help-13

[lean ledge]

@frigid sinew

[frigid sinew]

That’s good

So x is 3 and 0 right?

[lean ledge]

Ye so u do the brackets first? Then multiply?

[frigid sinew]

And replace x where it =0

In your case it’s 0 and 3

[lean ledge]

Thx ig sorry for being dumb

[cedar kiln]

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[subtle jolt]

how does -4^2x * ln(4) * 2 become (1-2*4^x)

[silver forge]

they r first factoring 4^xln(4)

[subtle jolt]

ooh

i see now

thanks

how does 4^2x become 4^x

.close

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[crimson sedge]

I don't understand part a i

So I intergrated the square root of 8x to get 16x/3

(I manipulated the square root of 8x to get (8x)^1/2 and then integrated that).

So it says that a=x So that means I can just replace my upper limit of a with x. I got :

16x/3 - 16(1)/3 = 10

Then i rearranged to find x and I did not get 20 root 2

[drowsy sage]

Your integral appears to be incorrect

When you apply the power rule to an exponent of 1/2, what is your new exponent?

[crimson sedge]

Ohhhh yeah...I think I see where I went wrong now. So i add one to the power of a half and then divide 8x by that so I actually get 16/3 x^3/2 , right ?

[drowsy sage]

Not quite.. notice that your 8x is being multiplied under the radical...

the 1/2 exponent applies to both terms

not just the x

And since we can remove constants to the front of the integral...

[crimson sedge]

i recommend just taking sqrt(8) out of the integral tbf

as you’ve done here (i didn’t read this far before i sent that)

Wait sorry how does that work ? If we're integrating 8^1/2 and x^1/2 then how can we times root 8 by the integral of x if we're integrating 8 along with x also ?

root 8 is just a constant

[drowsy sage]

Rules of integrals

[crimson sedge]

so you can take it out of the integral

Ah ok, but we can't take out the x right as that isn't constant ?

[drowsy sage]

Incidentally, you can do the same thing with derivatives..

[crimson sedge]

yes!

gives you the same answer either way

just makes ur life easier if you take it out

Ahhh alright, so we could keep the 8 as an integral but we just take it out to speed up computation ? I guess what i'm trying to say is; is it still mathematically correct if we keep 8 as an integral ?

[drowsy sage]

Yes, it's still correct if you keep the sqrt(8) on the right, inside the integral

You just have to remember that the 1/2 exponent applies to it, as well

But honestly? When you start playing with fractional exponents, it'll save you some headache to simplify

Especially when the integrals start getting harder

[crimson sedge]

Speaking of headaches... do you know how to integrate (8x)^1/2 ? Because I don't really know and it's giving me a headache lol. In my mind I keep getting 16/3 x^3/2 but that's wrong and I don't really know how to do it correctly...

[drowsy sage]

So, let's move the sqrt(8) outside the integral and just integrate x^.5

[crimson sedge]

1/6x ^6 ?

[crimson sedge]

oh wait I noticed a point 5 do you mean 1 half or literally x^5 ?

[drowsy sage]

1/2 = .5

[crimson sedge]

Ah ok

so sqrt(8) x 2/3x^3/2 ?

[drowsy sage]

Yep!

[crimson sedge]

oh Sweet !

That's it then right ?

[drowsy sage]

It is, but we can reduce things a little bit

2 * 4 = 8.. so sqrt(8) = sqrt(2 * 4)

4 is a perfect square, so we can take that out from under the radical

sqrt(8) = 2 sqrt(2)

[crimson sedge]

Ok, is that it for reducing it because i have no idea how we would reduce 2/3x^3/2 ?

[drowsy sage]

Can't reduce any further, but remember that your 2√2 can go in the numerator of 2/3

[crimson sedge]

Sorry I'm a bit confused on where this has come from:

[drowsy sage]

We're just combining like terms

combining our constants

[crimson sedge]

Ohhhh I see it's our fully simplified expression

[drowsy sage]

Yep

[crimson sedge]

Ahh alright that makes sense.

Well thanks a lot to you and @crimson sedge for the help. I really get this now and thanks for the tip about being able to remove the constants. Anyways, this was my last question of the day so I'm Gunna go and get some sleep but thanks again for all the help !

[drowsy sage]

That's your indefinite integral.. looks like the problem still has you doing a few other things with the definite integral

You good on that?

[crimson sedge]

Yep I should be. It's pretty much just the same process but with sqrt(x)

[drowsy sage]

👍

[crimson sedge]

So x^1/2 and that becomes (as an integral) 2/3x^ 3/2

[drowsy sage]

Yep

[crimson sedge]

And our lower limit is 0 so it's literally just:

2/3x^ 3/2 = 10 (I believe)

and then solve for x

[drowsy sage]

Not quite

[crimson sedge]

oh... In that case I didn't know this as well as I thought have lol. Let me go look at the question real quick

OHHHHH

[drowsy sage]

You plug the value of your upper limit into your indefinite integral, and then subtract [the value of your lower limit plugged into your indefinite integral]

[crimson sedge]

Yeah we can't make it equal to R because R starts at 1 and not 0

woops

[crimson sedge]

So have you integrated from 0 to a and taken the integral from 0 to 1 away from it ?

[drowsy sage]

Oh wait.. I don't think it's asking you to find the value of a

[crimson sedge]

.

[drowsy sage]

But.. that's what I was putting there

[crimson sedge]

Well I think since a=x then finding a value of a or finding a value of x should just be equivalent I think ?

[drowsy sage]

Well, you've got all the x values from 1 to a

I'm not sure exactly what your problem is asking you to find.. either the indefinite integral, which we did earlier, or the actual value of a, which is.. where I might be taking you too far down the rabbit hole

Have you guys done definite integrals yet?

[crimson sedge]

Definite integrals is when we integrate with limits right ?

Sorry it's been a while since i've done integration so I'm a bit rust with the terminology

[drowsy sage]

No.. it's when you integrate with defined upper and lower bounds.. a and 1 in this case

[crimson sedge]

Oh yes we've done that.

[drowsy sage]

ok.. it might be asking you to find the actual value of "a" then... in which case, that's the equation I wrote out above

If you can, ask your instructor exactly what is wanted..

If not? Ehh.. put both and ask for extra credit if there's too much info 😉

[crimson sedge]

Lol sorry this literally made me laugh out loud for a couple of seconds straight. Anyway, I just found this question from this google drive on the internet, here's the google drive if you're curious...https://drive.google.com/drive/folders/1QEmHel9tNGPnsiRRu0NgXGTA4HeTfawu

Also, in the mark scheme it integrates with a upper bound of a so I would say your approach is correct

because like why would they integrate with a and not solve for it right ?

[drowsy sage]

That was my thinking 👍

[crimson sedge]

Anyways, thanks again for your help, I'm Gunna go get some sleep and give my brain a rest from calculus...

[drowsy sage]

np - night

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[quartz crypt]

How do I solve for this?

[quartz crypt]

Says it need to be in a decimal

[ashen shard]

it can just be a whole number

it just has to be in the domain of f(x)

[quartz crypt]

um

so domaion would be (-infin, 6/7]

and [6/7, infin) ?

[ashen shard]

keep in mind that $\sqrt{x}$ is defined when $0 \le x < \infty$

[wraith dagger]

Dork9399

[quartz crypt]

I am so confused

[ashen shard]

so we know that 6-7x has to be inbetween 0 and infinity

aka 6-7x has to be positive

[quartz crypt]

okay

[ashen shard]

so $6-7x \ge 0$

[wraith dagger]

Dork9399

[ashen shard]

$6 \ge 7x$

[wraith dagger]

Dork9399

[ashen shard]

$x \le \frac{6}{7}$

[wraith dagger]

Dork9399

[ashen shard]

Just one bound

$(-\infty, 6/7]$

[quartz crypt]

I see

[ashen shard]

so you just have to choose values in that interval

[quartz crypt]

so any decimal in between would work

.25 and .50 for exmaple

and not in the domain would be outside

[ashen shard]

yes

for the first part you could have also chose numbers like -5 or -10

because they are also in the interval

[wraith dagger]

Dork9399

[quartz crypt]

yeah because it's on the outside

thanks broski that really helped my understanding

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[wraith dagger]

Ransik

Ransik

Ransik

[dusky peak]

you forgot something

the right term

product rule

it needs another 2x

otherwise you can do it piecewise and separate into 2 cases

ah sorry i mean chain rule

like this

[dusky peak]

also, this can be simplified a lot

yes

[wraith dagger]

Ransik

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

[cedar kiln]

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[wary fern]

so when looking for the missing parts of the triangle right the angles right, how do I know If im going to use sin,cos, or tan

[digital cliff]

see what information you know, then look at soh cah toa

if you have the angle and its opposite and want the hypotenuse, then you would want sin, since its sin, opposite hypotenuse

if you have two sides and want the angle, see what those sides are in respect to said angle, then choose the one which involves them

etc

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[wary fern]

okie thank you

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[simple seal]

Parametric to Rectangular conversion, I don't understand part (b) of this question, can someone explain it to me?

[simple seal]

is the answer, but I don't understand the process of finding the "z" variable at all

[cedar kiln]

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[simple seal]

<@&286206848099549185> 😭

Can someone explain why they took the trace to find a circle, and why the radius is x?

[cedar kiln]

@simple seal Has your question been resolved?

[gentle pecan]

@simple seal I agree that the solution key is a bit unclear. I have the following picture

so the idea is when we project onto the xz-plane we should get circles (since were working with a paraboloid) of the form x^2+z^2=r^2. The radii of the various circles are exactly x (the distance we move along the x-axis)

This gives the relationship that x^2 + z^2 = x^2

Using the fact that y = 1 + x^2/4, so x^2 can be replaced with 4y-4.

This gives $x^2 + z^2 = 4y-4$, or that $\frac{x^2}{4} + \frac{z^2}{4} + 1 = y$.

[wraith dagger]

twitch.tv/pencenter

[simple seal]

ohhhhhhhhhhhhh, that makes so much more sense

thats why it's a circle

thank u so much man 🙏

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[crimson sedge]

Last screenshot is the question

Would really appreciate some help

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@crimson sedge Has your question been resolved?

[crimson sedge]

<@&286206848099549185>

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[crimson sedge]

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[broken knoll]

do you have any practical application of eulers theorm (the mod one)

[rose olive]

yes in RSA encryption

[cedar kiln]

@broken knoll Has your question been resolved?

[broken knoll]

dude

don't left there

[rigid lava]

are you asking for a practice problem?

[broken knoll]

Yup

[rigid lava]

uh

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[rigid lava]

find the last two digits of 53^21348-71^34751

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[maiden venture]

"In which points and for which/which choices of the constant d does the ellipsoid have
2x^2 + 7y^2 + 3z^2 = 57 and the plane 2x + 14y + 9z = d a common tangent point?"

[maiden venture]

can someone help me solve this?

2x^2/57 + 7y^2/57 + 3z^2/57 = 1

is this the correct first step?

[dreamy void]

I think you figure out the gradient first and then equate it to the tangent plane

4x = 2k
14y = 14k
9z = 9k

2x² + 7y² + 3z² =57
2x+14y+9z = d

So we have to find a common point that has also the same slope

In both the function and plane

[cedar kiln]

@maiden venture Has your question been resolved?

[maiden venture]

Okay I get x=1/2, y=1 and z=3/2?

[cedar kiln]

@maiden venture Has your question been resolved?

[cedar kiln]

@maiden venture Has your question been resolved?

[wraith dagger]

𝔸dωn𝓲²s

[dreamy void]

This should be our system of equations @maiden venture

k is any constant

6z = 9k*

sorry i was gone

[maiden venture]

Yes okay but am I not supposed to solve out x y and z now?

[dreamy void]

We gonna go this through hold on

[wraith dagger]

𝔸dωn𝓲²s

[dreamy void]

Now how did I find this?

Well you need the gradient of f

Which is

[wraith dagger]

𝔸dωn𝓲²s

[dreamy void]

So here

[wraith dagger]

𝔸dωn𝓲²s

[dreamy void]

You follow?

[maiden venture]

I think so?

[dreamy void]

Ok

Now the same thing when we differentiate the tangent plane equation

Let's call it g

[wraith dagger]

𝔸dωn𝓲²s

[dreamy void]

And those gradients we set them to equal!

because we want to find a tangent point

[maiden venture]

Ahh okay!

[dreamy void]

but we need a constant k

are you familiar with vectors?

[maiden venture]

Yes I am

[dreamy void]

ok good

we need a constant because like in vectors

we need to find a certain scalar so that they not only have the same tangent plane

but also a common point

because the tangent planes can be parallel

but then not touch each other

[maiden venture]

Yes okay

[dreamy void]

hope u understand

[maiden venture]

I think I do

[dreamy void]

So this is how I came up with the first 3 equations

the other 2 are given

[maiden venture]

Yes that I understand!

[dreamy void]

Now we want to find a k so that they have at least the same tangent plane

So we solve the 3rst equations for x, y and z

$\begin{cases}
4x = 2k \iff x = \frac{k}{2} \
14y = 14k \iff y = k\
6z = 9k \iff z = \frac{3}{2}k \
2x^2 + 7y^2 + 3z^2 = 57 \
2x + 14y + 9z = d
\end{cases}$

[maiden venture]

Yes is x not k/2?

[dreamy void]

yes!

[maiden venture]

Oh yes!

[wraith dagger]

𝔸dωn𝓲²s

[dreamy void]

Now we plugin x,y and z into the 4th equation anyway

because in the last one we would have one equation and two variables

so first we figure out the "matching k"

[maiden venture]

Okay so we get k + 14k + 9k = 24k?

[wraith dagger]

𝔸dωn𝓲²s

[maiden venture]

Oh I put it in the wrong one

[dreamy void]

So we found our constant k, so now we can find x, y and z!

Can you tell me these?

[dreamy void]

You just need to plug them here

x = k/2
y = k
z = 3k/2

[maiden venture]

X=1?

[dreamy void]

go on

[maiden venture]

Or +-1

[dreamy void]

yes

[maiden venture]

And y = +-2

And z = +-3?

[dreamy void]

yes!

Now we plug them into the last equation in

and you should get 2 d's

[maiden venture]

You mean 2x + 14z + 9z = d?

Or the other one?

[dreamy void]

yes

this

to find

[maiden venture]

Okay the first one is d=57

[dreamy void]

yes!

[maiden venture]

And the other one -57!

[dreamy void]

yes!

[wraith dagger]

𝔸dωn𝓲²s

[dreamy void]

Also you already know your tangent points 🙂

Since we figured x, y and z

That should be it!

[maiden venture]

Okay so the answer is just when x= +-1 and so on for you and z?

[dreamy void]

yes

P(1,2,3)
P(-1,-2,-3)

[maiden venture]

And the common tangent point is +-57?

[dreamy void]

no that is the value for d

[maiden venture]

Oh okay

[dreamy void]

these are the common tangent points

look at the picture

[maiden venture]

Ohh I see!

[dreamy void]

yea you better see!

[maiden venture]

Haha thank you!

[dreamy void]

np!

[maiden venture]

.close

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[dapper heron]

!help

[cedar kiln]

To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.

[dapper heron]

so a^2 +b^2 = c^2 right?

[delicate elm]

for a right angled triangle with c as the hypotenuse, yes

[dapper heron]

i need to prove for natural integers

a^n + b^n = c^n when n is greater than 2

prove or disprove

[rigid lava]

here you go

https://people.math.wisc.edu/~nboston/869.pdf

[delicate elm]

https://en.m.wikipedia.org/wiki/Fermat's_Last_Theorem

[serene valley]

that was pretty famously proven lol

[dapper heron]

ty

[chrome delta]

Lol

[dapper heron]

just normal hw

[chrome delta]

might also want to clarify your definition of N

cuz 0^3 + 0^3 = 0^3 lmao

[dapper heron]

true

i sill got 1 more hw question

still*

[chrome delta]

let me guess

it's another conjecture

[dapper heron]

no

how do i type : properly ?

thats not the question

but still

like x/y

?

its there a better way?

[rigid lava]

how do you type a division symbol?

[dapper heron]

yeah so its horizontal

like you would normally write

2hrs of sleep

anyways...

2 natural integers N and Q in a way so N/Q= 1 - 1/2 + 1/3 - 1/4 ...-1/1318 + 1/1319

prove that N is divisible by 1979

[chrome delta]

ah yes an IMO problem

from fermat's last theorem to an IMO problem

[dapper heron]

care to show my how to solve?

i think is a telescopic series

right?

[delicate elm]

brute force compute the sum by hand

lmao

[dapper heron]

but i havent figured out the trick

i did

its close to logE I think

lan i mean

but sum/q isnt divisible by 1979

[chrome delta]

https://artofproblemsolving.com/wiki/index.php/1979_IMO_Problems/Problem_1

[dapper heron]

ty

[chrome delta]

any other question? maybe finding the nontrivial zeroes of the riemann zeta function that isn't along the critical line?

[dapper heron]

that too

[delicate elm]

pfft

[dapper heron]

we can split the prize money if you help me solve

[delicate elm]

i once read a book which says that anyone who is trying it for the money will 100% never solve it

[chrome delta]

that's the hardest way to earn a million dollars btw lol

[dapper heron]

i read a book that clams the red sea was split in half? and

if kai cenat has 10 grimace shakes and ishowspeed has 7 grimace shakes how many grimace shakes aidn ross get fanum taxed on?

only w answers get accepted

chess 1v1?

[delicate elm]

what elo are you

[dapper heron]

wanna guess?

[delicate elm]

too high

[dapper heron]

what is your lim

[delicate elm]

if your elo is any higher than 1600 im not playing

[dapper heron]

2160 ):

wanna pvp?

[chrome delta]

he's been playing for 1 year

[delicate elm]

no

[chrome delta]

based on his chat logs

[spring sage]

I am up for that

[dapper heron]

im hypixel lvl 247

[chrome delta]

damn the chess player got summoned

[dapper heron]

yay

[spring sage]

I am the chess player for a reason

[dapper heron]

let see what u got

[chrome delta]

poor guy lol

why delete the link?

https://lichess.org/41bqivB6/black#9

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[swift saffron]

Suppose s(n) denote the sum digit of n. If s(xy) >= y . s(x), can I say that s(yx) <= x . s(y)?

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[cedar kiln]

Please don't occupy multiple help channels.

[full forum]

i am solving b, and answer for a is 278

I can get the angle ADB and the lengh of BD

and I am not sure how i am supposed to solve for the depression angle

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@full forum Has your question been resolved?

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@full forum Has your question been resolved?

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[worldly cedar]

Why is the standard deviation based off of the mean and not the median? I am confused because the mean can be thrown off by an extreme outlier. Why isn't it based off of the median and then that extreme outlier is just deviating further?

[mighty shuttle]

as far as I know it's by definition

[worldly cedar]

But why. I want to know why.

[mighty shuttle]

https://stats.stackexchange.com/questions/448807/standard-deviation-around-mean-rather-than-mode-or-median#:~:text=I believe it is because,mean)%20of%20the%20random%20variable.

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[worldly cedar]

I feel like this is wrong. A margin of error associated with the mean can only describe the mean values of the entire population, right?

[worldly cedar]

Can anyone help

.close

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[crimson sedge]

How do I go from here?

I am trying to find out why the derivative of ln(X) is equal to 1/x by making use of integration by parts

Heya folks, I am trying to figure out why the derivative of ln(x) is 1/x and doing so by making use of integration by parts. Does anyone know where I can go from here?

<@&286206848099549185>

[onyx oyster]

hm

wait but aren't you already using the fact that ln(x) derivative is 1/x

v'(x)

[crimson sedge]

Yeah

But I should still get 1/x no?

Oh wait a moment, I'm integrating

[void glen]

you’re integrating ln(x)

[crimson sedge]

Right

I should do it the other way around

[void glen]

if you want to prove it, you can do it using the definition of the derivative

[crimson sedge]

Integrate 1/x to ln(X)

[onyx oyster]

use the limit definition

and then

[crimson sedge]

I must do it by integration by parts

[onyx oyster]

use the defintiion of e

[crimson sedge]

What is that?

[onyx oyster]

well you can't just assume the result you're tryna prove, to prove that result

[onyx oyster]

uh

first principles

so f'(x) = lim x --> 0 ((f(x+h)-f(x))/h)

[void glen]

i mean you can try but

[crimson sedge]

My teacher said so 😭

[void glen]

not sure if it’ll work

[onyx oyster]

even if it will work

[crimson sedge]

How come?

[onyx oyster]

this aint the approach

since if ur tryna prove lnx = 1/x

you have to assume that you dont know it

and hence you really can't be using it in your proof

[void glen]

i mean you can def try it

choose 1/x to differentiate and 1 to integrate

but i’m not sure if that’ll go anywhere

[onyx oyster]

yh

[crimson sedge]

Ah man

Look at this

[onyx oyster]

the ln won't come anywhere

so integration by parts

ehh

I don't think it'll work

[onyx oyster]

you need to use the definition of e somewhere

or the fact that e^x derivative is e^x

[crimson sedge]

Why is my teacher like this

[void glen]

yeah so see, this doesn’t work

[onyx oyster]

but this feels like cheating

[crimson sedge]

But how can I do that to prove this

[void glen]

you are assuming that you know the derivative of ln(x) is 1/x

[crimson sedge]

Yeah

[void glen]

definition of the derivative

[crimson sedge]

That was what my teacher said

[onyx oyster]

oh btw

if you're allowed to use the fact that the derivative of e^x is e^x

[crimson sedge]

to clear it up, I have a presentation about partial integration

integration by parts*

[onyx oyster]

there's a VERY elegant solution

[crimson sedge]

which is why im doing all of trhis

[crimson sedge]

I know I know

[onyx oyster]

ok here

x = e^t
y = t

dx/dt = e^t
dy/dt = 1

(dy/dt)/(dx/dt) = dy/dx = 1/e^t = 1/x

use parametric equations

[crimson sedge]

I already involved xe^x

[onyx oyster]

boom

[crimson sedge]

in my presentation

[onyx oyster]

again I feel like this is cheating since ur basically getting the derivative of e^x for free

[crimson sedge]

but i that the same as ln(x)..?

[onyx oyster]

but it works

[onyx oyster]

yes

[crimson sedge]

Could you please explain how?

[onyx oyster]

convert those parametric equations to cartesian coordinates

[crimson sedge]

I see that you got 1/x as a solution

[crimson sedge]

uhh

im not sure my class knows what that is

[onyx oyster]

so the 2nd equation

[crimson sedge]

im still in grade 11

[onyx oyster]

uhhhhhh

nvm that solution then

[crimson sedge]

arent cartesian coordinates the thing with complex numbers?

I saw it in a uni course at some point

[void glen]

just the xy plane really

[crimson sedge]

but i dont think its a level material

[onyx oyster]

do you know this formula

[onyx oyster]

wait ur in a level

how is grade 11 a level

[crimson sedge]

uhh

my a levels are next yeart to be fair

but here in Germany grades 11 and 12 are prep for a level

so you just say it like that

[crimson sedge]

I don't think so

not in this form

[onyx oyster]

ok then I literally don't know how to help you prove lnx

[crimson sedge]

ngl I honestly do not know what my teacher meant

either I misunderstood her

[onyx oyster]

you need either the limit definition

[crimson sedge]

or she is incorrect

eitherway, I will make sure to ask her tomorrow

[onyx oyster]

or the parameters

yh

[crimson sedge]

what other uses are there for integration by parts?

[onyx oyster]

uh

[void glen]

how do you know integration by parts but not the limit definition of a derivative?

[crimson sedge]

other than integration products in which one is a polynomial

[onyx oyster]

literally any integral with a product

[void glen]

that’s kinda weird ordering

[crimson sedge]

yep

[onyx oyster]

THIS IS WHAT IM CONFUSED ABOUT

[crimson sedge]

but it kinda sucks unless one is a polynomial

[onyx oyster]

prob weird syllabus

[void glen]

fr

[crimson sedge]

I think I know the limit definition

but I learned it differently

one sec let me show you

[void glen]

there’s 3 versions

but like the other two are only for a specific coordinate

[crimson sedge]

(f(b)-f(x))/b-a

something like that

this I did last year already

with the lim x->a

[void glen]

that’s for f’(a)

[crimson sedge]

yes

is this what you guys are referring to?

[void glen]

but the one 2cap posted is more general

[crimson sedge]

See me personally

[void glen]

the one you have wouldn’t help

[crimson sedge]

I would love to use it

but idk if my class would understand it

we never did it

[void glen]

odd

[crimson sedge]

and my class is not exactly the brightest

[crimson sedge]

our teacher is leading us into failure ngl

my grades have been dropping like crazy with her

but it is what it is

[void glen]

with a syllabus like that, yeah

just gotta do self studying ig

[crimson sedge]

I have been sitting here brainstorming like this for the past hour or so

[crimson sedge]

seems to be my only option

[crimson sedge]

so in conclusion, there is no way to prove or exemplify the integration of 1/x to ln(x) via integration by parts whatsoever?

[void glen]

don’t think so

[onyx oyster]

uh

if there is

it's prob WAY out of the spec

[crimson sedge]

Probably

[onyx oyster]

I'm pretty confident it's impossible to prove the derivative of lnx without either knowing the derivative of e^x and using it in your proof, or without the definition of e

[crimson sedge]

so I should just stick to the integration of xe^x and xsin^x etc

[crimson sedge]

What do you mean by the definition of e?

[onyx oyster]

uh

the limit as x approaches infinity of (1 + 1/x)^x

[crimson sedge]

oh yes

I did this in uni

but idt my class knows this 😭

[onyx oyster]

yh so if you use the definition of derivative

[onyx oyster]

this

and instead of f(x) if you put lnx

you can achieve a form similar to the definition of e

[crimson sedge]

Very interesting

while we're on the topic though#

[onyx oyster]

I can send you a vid of how to prove it in dms

[crimson sedge]

could you explain to me how integrating x^2sin(x) works if I do integration by parts twice?

[crimson sedge]

yes please

[onyx oyster]

uh

[crimson sedge]

ill definitely look into it for my personal understanding of the topic

but I doubt my class will be able to do anything with it

[onyx oyster]

when you do the 1st IBP

u is x^2 and dv is sinx

[crimson sedge]

ohhh

u' becomes x

then do it again

[onyx oyster]

and you get uv - integral of vdu

[crimson sedge]

x becomes 1

[onyx oyster]

then the integral of vdu can be done by another IBP

[crimson sedge]

oh yes

could I try that then send you my solution?

just to make sure that it is correct#

[onyx oyster]

sure

though you can always verify with wolfram alpha

[crimson sedge]

thanks man

that video would also be highly appreciated

[crimson sedge]

true

[onyx oyster]

yh ill send you in dms

[onyx oyster]

wait is this the same question

oh it isnt

[crimson sedge]

Wdym?

[onyx oyster]

...

whatever ima still check

[crimson sedge]

Oh

My bad

Same concept

[onyx oyster]

yh

tho with e^x it's easier

[crimson sedge]

yup

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[sweet marsh]

what is the area in the first quadrant bounded by y=e^(-x^2/4) and y=0

[onyx oyster]

integration question

hm

[sweet marsh]

can use calculator

[onyx oyster]

so write this as an integral

no need for calculator here

it's the integral from 0 to inf of e^(-x^2/4)

[sweet marsh]

what do you mean?

[void glen]

you definitely need a calculator for this

the function is a non elementary integral

[sweet marsh]

yeah this is like a weird fresnel integral thing

[vagrant elbow]

unless op is aware of the gaussian integral

[onyx oyster]

yh

[sweet marsh]

or gaussian

[vagrant elbow]

fresnel is sin(x^2) or cos(x^2)

[sweet marsh]

the answer choices just arent lining up though

[sweet marsh]

yeah

[onyx oyster]

firstly you want to make this like the gaussian integral

so you wanna take x/2 to be u

so that u^2 is x^2/4

[void glen]

it’s not from negative infinity to infinity

[onyx oyster]

yh it's from 0

[sweet marsh]

is it not just the integral of 0 to infinity of e^(-x^2/4)

[void glen]

oh

i didn’t read the question

[onyx oyster]

yep

so u = x/2 will make this in the form of the gaussian integral

so this is 2 * the gaussian integral but from 0 to inf

[sweet marsh]

ok can you guys compute it too, since its not coming up on any answer choice here

[void glen]

i mean then you can technically use the fact that $\int_{0}^{\infty}e^{-x^2}dx=\frac{\sqrt{\pi}}{2}$

[wraith dagger]

y0shi

[onyx oyster]

ye

that's what you'll end up doing

but then it's twice of that due to our substitution

[onyx oyster]

no calculator needed since this is a standard result

[sweet marsh]

whats the answer then, I dont even know what you guys did

[void glen]

so use u sub to get the integral we have into this form

[sweet marsh]

ah gotcha

so the answer choices are wrong then, since none of these show up

1.7724538

[void glen]

that’s square root of pi yeah?

that should be right if so

[cedar kiln]

@sweet marsh Has your question been resolved?

[sweet marsh]

yes

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[half vigil]

Could someone please explain this to me in an easy manner, I don’t understand what to do at all

[royal loom]

Is this for a test?

[half vigil]

No it’s for an assignment I have 3 tries on

I’ve been at it for an hour

[royal loom]

Are you allowed help with it?

[half vigil]

Yes I am

She told us to reach out to her if we don’t understand but she’s not responding to my email and it’s been over an hour 💔

[royal loom]

I don't think that the circle is relevant here

imagine it as just the triangle, redraw it and label the sides

do you know SOHCAHTOA?

[half vigil]

Yes

[royal loom]

Okay, so once you've done what I've suggested, let me know if you are still stuck

and actually, this will just use the pythagorean theorem I believe, so not much trig.

Wait..

10 is the hypotenuse of this triangle, so how is the other side length 14?

that is not possible

[half vigil]

Hmm

[royal loom]

Perhaps they mean that the total side length of the hypotenuse is x+10

and 10 is just the upper portion outside of the circle

[half vigil]

Oh wait that makes so much sense

So then would I just use SOHCAHTOA

[royal loom]

pythagorean theorem

[half vigil]

Ohhhh ok

Thank you so much

[royal loom]

np

[half vigil]

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[onyx oyster]

just ask the question bro

plug in 3 into x

and see if the LHS is equal to the RHS

for x = 3 to be a sol

putting 3 instead of x should make the equation true

I guess that's one way to do it

but

yes

upon replacing x with 3

if the left hand side does = to the right hand side

then x =3 is indeed a sol

otherwise no

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[pulsar turtle]

Why doesnt it equal purple? Purple is what i thought it should equal below that is what it equals

[void sand]

the -x in the numerator and the x in the denominator cancel out

I don't exactly know why you think it's what's written in purple, but that's what the next step was

combining the fractions and cancelling the x's

[pulsar turtle]

Alright what about this? I dont get how they get all of that. I would understand if it was just 2+x repeated 3 times and then -8 on all of it but this makes no sense

@void sand ^

[void sand]

2-2 = 0

then you have an x in the numerator and denominator

so they cancel

[void sand]

2 + x - 2 = x

that's what I mean

[pulsar turtle]

Yeah but where did that even come from? How was (2+x-2) derived from (2+x)^3 -8

[void sand]

looks like a difference of cubes

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

that's where it came from

[pulsar turtle]

man wtf our teacher didn't teach us that

[void sand]

well uh

now you know lmao

[pulsar turtle]

lol

thanks for the help 🙏

[void sand]

you can close this channel if you're done

[pulsar turtle]

.close

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[smoky barn]

claim

[smoky barn]

Can I please get some help. I got suspened from school and now I have a test tomorrow

I dont undertand anything please help

[thick forge]

Hey, for a quadratic we typically want it in the form ax^2 + bx + c

where a, b, c are constants

[smoky barn]

so a=1
b=2 (or 4)
c=9?

[thick forge]

not quite

try and expand (x-2)^2

[smoky barn]

like solve?

[thick forge]

essentially, just simplify it

(x-2)(x-2) =?

[smoky barn]

when i put it in my calc i got (x-2)^2

They also put it as x^2 -4x +4

[thick forge]

Just do the math, no sense using a calulator

but yes (x-2)^2 is x^2-4x+4

add that 9 we had

and we now have ax^2 +bx +c

[smoky barn]

x^2-4x+4 +9

?

[thick forge]

to get a vertex we plug in our a and b values into
-b/2a

thats for the x axis

[smoky barn]

is -4 b?

[thick forge]

b is a constant not a variable

plug in what a b and c is

[smoky barn]

wait

one second lemme write this down

[thick forge]

the y axis you take what number you got from -b/2a and plug it in for x in this

thus we get the vertex

[smoky barn]

So like this?

Im a little confused on what a b and c is

[thick forge]

we have x^2 -4x +13 right

just generally agree

a b and c are the constants right out front of the variables

so
a=1
b=-4
c=13

(ax^2+bx+c)

[smoky barn]

Got u

[thick forge]

cool

[smoky barn]

So just plug b and a into the vertex

And then pluf whatever into x?

[thick forge]

yup

yup

[thick forge]

for x coordinate

[thick forge]

for y coordinate

[smoky barn]

And then solve and get y

Ohh okay one sec lemme cook

[thick forge]

for maximum and minimum its also super straightforward

[smoky barn]

Yo I think i am the one thats cooked lmfao

[thick forge]

-b/2a

so we get -(4)/2(1)

which is

[thick forge]

and then we take this and plug in for x

[smoky barn]

So (2-2)^2+13

So 13 is y?

[thick forge]

i mean you can plug it into that yeah but its (2-(-2))^2 + 9

keep track of ur negatives

and remember it originally had a 9 on the outside

[smoky barn]

Oh so u plug

X into the very first quation

[thick forge]

you can

[smoky barn]

Okay so just 2,9?

[thick forge]

no

[thick forge]

what is negative 4 divided by 2

[smoky barn]

-2

[thick forge]

there ya go

[smoky barn]

So -2,9

And because its a negative its a minimum?

[thick forge]

^

why 9?

is (2-(-2))^2+9 = 9?

[smoky barn]

The answer key said it was 2,9 😭

[thick forge]

wack what

[smoky barn]

Am i cooked

[thick forge]

whyd they make a y intercept the y coordinate 😭

ur school is cooked bro

[smoky barn]

I hate cpm so much bro like

[thick forge]

anyways to solve b

[smoky barn]

Its so confusing

[thick forge]

x^2 is a parabola, if the constant out front the x^2 is positive, it opens up (because it is all the positive y values) and if its negative, its all the negative y values so it opens down

so if a>0 it opens up, a<0 it opens down

imagine a parabola in your head. You just found the vertex, if it opens up is the vertex the minimum or maximum? If it opens down is the vertex the minimum or the maximum?

Is our vertex a minimum or a maximum?

sorry not b

part 2 of a

[smoky barn]

Its a minium

[thick forge]

and b is just because a is positive it opens up, so our vertex is a minimum

[smoky barn]

Minimum

[thick forge]

and minimum with positve x value coordinate mean it never cross x axis

ez stuffs

lastly

to find the roots of a quadratic equation we use to quadratic formula

classic equation, plug and chug

not fun by hand but most teacher allow you to use a calculator

useful to know so u can plug it in on a calculator without a button on it like a TI-30

anyways I hope you took a little away from this

if anything

ax^2 + bx + c for quadratics

[smoky barn]

Type shi

Do you want to do another one?

[thick forge]

i gotta do some limits for lin transforms of multi var calc problems sry, tedious stuff

i just hop on rarely to help a lil

[thick forge]

i believe in you man

[smoky barn]

Its okay ill attempt and make another ticket ltr

Thanks for everything bro

[thick forge]

bro u just got all the info u need 😭

try googling a little

"how to find vertex of a quadratic equation"

bam

[smoky barn]

bam

Im just worried about this test ngl

If i fail i have another unit but I dont want to fail bro 😭

[cedar kiln]

@smoky barn Has your question been resolved?

[cedar kiln]

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[misty thistle]

Why?

[mental trail]

There's no minus

4u = 6v = 6*3 = 18

[misty thistle]

Idk, this is the mark scheme

[mental trail]

Sure but there's no minus, it was a mistake

[misty thistle]

But I still don’t get the +/- 6 x 3

@mental trail

[mental trail]

It's just a +.
You do the conservation of momentum
Since the initial momentum is 4u, and new momentum is 2v + 4v (see graph), those momentums are equal

So 4u = 2v+4v = 6v

You computed v = 3

So 4u = 18

[misty thistle]

Are you able to help me?

[misty thistle]

There was no mistake, the CoM is in the direction to the right whereas CoM of A is pointed to the left

Slow internet

Ohhh

TYSM

@mental trail Thank you!

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[kindred granite]

how would i do this

[kindred granite]

i wqas studying for my calc ab finals for saturday and doing practice problems and came across this

it doesnt seem to simplify nicely with u substitution

[mighty shuttle]

try IBP

I doubt u-subs will work here

[kindred granite]

oh thuis is calc 2

thats why ni couldnt do it

wait

yeah this is bc material

idk how to do this then

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[old zephyr]

Is there a way you could do it with trig identity?

[mighty shuttle]

I don't think so

[old zephyr]

Okay. Thank you.

[kindred granite]

oh wait it didnt close]

.close

wait did it close

idk

[cedar kiln]

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[formal pasture]

,rotate 270

[formal pasture]

how do i do 10

where do ik where to plug what points

like in which derivative

[livid hound]

from the definitions

the relationship between first derivative with local extrema
and
inflection with second derivative

[formal pasture]

okk

[formal pasture]

i duno what to do after

this step

[livid hound]

don't forget that (2,11) and (1,5) are also points on the graph

[formal pasture]

i did this

[livid hound]

I.e.
f(2) = 11
f(1) = 5
giving you two more equations to work with

[formal pasture]

i alr got all 4 equations

those r two others

[livid hound]

solve the system

[formal pasture]

but the point of the system is to eliminate a variable

[livid hound]

you have 4 equations with 3 variables which should be more than wnough

use a combination of Gaussian elimination or substitution

[formal pasture]

whats that

💀

how do u usr both

[formal pasture]

is this wrong

hrlp

[cedar kiln]

@formal pasture Has your question been resolved?

[livid hound]

you have two equatiions
7a + 3b = 6
6a + 2b = 0
solve that to get a and b

[formal pasture]

how do u get b

.close

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[formal pasture]

wheres the point of inflection

at 3/2

[orchid panther]

Point of inflection is (3/2, 30) because its the point where the second derivative is zero @formal pasture

[formal pasture]

but usually

its like cubic

and it shows how it changes

[orchid panther]

Yea original function is cubic, so the first derivative is quadratix and second derivative is linear

Quadratic*

[formal pasture]

no like

the point

of intersection

is usuallu shown

as a cubic

or smthn

to show it goes from smthn to smhtn

[orchid panther]

I dont understand what you mean could you explain

[formal pasture]

its ok

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[bronze sparrow]

can someone walk me through this?

[pallid moon]

do you know law of sines

$\frac{x}{\sin\angle X}=\frac{y}{\sin\angle Y}$

[wraith dagger]

BuboBlakistoni

[pallid moon]

i think you can plug the values into that

[bronze sparrow]

alr ty lemme try

[rose olive]

law of cosines

[bronze sparrow]

I got 170 degrees for angle y using that equation which isnt possible

any other ideas?

[pastel vault]

So recall that sin(180 - x) = sin(x): this is true for any x

jesus christ, the deleted msg

Also you should get 160 degrees and not 170

[bronze sparrow]

so wouldnt that make this triangle impossible?

[pastel vault]

Yes, the triangle is impossible with 160 degrees

But sin(160) = sin(20)