yet

[sturdy escarp]

I was about to ask lol

okay

so what do I take derivative of?

[lyric plank]

your power series

[sturdy escarp]

okay

uh what's the power series the (10x)^k?

[lyric plank]

yup

[sturdy escarp]

so derivative is

10k(10x)^(k-1) ?

[proper palm]

yeah

[sturdy escarp]

okay so we need IOC now?

[lyric plank]

now we can do it

[sturdy escarp]

okay how though, ratio test?

[lyric plank]

yup

[sturdy escarp]

bruh 😭

long problem lol

okay but wait

why did we find R before

[lyric plank]

we didnt need to

[sturdy escarp]

ah okay

[lyric plank]

we just need to find R now

[sturdy escarp]

okay

[lyric plank]

this is the actual power series

[sturdy escarp]

right

okay yeah I see the path

ima write this all out neatly when we're done

can we try the last problem really quick?

I'll try to speedrun it see if I understand

[lyric plank]

ya sure

this one has a few extra steps bc of the reverse direction so itll be good to walk through

try to walk through it and ill just correct as we go

[sturdy escarp]

okay

first thought is take derivative and get -4/(1-4x)

then you get -4 * 1/1-4x

-4 * series k=0 to inf of (-4x)^k ?

wait

take it inside

so you get

series k=0 to inf of -4(-4x)^k ?

[proper palm]

there is a sign problem no?

this is -4/(1+4x)

[lyric plank]

oop

yeah

mb

[sturdy escarp]

oh

it's x^k

-4(4x)^k

[proper palm]

yeah exactly

[sturdy escarp]

okay cool

so now integral of -4(4x)^k ?

[lyric plank]

yup

[sturdy escarp]

-(u^(k+1) / (k+1)) ?

u=4x

[proper palm]

[sturdy escarp]

then just ratio test and get R then find interval then test endpoints and gg?

[lyric plank]

yup

one thing we do wanna check here is the constant

[sturdy escarp]

yooooo

[sturdy escarp]

okay

[lyric plank]

it might just be 0 here i havent checked

but f(0) should be the same for both functions, that's an easy way to check what the constant term should be in your power series

[sturdy escarp]

wdym both functions

[lyric plank]

like the original function vs your power series representation

since we took an integral at the end, we have a +C floating in our series

and that gets fixed by whatever the original function is

[sturdy escarp]

ah

so ln(1-4x) and -(4x^(k+1) / (k+1)) ?

[lyric plank]

yup

specifically the sum of that thing over k from k = 0 to inf

[sturdy escarp]

so whatever f(0) is for the original function will be the +C for the power series?

[sturdy escarp]

yeah

[lyric plank]

yes, that'll be the +C which is just the constant term in the power series

since our current power series starts at the x^1 term

[sturdy escarp]

can you elaborate on this please

[lyric plank]

this is why i reiterated this

k goes from 0 to inf

what's the first term in our series now

[sturdy escarp]

-(1 / (k+1)) ?

wiat

-(4x^(0+1) / (0+1))

-4x?

[lyric plank]

yup

so our smallest degree term is x^1

[sturdy escarp]

I see

[lyric plank]

which means we don't have a constant specified

and we know after integrating there's a +C

we need to figure out what that +C is, to line up with the original function

[sturdy escarp]

okay makes sense

[wraith dagger]

The following error occured while calculating:
Error: Undefined function ln

[sturdy escarp]

so 0=-4x?

[lyric plank]

not exactly

think of the current power series as

C + sum from k=0 to inf of -4x^(k+1)/(k+1)

when we plug x = 0 into this, what do we get

[sturdy escarp]

C + sum k=0 to inf 1/(k+1) ?

[lyric plank]

if x = 0, and k >= 0, every single term in the summation equals 0

[proper palm]

its x^k+1 *1/(k+1) dont forget

[sturdy escarp]

oh yeah

[sturdy escarp]

so C=0 ?

[lyric plank]

yup

[sturdy escarp]

I see

[lyric plank]

which is super convenient lol

we dont have to do anything

[sturdy escarp]

lol

if it wasn't zero then what?

[lyric plank]

then you would plop the constant in front of your power series

[sturdy escarp]

I see

[lyric plank]

that would be the first term, followed by your existing powers of x

[sturdy escarp]

yeah makes sense

yeah a lot more work for these last couple problems than the other ones

[lyric plank]

in general for this context, f(0) will always be the +C, because integrating a power series will make the smallest degree term x, meaning the entire series equal C after plugging in 0

this is also just how taylor/maclaurin series work

[sturdy escarp]

interesting

okay I'll make note of that

tysm for the help, everything is much more clear now

[proper palm]

great methodology 👏 @lyric plank

and gg paul !

[sturdy escarp]

.close

[cedar kiln]

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[crude brook]

Can someone please help me find the partial solutions of this ODE please?

[crude brook]

By using lagrange

Someone please

Can riemann help with this?

I bet he can

exp(-3x)

<@&286206848099549185> riemann help please

Noooo riemann is asleep

Can you help brandon?

[inner pendant]

Do you mean laplace?

[crude brook]

Lagrange

[half apex]

Write agian

I cant see your words

[crude brook]

$9y''-12y'+4y=e^{-3x}$

Ye that

[wraith dagger]

brandon

MichaelRafto

[crude brook]

Ye

There are 2 ways to solve this, one is with lagrange and the other is with the μέθοδο προσδιοριστέων συντελεστών (sorry idk how to translate that give me one sec)

Determinant coefficient method

The original exercise wants it solved with the second method

But as of now i am trying with lagrange

I want to learn it

Some exercises can only be solved with lagrange and lagrange always works, thats why

[half apex]

$9y^n-12y^2+4y=2^{-3x}?

[crude brook]

This please

Fixed it

So

I have $c1e^{2x/3}+c2e^{2x/3}*x$

[wraith dagger]

MichaelRafto

brandon

[crude brook]

$c1e^{2/3x}+c2e^{2/3x}*x=0$ is the first equation i take

[wraith dagger]

MichaelRafto

[crude brook]

The second one is gonna be a bit messy isnt it?

$c1*2/3e^{2/3x}+c2(e^{2/3x}+2x/3e^{2/3x})=e^{-3x}$

Is this correct?

No wait

Format is kind of messy

[half apex]

Wait you are Greece?

[wraith dagger]

MichaelRafto

[crude brook]

Ye

[half apex]

Ah

[crude brook]

That is the homogenous solution

[half apex]

Sorry i don’t know about Darivative

Bye ^^

[wraith dagger]

brandon

[crude brook]

Well i got double root of 2/3 that is why i got that result

[crude brook]

Wait so this is correct?

Cause im going to put this in a determinant

Is this determinant the wronskian you are referring to?

I will take |(y1 y'1)T,(y2 y'2)T| if that made any sense

Sorry idk how to use texit that well 😅

[wraith dagger]

brandon

[crude brook]

Ye

[crude brook]

So i take values from this? This correct, right?

[wraith dagger]

brandon

[crude brook]

Good so it is correct

[wraith dagger]

brandon

[crude brook]

One sec

$W(y_1,y_2)=e^{4x/3}$

No wait

Wait

[wraith dagger]

MichaelRafto

brandon

brandon

[crude brook]

Ahhhhh crapp

I got $3/11e^{-3x}(1-x)$

[wraith dagger]

MichaelRafto

[crude brook]

As partial solution

From replacing the c1 and c2 i found

But yes you said that y" must have 1

Not 9

So that is a problem

What do we do now?

I took $c1'=1/Det*W((0, e^{-3x})T, (y_2, y_2')T)$

And i took $c2'=1/Det*W((y_1, y_1')T,(0, e^{-3x})T)$

And then i integrated and found c1, c2 and then i substituted in the homogenous

Idk man im just trying to do what my professor did before me

Also i wrote those wrong one sec

[wraith dagger]

MichaelRafto

MichaelRafto

[crude brook]

Det being the first wronskian

[wraith dagger]

brandon

[crude brook]

Hmm

[wraith dagger]

brandon

brandon

[crude brook]

?

Is the partial solution?

[wraith dagger]

brandon

[crude brook]

How do we do that?

[wraith dagger]

brandon

[crude brook]

Huh that is the method chatgpt showed me, ig its the same method we just do it a little bit differently

Idk

Wait what is f(x) though?

Ahhhhh

[wraith dagger]

brandon

brandon

[crude brook]

Can we leave this for tomorrow please i am tired

Sorry

Thanks man

It says the solution is $\frac{e^{-3x}}{121}$

[wraith dagger]

MichaelRafto

brandon

Results provided by WolframAlpha

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

Requested by brandonghg

[crude brook]

Nice!

Ok imma finish it tomorrow

Yup

Aight gn man

You did help man

.close

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[opaque oxide]

hello

[opaque oxide]

does this question make sense to anyone

if so

how do i go about with it

Draw two rays starting from a common point P, forming an angle. Label the vertex and arms of the angle. Then, add a point Q on one of the rays, and draw an interval between P and Q. Next, add two more points, R and S, such that P, R, and S are collinear. Finally, draw a second angle with vertex S and label all points, lines, intervals, and rays.

im confused because is common point P and the vertex of the angle not the same thing?

does it have 2 names or is there something wrong here

and how would i drawn an interval between P and Q if they are already on a ray together?

<@&286206848099549185>

[dire geode]

!15m

[cedar kiln]

Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.

[opaque oxide]

oops

sorry

[fossil dawn]

could you sketch what you understand from the question?

[opaque oxide]

yea sure

bad sketch my bad !

i just want to know if they explained the question wrong or if im not getting something

they tend to incorrectly format questions

[opaque oxide]

^^

[fossil dawn]

,rccw

[fossil dawn]

i think P is the vertex of the angle and the common point as far as my understanding tells me

[opaque oxide]

why would they ask me to label the vertex is p IS the vertex

just confused

[fossil dawn]

i am just as confused as you are mate

maybe others would have a better idea of what's going on

but from my reading through this question P is the vertex

[opaque oxide]

im just gonna skip this question and report it to them later

thanks anyways

[fossil dawn]

nps, but ig you can try asking for a second opinion

[cedar kiln]

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[pseudo merlin]

HOW DO I do this

Sorry cpas

Caps

[tropic oxide]

work out both derivatives of y and plug them in??

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[torn marsh]

Did I do something wron

wrong*

Im tryna solve part b

[latent bloom]

Hmm

[civic hedge]

you should try and subsitute (x^2+9) as t or u

it would simplify a lot

[latent bloom]

Use a trig sub

Wait nvm

[civic hedge]

there's no need for that

[fossil dawn]

there's no integration needed i think?

[rocky geode]

no really understand this mean but i do understand the caculation

[civic hedge]

the final answer should be 2 i guess

[latent bloom]

Part b

[fossil dawn]

yea but they already gave the antiderivative

[latent bloom]

Sorry, but this channel is now occupied

[upper laurel]

go look at #❓how-to-get-help to see how to have people answer your question

[civic hedge]

make a new channel

[torn marsh]

she's prob right

[nova snow]

Y’all forgot the FTC exists 💔 🥀

[twilit sierra]

you plugged in the values to the wrong equation

[nova snow]

pmo

[fossil dawn]

clarification: by no integration needed, i mean he doesn't have to find the antiderivative

[civic hedge]

oh ye

[upper laurel]

yall are taking too long already

the only problem here is a major typo on line 2

[civic hedge]

$(\int {0}^{4}\frac{x}{\sqrt{x^{2}+9}}dx=\left[\sqrt{x^{2}+9}\right]{0}^{4}).

[wraith dagger]

Nobita18
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)

[tropic oxide]

... because he's already given it for free.

[twilit sierra]

yeah

[civic hedge]

now just put the values

[twilit sierra]

h do you remember the fundemental theorem of calculus

[torn marsh]

the theory behind it

or

the formula whatever you call it

[fossil dawn]

one question.

you substituted the upper bound

where's the part where you sub the lower bound?

[torn marsh]

it would be 0

[fossil dawn]

i think this is what everyone else is pointing out to you

[civic hedge]

no it would be 3

[fossil dawn]

you sure it would be 0?

[twilit sierra]

the formula

[fossil dawn]

sub in your x = 0 and calculate

[civic hedge]

the 9 is still present

[twilit sierra]

no in that photo they used the wrong equation

[torn marsh]

the 2x is being multiplied with sqrt(x^2+9)

[fossil dawn]

oh my goodness.

[torn marsh]

(2(0)sqrt(9))/1

[tropic oxide]

heres something you should know

x is NOT a constant

and the antiderivative of x*f(x) is NOT equal to xF(x) in general

[civic hedge]

nvm

[twilit sierra]

the formula i guess

[torn marsh]

what is it again?

[twilit sierra]

in this case f'(x) is x/(sqrt(x^2 + 9))

[torn marsh]

isnt that ftc2?

[twilit sierra]

maybe

i lowkey forgot

but you know f'(x) and f(x)

[torn marsh]

yea

[twilit sierra]

so you can just plug in the values into f(x)

for the answer

like the f(b)-f(a) part

does that make sense kinda

[torn marsh]

Im a bit lost ngl

one sec

[split scroll]

When you integrate a function you find an anti-derivative. But you already know what an antiderivative is by part a).

[fossil dawn]

you don't need to find it again, as mentioned

the hard part is already done for you

[torn marsh]

from here

[torn marsh]

so i dont need to integrate

[fossil dawn]

that's what i said, yes.

[twilit sierra]

you have the anti derivative

[torn marsh]

I need to interegate the deritive then

[fossil dawn]

???

how is "you don't need to integrate" = "you need to integrate the derivative"?

[torn marsh]

I need to use the derivitive to find the anti derivative

[twilit sierra]

well if you think about it you have it already

[torn marsh]

yea but I dont have the anti derivative

[fossil dawn]

??

[twilit sierra]

did your class teach you how integrals and derivatives are inverses

[fossil dawn]

what did part a give you then

[split scroll]

Do you remember what makes something an antiderivative?

[torn marsh]

the derivative

[torn marsh]

yes

[twilit sierra]

so what would the original function be in relation to the derivative

[torn marsh]

just add 1 to the exponent divided by n+1

[fossil dawn]

okok look.

[twilit sierra]

that's only for polynomials

[fossil dawn]

in this part, you showed that the derivative of that thing on the left is the thing on the right

correct?

does this thing remind you of the right side?

[torn marsh]

well yea

[fossil dawn]

and what is the right side in relation to the left side in part a?

the right side is the ___ of the left side (fill in the blank)

[torn marsh]

derivative

[twilit sierra]

yes so the function inside the integral would then be f'(x)

[fossil dawn]

so the left side is the ___ of the right side

(fill in the blank again)

[torn marsh]

original function

[fossil dawn]

so now you are integrating the derivative

that gives you the ___

fill in the blank

[torn marsh]

btw if with the d/dx in front, it woudnt be the original function

it would be classified as a derivative but not yet evaluated

[fossil dawn]

don't care about the d/dx first

[fossil dawn]

this one

[torn marsh]

anti derivative

[split scroll]

This is the definition of anti-derivative, with an example.

[fossil dawn]

and the antiderivative of a derivative would be the ___

(last blank. fill in with two words that you've answered with before)

[torn marsh]

original function

[fossil dawn]

good

[fossil dawn]

so you noticed that the left side is the original function

and you are told to integrate the derivative

therefore its antiderivative is the original function, which has been given to you already!

[twilit sierra]

(ignoring the d/dx)

[fossil dawn]

yea ignoring the d/dx

so you just need to apply the F(b) - F(a) part of the FTC

why go through all the trouble integrating again?
part b even gave you the permission to reuse the result from part a by starting with "Hence"

[torn marsh]

anti derivative of what?

[fossil dawn]

oh my god. do you read?

the antiderivative of the derivative

ok let me make it simple

[split scroll]

sqrt(x^2+9) is an antiderivative of x/sqrt(x^2+9).

[fossil dawn]

if differentiating the left side of part a gives you the RHS of part a as the derivative, then integrating the RHS of part a (which is the integrand in part b) must give you the original function back

idk how i can make this more obvious than it is at this point

[torn marsh]

"if differentiating the left side of part a gives you the RHS of part a as the antiderivative", how would it give the anti derivative

its derivative

[fossil dawn]

woops wording

corrected

my bad

hey how could you spot these tiny mistakes but not the whole picture

[torn marsh]

idk ngl

[dreamy void]

focus on the wrong things

[torn marsh]

Ig im good at finding little mistakes

[fossil dawn]

should i start saying all the wrong stuff so you learn by correcting me

but do you get my point?

[torn marsh]

" then integrating the RHS of part a (which is the integrand in part b)", how will it be the integrand in part b if it is the derivative in part a

[dreamy void]

What's the question again

Or your confusion

[fossil dawn]

here

[dreamy void]

Thanks guys

[fossil dawn]

he doesn't understand why the antiderivative for part b is already given in part a

[fossil dawn]

because there's that, idk, big integration symbol next to it?

[dreamy void]

Oh

[torn marsh]

oh right yes

[dreamy void]

Well, it's indeed weird, reverse order, meaning (b) to (a) would make more sense

[fossil dawn]

a to b makes sense as well, because then you can skip the hard part of finding the antiderivative

because it's been proven in a

[dreamy void]

Of course

[torn marsh]

I understand this

[dreamy void]

If you didn't encounter substitution before and are studying the FTC rn then the exercise makes sense in that order

[torn marsh]

but didnt you say earlier that theres no point of interegating because we were given the derivitive?

[dreamy void]

You could still try to integrate it by yourself and then verify your answer

[fossil dawn]

you can if you want, but if you want to, make sure you do it right
though in this case, you're expected to use the result from part a anyway

[dreamy void]

So it's an FTC task?

[fossil dawn]

yes, i presume. the last few questions he had asked are all about FTC2

[dreamy void]

I guess this sums it up then

[fossil dawn]

did OP get it

[dreamy void]

I don't know, but I think there is no point in questioning the exercise, and rather just move on

Use FTC as most likely intended, if you wanna go harder, try to solve the integral completely by yourself

[fossil dawn]

i would be more tempted to encourage him to take this as a free pass if not for the fact the past three or four questions he asked were all on the FTC

but this seems to be a more fundamental problem than just the FTC

(pun not intended but i'm sticking with it)

[magic solar]

I mean that integral really isn’t the most difficult anyways

[torn marsh]

so bassically what i need to do is integrate the

derivative

to get the original function?

[dreamy void]

Yea you can do that

[torn marsh]

I skipped the other question for now but with 13. b

what If im trying to interegrate the derivative and im getting -1 as the exponent

[dreamy void]

I don't understand your question

[torn marsh]

wait dw

give me a sec

I have question, why is u sub used?

is it suggested to use it in complex integral problems?

[dreamy void]

Yes

It's especially used to simplify integrals or identify hidden chain rule

[torn marsh]

identify hidden chain rule?

[dreamy void]

Let's do a concrete example

[torn marsh]

okay

[dreamy void]

,, \int \sin,!(x^2) \cdot 2x , dx \overset{u=x^2}{=} \int \sin(u) , du

[dreamy void]

See we simplified the integrand way nicer

sin(u) is obviously nicer to integrate

An experienced integral try hard would of course see it right away

but you don't always see this instantly, and that's why I said hidden chain rule

[torn marsh]

I get what you mean but why would you call it hidden chain rule

[upper ruin]

Because it should remind you the chain rule (hoping you know what chain rule is...)

[dreamy void]

because it's not always obvious

[upper ruin]

But usually it's called u-substitution

[dreamy void]

you don't always know right away what you may substitute

[torn marsh]

so

when should I use this method?

[upper ruin]

When you see the derivative of some part of the function. Or when you see multiple things being repeated

[dreamy void]

kinda difficult to generalize it

but let's say it's never bad to attempt a substitution, they mostly work out intuitively

[twilit sierra]

don't over think it rn if you haven't learnt it in class

[fossil dawn]

i wonder if i should post an example of a harder-to-spot u-sub

[dreamy void]

When you are more experienced you will get a feeling of it when it's worth to apply a sub, as you will be able to tell what terms will cancel and how it may simplify the integrand

[dreamy void]

Yes!

[torn marsh]

ykw send it haha

[upper ruin]

He's struggling to understand FTC so maybe not 😅

[fossil dawn]

here you go

that power's a 3

[fossil dawn]

true, i guess just to serve as an example hahah

[torn marsh]

fair

so I should use this method in harder integraration problems

but how will i know if its

hard?

[dreamy void]

You will have to go through a decent amount of examples

in order for you to know

[fossil dawn]

if your first thought is "shit i need to ask this on mathcord help channels" it's probably hard

[dreamy void]

As Kirby mentioned, I wouldn't overthink it for now

[upper ruin]

Also easy integrals can be done with u sub
Such as the integral of 2x•cos(x²)
Even though if it is this easy you will do the substitution or recognize the primitive by eyeballing it

[fossil dawn]

since we are on the topic of u-subs

[upper ruin]

But you need to know derivatives very very well and also doing a lot of exercise. In general , in order to understand integrals, make sure to master the derivative topic perfectly

[fossil dawn]

#help-36 message
do note that this is a very special instance of a u-sub

[torn marsh]

I feeel like if i do u sub (especially with easier problems), there is extra steps involved

[upper ruin]

Such as?

Send the problem

[upper ruin]

I feel you're either overthinking or you didn't understand u substitution...

[fossil dawn]

what was the original problem that led to this u-sub in the first place

[torn marsh]

Im getting to know it

[upper ruin]

That's a very important step 😬

[torn marsh]

I just had a general inquiry

i thought if i used this method, it would make my life easier

[upper ruin]

Sure it does, especially if you can't solve the integral with other methods

[fossil dawn]

this is not a be-all and end-all method

for a lot of complex integrals like those with roots and polynomials under those roots, or weird trig functions, or stuff like that, sure

[torn marsh]

Ig ill need to be exposed to more integral problems

[fossil dawn]

!done

[cedar kiln]

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[wicked mantle]

.close

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[crimson sedge]

whys I. sufficient to show that the two triangles are similar? I lowkey dont get it

[loud gyro]

check which similarity tests would get satisfied

[crude brook]

You chose A?

[crimson sedge]

the answer key said its A

[crimson sedge]

uhhh im not sure i think both doesnt satisfy the condition

[loud gyro]

do you know what are the tests?

[crimson sedge]

yea like all three sides having same proportion or all angles having same proportions or two side and one angle verified or like right triangle with pairs

[inland zealot]

Hi

[bitter trout]

yes, two side at the same length ratio + one angle, but the angle between them

so like I. is okay, in II. you have wrong angle/wrong side

so the answer key is actually alright

[crimson sedge]

so like the info about the proporiton of LM and PQ shows that the two angles and one sides are matching?

[loud gyro]

one angle and the two sides that form the angle

yea

however, in statement II, you have the angle that is not between the two sides that you know, so its not gonna work to claim similarity using II

So, I is sufficient

[crimson sedge]

we alr know one of the angle is matching but the proportionality of the side gives info that the other angle is also gonna be matching(?) too so we have enough info to know that the two triangles are similar???

[loud gyro]

how does side tell you about the angle?

[crimson sedge]

idk man 😭

is it just one sides + one angle

?

[bitter trout]

like you have the info: angle LMN and PQR being similar and and QR being 2/3 of MN
so look, you need to use LM and PQ sides, so angle can be between LM-MN and PQ-QR, cuz thats the angle you know

[loud gyro]

one side + 1 angle is the info that you already have

you are getting a 3rd piece on info from the statements

so, using those 3 pieces you have to determine if it matches the tests you know

info about the side only tells you about the side, and not anything else

[crimson sedge]

oh wait

i didnt see that i was given the lenght of the bottom of triangle

😭

thanks yall 🙂

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[hallow quartz]

Need help with cyclic groups, recently started group theory

[hallow quartz]

Question comes from an exercise left by lecturer

Let G = <a> be a cyclic group and H a subgroup of G. Prove H is a cyclic group. Hint: work with exponents and use the description of subgroups of (Z,+)

I got stuck defining H, where h is the elements in the form a^m. I feel like m should be larger than n for group G because Order(G) needs to be less than Order(H)

[cedar kiln]

@hallow quartz Has your question been resolved?

[hallow quartz]

<@&286206848099549185>

.close

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[vocal hemlock]

I’m trying to figure out how this will go... Is there any way to solve this cos ax/sin x?

[fossil dawn]

hey, just dropping by to say you have nice handwriting

[vocal hemlock]

Thanks 😭😭

[native heath]

shouldnt it be infty

[civic hedge]

my bad wait wrong one

i forgot the name

[random swift]

I don't think L'Hopital's rule applies here because the limits of the numerator and denominator aren't 0 or infinity

[native heath]

it doesnt exist actually

which makes sense

[iron wren]

is there a way to use the squeeze theorem here

[random swift]

Not if the limit doesn't exist

[vocal hemlock]

My teacher banned l hospital since we havent reach there

[native heath]

lhop doesnt even apply

[opal hinge]

make sense

[iron wren]

the limit does exist ill say that

[native heath]

no it doesnt

[vocal hemlock]

Soo I should leave it with the limit doesnt exist?

[native heath]

if you wanna be cursed, this is the same as $\lim_{x \to 0} \frac1{x}$

[wraith dagger]

Percy

[iron wren]

oh i put 3cosx instead of cos3x

my bad

[native heath]

because sin(x) \approx x lol

but DONT write that

[random swift]

Can you aproximate sin(x) ≈ x and cos(x) ≈ 1 when x ≈ 0?

[civic hedge]

yep

as x is very very small

[native heath]

you can but its not very rigorous lol

[vocal hemlock]

Is this the same case where left & right limit doesnt match? [Saw it but i dont understand lol]

[native heath]

yes!

[random swift]

Or apply the sine subtraction formula

[nova snow]

percyyyyyyyyyyyyyyy

[native heath]

hey water beam

long time since ive been in help channels lol

[nova snow]

true truee

[vocal hemlock]

Kkay thanks for the help 👋🏼

.close

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[drowsy obsidian]

how?

[candid mist]

multiply on top and bottom by cos(x)

[drowsy obsidian]

ok

[opal hinge]

probably a typo

[worldly edge]

Yes phy do that all the time

[opal hinge]

the 1 just disappear to limbo

[tropic oxide]

this looks incorrect as-written

[drowsy obsidian]

ohhh

ok

yh

thanks

.close

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[stray crypt]

hey i need to proof the red brackets, any hint will be cool

no full answer plz

ive tried few things but it didnt seem to work

[opal hinge]

using the unit circle ig

that might work

[lyric widget]

yeah it's pretty evident using the unit circle but I think that might be a general result for an invertible odd function

[stray crypt]

@lyric widget can u explain more about using the unit circle here?

[stray crypt]

by using trig identites or smth else?

[lyric widget]

well if you flip the y coordinate then by symmetry the angle also flips

because arcsine is defined from -pi/2 to pi/2 iirc

[stray crypt]

by flipping u mean -y or pi-y?

[lyric widget]

I mean multiplying by -1

so the former

[frail citrus]

it is an inherent property noticed by our dear scientists and mathematicians

well proving it will be interesting too

[stray crypt]

thank you this helped !

[lyric widget]

glad to hear!

[stray crypt]

.close

[cedar kiln]

Channel closed

Closed by @stray crypt

Use .reopen if this was a mistake.

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[mystic turret]

Ignore the names

<@&286206848099549185> sorry for the ping but I need help ASAP

[meager kettle]

which part is the problem?

choosing graph type?

[mystic turret]

How to make the graph

[mystic turret]

2 mm

[meager kettle]

what does it mean?

[mystic turret]

I have to make it in a 2mm graph. How to make it

[meager kettle]

isnt 2mm graph paper type?

not a type of graph

[mystic turret]

Ya mb

Wait nhm

*nvm

I got it

.close

[cedar kiln]

Channel closed

Closed by @mystic turret

Use .reopen if this was a mistake.

[cedar kiln]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[rich blaze]

Functions