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[leaden yarrow]

How am i supposed to solve this exercise with only BASICS of integrals (No substitution or whatever but only reverse formulas and first 2 linear proprieties):

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Please don't occupy multiple help channels.

[thick hedge]

Write sin(x) in terms of tan(x/2)

[leaden yarrow]

Book's result: (sinx - 1)/(cosx) + c

[torn jolt]

@thick hedge

[leaden yarrow]

That's substitution

[thick hedge]

ah, right

[thick hedge]

yes?

[torn jolt]

#help-39

[torn jolt]

that is expansion

not substitution

[leaden yarrow]

What is it?

[torn jolt]

sin(x) = 2tan(x/2)/(1+tan(x/2))

[leaden yarrow]

Hmm, idk if i am supposed to do that

[torn jolt]

no other way

simpler way*

[thick hedge]

I mean, you could multiply and diide by 1-sin(x)

that should help

[leaden yarrow]

that's what i tough but my mind's calculus went to an unresolvable solution

let me try again

omg

i solved it but in a strage way

i mean, i don't think that i am supposed to do the last operation:
Writing that integral of sinx/cos^2(x) is equal to -1/cosx

i mean, i shouldn't solve integrals by thinking " is that expression the derivate of 1/f(x)?"

[torn jolt]

you did it right

[leaden yarrow]

However that's a clever way to solve it, i should memorize that formula

[leaden yarrow]

that's my book of 2 years ago of goniometry

[torn jolt]

but this one is more clever

becuase you have to ge that idea

[torn jolt]

this is what normal people do

[leaden yarrow]

yeah

[leaden yarrow]

Also my teacher said that inside integrals i cannot multiply and divide by x (and i dind't understandy why) or maybe i didn't understand what she meant.
Can i do it?

[torn jolt]

that won't help in this case

[leaden yarrow]

but can i do it?

[torn jolt]

yes

[leaden yarrow]

Ok, but if i do it then x must be != from 0.
When i do integrals, do i have to mind about Domain?

[torn jolt]

but of all the integral questions i have come across, i have never used the above method

[leaden yarrow]

I mean, when i get the main function, what happens to the Domain of the function i have to integrate?

[leaden yarrow]

ok

[torn jolt]

i didn't ge tyou

[leaden yarrow]

I have to integrate that function f(x) which integral is F(x), yeah?
Initially, f(x) has it's domain (such as x != 0), but when i integrate f(x) to obtain F(x) (which may not have x to the denominator and so x can be 0), what happens to the domain of f(x)?

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[tribal pagoda]

I need to prove that the recursively defined sequence $(an)$ which is is defined as $a_1 \coloneq 2$ and $ a{n+1}\coloneq \sqrt{1+a_n}$ for $n\leq 1$
converges, and I need to find it's limit.

[glossy valve]

Lukas

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[torn jolt]

what do you mean by "x != 0"

[torn jolt]

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[indigo osprey]

So I drew a circle and chord and constructed triangle PAB

If we take area of triangle it's 1/2* PA PB sin P

So PA PB = 2* area / sin P

As P is constant

We need to maximise area

Which is maximise height of traingle

How to select P in such a way?

<@&286206848099549185> sorry for ping

<@&286206848099549185> please ?

[torn jolt]

which one

a)

?

[indigo osprey]

Yes?

[indigo osprey]

And b too it's all same question

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[indigo osprey]

Nope

<@&286206848099549185>

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[analog ridge]

Where is the 6 coming from? since HA is at y=0?

[solemn gate]

im not sure how this 6 correlates to the asymptote

[analog ridge]

could it be that they're multiplying the intercepts?

but that wouldnt make sense since 2*-3 = -6

[vast fossil]

Simply plug in the value of x where the function is nonzero

[brisk obsidian]

When you come up with your initial function, there is a k constant that it is multiplied by.

You then use a known value to determine k.

[analog ridge]

how is 6 a known value though?

[brisk obsidian]

There are two given points on the graph.

(0,-3) and (2,0)

[analog ridge]

correct

the intercepts

[brisk obsidian]

Substitute either points into the equation above and solve for k.

Hmm, (2,0) would not be a good choice since it is an x-intercept.

edit : intercept to x-intercept

[analog ridge]

(0,-3) is also an intercept

[brisk obsidian]

The x-intercept will not be affected by a vertical stretch.

However, the y-intercept will be affected by a vertical stretch.

[analog ridge]

ohhh

I just did that

and got -12/-2

which is 6

[brisk obsidian]

👍

[analog ridge]

thank you y'all

appreciate the help

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[plush egret]

Here's just the full problem for context, my question is more specific

We are able to get to an 'ODE" $$ \pdv[2]{\tilde \phi }{ z} = s^2 \tilde \phi$$

[glossy valve]

jan Niku

[plush egret]

now I lost a few points on the old final because i didnt consider all the cases but im curious how exactly this works now

say we take s=0 does that mean that we get the solution $\tilde \phi = A(s,t)z + B(s,t)$?

[glossy valve]

jan Niku

[plush egret]

and then how do we reject this solution given its not present in the answer

solution mentions this method but ignores s=0 as well

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@plush egret Has your question been resolved?

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@plush egret Has your question been resolved?

[plush egret]

$\pdv[2]{\tilde \phi}{z} = s^2 \tilde \phi$

[glossy valve]

jan Niku

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@plush egret Has your question been resolved?

[torn jolt]

Which part of the question it is?

[torn jolt]

So the fi = to integral of something is the same fi you find in first part by solving differential equation?

[plush egret]

sorry, i didnt realize you'd replied

the question is about the s=0 case

like, it is possible for s=0

you end up with $\pdv[2] z \tilde \phi(0,z,t) = 0$

[glossy valve]

jan Niku

[plush egret]

so $\tilde \phi(0,z,t) = A(t) z + B(t)$

yea?

[glossy valve]

jan Niku

[plush egret]

i mean theres more but agreed up to here?

[torn jolt]

Yeah but is this part 1 or part 2?
So you want to get rid of dfi/dz

[plush egret]

we just want to pick up every component of the solution which satisfies the underlying model given by the BC and IC

this is part ... there are no parts

this is the part "show that"

[torn jolt]

You mean you solve differential equations for all variables of fi?

[plush egret]

have you solved a pde via fourier methods before?

[torn jolt]

No.......

[plush egret]

hmm

[plush egret]

it creates an 'ordinary' differential equation

rather its a partial differential equation but only one variable is present

$\pdv[2]{\tilde \phi}{z} = s^2 \tilde \phi$

[glossy valve]

jan Niku

[plush egret]

Now you need to solve this

so s can be anything

then theres 2 cases here

s^2 can be positive or 0

each contributes a solution, m aybe

I'm curious about the case where s=0

this requires solving $\pdv[2] z \tilde \phi = 0$ and applying the boundary conditions

[glossy valve]

jan Niku

[torn jolt]

How is s defined here?

[plush egret]

its a transform variable

have you done integral transforms before?

its just a fourier variable weve shifted one of the variables to another domain

[torn jolt]

Is there a link for this paper with solutions or any other context ?

[plush egret]

https://math.libretexts.org/Bookshelves/Differential_Equations/Introduction_to_Partial_Differential_Equations_(Herman)/09%3A_Transform_Techniques_in_Physics/9.11%3A_Transforms_and_Partial_Differential_Equations

[torn jolt]

Alright I will try to brute force it so it probably take whatever amount of it.
Have you tried any calculus subredidts?
Maybe studentroom forum

For me onky way to get sensible answer is to trace whole context which will take a lot of time.

So yeah, that's kinda how it is the harder the question the less possible that you will get answer on it

[plush egret]

lol

yea

i mean

its pretty specific subset of a larger problem

[torn jolt]

That's why I am saying that someone good at calculus might give you an answer straight away. But I need to rely on context

[plush egret]

its just that we transform to another function

[torn jolt]

So that's kinda the problem by itself how to get answer evn on this specific question

[plush egret]

the solution is some say $\phi(x,z,t)$

[glossy valve]

jan Niku

[plush egret]

so we instead solve a pde for some transformed function

$\tilde \phi (s,z,t) \triangleq \int _\mathbb R \phi (x,z,t) e^{isx} \dd x$

[glossy valve]

jan Niku

[plush egret]

we transform each equation and boundary condition the same way

for example youll find that $\int _{\mathbb R} \nabla^2 \phi e^{isx} \dd x = \pdv[2] z \tilde\phi - s^2 \tilde\phi$

[glossy valve]

jan Niku

[torn jolt]

I mean if s is constant you can pick any value and it should be valid solution.

But again I just had question with dummy variable in differential equation and question had no specific context so I could have just used special case for my differential equation like n=1

[plush egret]

s is a variable

ultimately we re-transform out by integrating over all reals of s

[torn jolt]

Well I ment that n was a free variable so you can pick any value for show part of the question. So like to solve for particular value of n you pick

[plush egret]

so we have to check each possible value of s to see if we get a solution that we need to accumulate

[plush egret]

s is not free

its no more free than x is in y'(x) = y(x)

[torn jolt]

Alright so what can you do with solutions when s=0
Are they useful for you in any way?
If yes
Then is it contradict with any other solutions
If no
Then is it complete the answer.
?

[plush egret]

no, its like this

you de transform

$\int _{\mathbb R} \tilde\phi e^{-isx} \dd s = \phi (x,s,t)$

[glossy valve]

jan Niku

[plush egret]

something like this

you see how s sweeps from -inf to inf

so if for some value of s there is a solution, we accumulate it as we un-transform

@torn jolt this is all big picture which is not really required for my specific issue

[torn jolt]

.

[torn jolt]

.

[plush egret]

Correct so here's what I'm getting

You solve $\pdv[2] z \phi = 0$

[glossy valve]

jan Niku

[plush egret]

you get $\phi (0, z, t ) = A(t) z + B(t)$

[glossy valve]

jan Niku

[plush egret]

sorry, this should be tilde

but lets drop it for a second

you aply the boundary condition $\pdv \phi = 0$ on $z=-h$

[glossy valve]

jan Niku

[plush egret]

then $A(t) \equiv 0$

[glossy valve]

jan Niku

[plush egret]

so $\phi (0,z,t) = B(t)$

[glossy valve]

jan Niku

[plush egret]

yea?

[torn jolt]

Alright
So for particular value of s there is particular solution that is not listed in markscheme as a possible solution and you want to reject it?

[plush egret]

idk how much more to explain

[torn jolt]

Alright what will happen if s=1?

[plush egret]

i'm just explaining my solution in this specific context

[torn jolt]

Yeah I know.

[plush egret]

the solutions are not different here

we can consider s != 0 collectively

since they vary continuously with s

particular and general idk what you mean

[torn jolt]

Their method Makes more or less sense
Since - inf +inf not really defined so they combine it in such way to suit argument

I don't think setting s=0 contradict with their solutions.

<@&286206848099549185>

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[hidden delta]

someone teach me

[hot herald]

approach with rational root theorem

[hidden delta]

im bad in math

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[edgy osprey]

how to solve this using comparison test

[torn jolt]

wdym by "solve this"

send the complete question

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[edgy osprey]

oh sorrry

find if the limit converges or diverges

its a limit n to inifinity

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[distant swan]

.rotate

Wouldn’t f_xy and f_yx yield different results?

[shrewd hamlet]

if its continuous then we're good by clairauts

or young's, whatever u call it

f_xy = f_yx

[distant swan]

Hahaha, i’ll search that up

[shrewd hamlet]

👍

i learned it as young's, but after coming here, i learned it was also called clairaut's lol

and apparently theres another name too

anyways, its the theorem of equality of 2nd mixed partials

[shrewd hamlet]

yep

[distant swan]

Ahhh, so we need to check if its continuous then for this to hold.

What if it isn’t?

Then we do xy and yx manually and multiply them?

[steady falcon]

I didn't even know it had name lol

[shrewd hamlet]

probably something like that. but just lacking continuity doesnt necessarily imply that f_xy != f_yx. it just may not be true, but we don't know for certain unless we check. young's just guarantees it when we do have continuity. without continuity, we may or may not have equality of 2nd mixed partials

u can quote me on the second sentence, but im not 100% on "Then we do xy and yx manually and multiply them" so

but it makes sense so

[distant swan]

Ahh that’s fair

Thanks!!

[shrewd hamlet]

np

[distant swan]

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[distant swan]

Another one, finding max and min

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[distant swan]

How to find max min and saddle points of this one?

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[fast agate]

how do i solve this? i'm not really sure where to start

[crisp jetty]

Do you know what the maclaurin series is?

[fast agate]

i know that the maclaurin series for sin of x is x - x^3/3! + x^5/5! etc

[crisp jetty]

Good

So with the given properties the first two terms are both zero

The first non Zero term would be

0.5×f''(0)×x^2

So you wanna think what the derivative of f'(x)=sin(x)^2 is

Sorry for all the typing but I'm on my phone

[fast agate]

no worries!! thanks for all ur help in the first place

[fast agate]

2xcos(x^2)

[crisp jetty]

Good

And then you put that in the term I have sent: 0.5×f''(0)×x^2

[fast agate]

woulkdn't that just result in 0?

if i were to plug in 0

[crisp jetty]

I don't think so

[fast agate]

2(0)cos^(0^2) = 0

[crisp jetty]

Oh true I didn't see the x, I should go to sleep 😴

[fast agate]

haha no worries

[crisp jetty]

Okay then we have to calculate the next one

[fast agate]

but i think it would just keep giving us 0 forever

if we keep taking the deriv

cause you're always multiplying by 0

[crisp jetty]

No

Let's get the next derivative

Because we need the product rule it will work 🙂

[fast agate]

ohh oops i forgot about that

alr let me try and take the product rule

2cos(x^2) + (4x^2)cos(x^2)

so it would be 2?

[crisp jetty]

Looks.good

Yeah but

The entire term would be
1/6 × 2 × x^3 then

So then we have found the first non-zero term of the solution! That corresponds with answer 3 and 4, so we have to go one further

[fast agate]

ohh okay i see, but why'd you multiply by 1/6?

[crisp jetty]

Because of the maclaurin series

[fast agate]

SORRY i keep thinking thats the second derivative omg

okok got it so 2(x)^3/3!

which is x^3/3

[crisp jetty]

Yeah

[fast agate]

makes sense

[crisp jetty]

Judging from the solutions given, the next term would be at x^7 which means you need a few more derivates first

[fast agate]

ah okay

would there be an easier way to do this?

seeing as taking the deriv each time takes quite a while

[crisp jetty]

I am not sure about that

However you could just calculate it with a computer

[torn jolt]

no

it will lead to same thing

[fast agate]

yeahh unfortunately this is from the ap calc bc exam on the non-calc portion so i'd have to do it manually

[torn jolt]

ultimately

computing the derivatives

[fast agate]

ohh okay, so there's no shortcuts or anything?

[torn jolt]

not that i am aware of , sorry

[fast agate]

no worries!! thanks for your help, vien and zue :))

[crisp jetty]

You're very welcome

[fast agate]

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[brisk grotto]

Hello ! I would need some help with a double integral, please

I've made the first part of an exercise but then I need to deduct something from my result and I don't quite see how it should be done... My first guess is to use the Fubini theorem, but my attempts lead to nowhere

You'll find my work below 👇 (I'm sorry if the redaction isn't accurate, it's the first time I do math in english)

[scarlet herald]

Heres an idea

It might be kinda reaching considering we already know what the bottom integral should be

But

Integrating e^(-x²+y²) with respect to dx and dy

Is the same as integrating e^-x² dx and integrating e^-y² dy

Since the bounds are same, and the function is the same, it's just the square root of π/a

[brisk grotto]

Is that how I should write it ?

[wide sundial]

What exactly is the question

The math looks good but I’m not sure if you’ve answered the question (cos you haven’t posted what the question you’re solving is)

[brisk grotto]

Calculate the double integral... where a>0. Deduce the value of...from it

Is that in general that if the x part and the y part of
int (f(x,y) dxdy) are the same, then it's just (int (f(x) dx))^2 ?

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[craggy tapir]

When manufacturing aluminum funnels (cones without the base), designers always aim for least materials cost, that is least surface area of the cone. If we want to manufacture a funnel with a volume of $2dm^3$, what should be the minimum value of its surface area?

[cloud furnace]

dm being decimetre?

[craggy tapir]

not sure how to approach this, i have pi*r^2*h=6, but not sure how to find least surface area from it without at least the base radius or height

[cloud furnace]

volume of cone = pi x r^2 x h/3 = 2

[craggy tapir]

cube? you mean cone

[cloud furnace]

you want to minimize its csa = pi x r x l

[cloud furnace]

yeah mb

[craggy tapir]

yeah thats the whole point

[cloud furnace]

one moment

h = 6/pi * r^2

[cloud furnace]

sub in here

and differentiate to find the minima

[craggy tapir]

id have a 5 degrees

pretty ugly

but let me try

doesnt seem correct, the min value doesnt exist

[cloud furnace]

r^2 + 36/(pi^2 x r^4) needs to under a sqrt

[craggy tapir]

oh right, my bad

still doesnt give a minimum value

oh wait

solved, its 6,65dm^2

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[twin wolf]

how come median not 5? when i draw a line at 50 and line it up with the graph it looks more like 5

[celest pewter]

what's the actual median

[twin wolf]

it says 3.5

[celest pewter]

tbh i'm pretty sure the scale is just broken

[twin wolf]

ohh yeahh

thats probably it lol

something looked off

thanks

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[covert wraith]

What's the easiest way to solve 1200/0.63 without a calculator? Is this ability reasonable to expect on an Algebra I level? I asked chatgpt and it fully told me to use a calculator

[hot herald]

wdym by "solve"

what format is your desired result

[covert wraith]

A decimal

[solemn gate]

being able to calculate this isnt useful in algebra at all

like its not what algebra is about

[hot herald]

the concept of long division is actually useful though

[solemn gate]

i mean it is

[hot herald]

due to relevance with euclidean division and polynomial division

[solemn gate]

just dont expect to see things like: calculate 1200/0.63

[hot herald]

you could first multiply numerator and denominator by 100 to rid yourself of that initial decimal
then divide numerator and denominator by 3

[covert wraith]

I've been brushing up on my long division for like a week and it got me to solve the actual problem another way (and it feels so GOOOOOOOOD) but if I should know how to do it this way too I wanna be able to do it

[torn jolt]

how do you say 1200/0.63?
one thousand two hundred point sixty thirdths?

[hot herald]

resulting in 40000/21
which is somewhat less intimidating to divide

[covert wraith]

I'd gotten this far but floundered here

[hot herald]

then approach it like you would any other long division problem

[covert wraith]

Huh

Yeah that worked lol

Idk why I felt like I couldn't do that

Thanks for clarifying my thoughts after I've been so math-forest-addled

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[forest jackal]

Can someone help me understand what three equations can be formed with this information.

[fast peak]

what they wrote down has to be true for n=1, n=2, n=3, n=4, ... and so on

[forest jackal]

and...

[fast peak]

how does the equation look for n=1

[forest jackal]

oh you need context

[fast peak]

no

I am asking you to plug n=1 in

[forest jackal]

alright

ah i see

so justplug in different values

to form three equations

got it

tysm

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[delicate oasis]

I dont know how my teacher got from k(x) to k'(x). I know that the this function has to be treated as two and I understand how to go from 250x to 250 and e^(-o,5x) to -0,5e^(-0,5x), however I am struggling to understand why I can exclude the twenty. Is it not linked to the e part of the function?

[swift talon]

The 20 is just a constant being added

It doesn't form part of the derivative

[delicate oasis]

But isn't it linked to the e and has to stay with it in the first part like : 250e^(-0,5x) +20 because I only derive the 250x?

[swift talon]

It isn't linked to the e no

It's just added at the end of the function

Different terms

[delicate oasis]

oh okay. So I can always leave it out completely like when I do linear derivatives during linked functions?

[swift talon]

Well if it's separated by a plus or minus it's a separate term

So it's not linked to the e here and is its own term

The derivative of a constant is 0

[pearl trout]

If you have a sum of two things, you can differentiate them separately, and a constant always differentiates to 0

So for example the derivative of x+x would be 1+1

The derivative of x+1 would be 1+0

[swift talon]

If it helps you to think about it, 20 is basically just $20x^0$

[glossy valve]

DJJ

[delicate oasis]

I understand that numbers without an x always turn to 0, I was just a little confused because of the links and different rules that were introduced with them. But I get why it is still seen as a single term and can thus be left out in derivitating. Thanks for the help :)

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[pearl trout]

np ^^

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[subtle prairie]

stuck on integral

[subtle prairie]

Tried substituting cos x here but it solves nothing at all

Looking for a way to solve this with high school integration methods

[lofty vine]

no elementary antiderivative

not sure where you saw this, but ive seen this integral with limits of 0 and pi/2

the definite integral is solvable with those limits. but not this one

[subtle prairie]

Huh

[lofty vine]

dont believe me?

[subtle prairie]

💀💀💀

Holy

😹😹😹

[solemn gate]

yeah you not gonna solve that boy

[lofty vine]

yeh where'd u see this integral?

[subtle prairie]

Online lol

But it is possible with boundaries

[lofty vine]

it wasnt a definite integral?

[solemn gate]

can you show with full context?

[subtle prairie]

Wait a sec

[lofty vine]

aha

x -> pi - x then add the resulting integral with the original

[subtle prairie]

Alright thx 💀💀💀

. close

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[torn jolt]

So a bit confused about something

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[torn jolt]

hey

whats up

for proving [
\grad\8{fg} \overset?=f\grad g + g\grad f
]
How am i meant to treat [
\8{\vc x,\pdv{!x} +\vc y,\pdv{!y} + \vc z,\pdv{!z}}(fg)?
]
I dont think this is like multiplication so idk how to progress

[torn jolt]

dude

i am in grade 10

but wait i can try

[kindred grove]

what is not like multiplication here

you mean the operation between x d/dx + y d/dy + z d/dz and fg ?

@torn jolt

[torn jolt]

@kindred grove

dude

[torn jolt]

yeah

like thats just composition rather, right?

which level of education u curently doing ?

[kindred grove]

yeah it's like function application

[torn jolt]

yeah

[kindred grove]

(x d/dx + y d/dy + z d/dz) takes a (scalar) function as input and spits you out another (vector) function as output

[torn jolt]

so like how do i go about proving it if i cant directly modify the expression grad(fg) by writing it in terms of its partials

[kindred grove]

you can distribute that operator tho

(x d/dx + y d/dy + z d/dz) (fg) = x d(fg)/dx + y d(fg)/dy + z d(fg)/dz

[torn jolt]

ok yeah fair enough

[kindred grove]

so you're left with just the three partials

[torn jolt]

and just product rule it

[kindred grove]

yah

[torn jolt]

ok this was uh just like a stepping stone for the actual problem i want to solve

that is

[kindred grove]

phase 2

[torn jolt]

,, \grad\8{\vv A \vd \vv B} = \vv A \cross \8{\curl \vv B} + \vv B \cross (\curl \vv A) + (\vv A \vd \grad)\vv B + (\vv B \vd \grad)\vv A

[torn jolt]

so uh

how do i do this without crying inside

$\vv A \vd \grad$ is the directional derivative btw so nothing fancy over here

[topaz valley]

[torn jolt]

[topaz valley]

are you just verifying this in cartesian

[torn jolt]

i mean

thats a question of if i want to be a normal level masochist or an above average l evel of masochist

[topaz valley]

[torn jolt]

but yeah fuck it

cartesian we ballin

[topaz valley]

i mean

its not actually enlightening at all

you just uh

expand the whole thing out

and verify that the LHS and RHS are equal

[torn jolt]

bruh thats why this fucker is so useless

who the frick cares about the GRADIENT OF THE DIVERGENCE

the divergence of the gradient is where its at

ok anyways ill just algebruh it out no biggie

thanks @kindred grove @topaz valley

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[fallen osprey]

you see on the right

3rd line up

what is that symbol with a circle and line

[minor crater]

$\Phi$

[glossy valve]

artemetra

[minor crater]

this?

[fallen osprey]

yh

what is that

[fallen osprey]

what is that

$\Phi$

[glossy valve]

chriseubjr

[minor crater]

cdf

cumulative distribution function

it's what you use to find P(Z > ...) and stuff

[fallen osprey]

ohh

ok

thanks

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[maiden current]

if f(x) = x^3-3x^2+1 i need to solve the inequality f(3+x^2) > f(2+2x^2) and the previous question was to examine the monotone of the function so i thought i need to take cases on each monotone interval but i may be doing something wrong

[maiden current]

can someone help me

[graceful bane]

did you try to develop the expression ?

[maiden current]

it is going to be a high degree polynomial

i may have found the solution. x^2 + 3 >= 3 and 2+2x^2>= 2 so both terms belong to the interval [2,+inf) where f is increasing so i can solve the inequality from there?

can you check

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[maiden current]

ty

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[acoustic sphinx]

https://www.chegg.com/homework-help/questions-and-answers/abc-company-expects-cash-inflows-investment-proposal-undertaken-time-period-0-tk-2-00-000--q82247215

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[acoustic sphinx]

no

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[silent moss]

Could you please help me work out the general equation for this pleas

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[round rapids]

can sinx be considered a constant?

and move it out front?

probably not right cuz it has an antiderivative...?

or you cant because sinx =/= cosx?

[spark igloo]

Im not completely sure here but try using U substitution for this, and no you can't move sin to the front

ok yeah its actually really simple using u sub, if you aren't looking for a way to solve this and just wanted to ask the sin question, then yeah its not possible to pull it out of the integral.

[round rapids]

yeah i was just wondering the sinx thing

thanks

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[pearl bridge]

Can you help me with this

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[drowsy rain]

Hello, The AP calculus BC exam is coming up and I am struggling with the content. I only got a 2 on my practice exam, and want to know the best resources to study and help me understand. Any websites, books, or particularly helpful youtube videos that people used would be great!

[serene aurora]

I know that if there's any specific things you're struggling with Kahn academy is a really good resource

Which is what I have been using all year

also I know it sounds self explanatory but just doing more practice tests can be super helpful as well

[drowsy rain]

Khan hasn't been working very well for me personally, I think I want to do another practice test, but I want to try to change things first, I don't think I'll get a very different result if I try it now...

[serene aurora]

hmmm okay good to know

are there any specific things you're struggling with

[pearl bridge]

@drowsy rain help me with this

[drowsy rain]

no not really, I know that isn't very helpful, but I took on too much this year so I'm just generally behind.

[drowsy rain]

I'm not really qualified to be a helper but you could try multiplying everything by (x-9)(x-3)(x-4)

[serene aurora]

no that's super fair, how about different parts of the test

was it the mcqs frqs?

[drowsy rain]

we weren't allowed to keep the tests since its collegeboard material, but I think I struggle more on FRQs than mc

[serene aurora]

I think

somewhere I have a test from a couple years back in document form

that you could go through stress free

if that would help

[drowsy rain]

that would be helpful for later, thank you!

[serene aurora]

Let me grab it rq

that should be the whole thing

and I also have a practice FRQ

[drowsy rain]

Aren't all of those released by collegeboard?

[serene aurora]

they might be

I'm not entirely sure this is just what I've been using

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[peak ember]

hey, can someone help me answer like why does (1+a^2/b^2+a^4/b^4...)=1/(1-a^2/b^2)

[peak ember]

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[pearl bridge]

Tysm I got the answer 🫶🏻

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[gentle coyote]

In a triangle of vertices A, B and C, consider the point X on the segment AB which AX = 2XB and point Y on the segment BC which BY = 3YC. Let P be the point of intersection of CX and AY. Write vector AP as a linear combination of vectors AB and AC

[gentle coyote]

A: ||2/9AB + 2/3 AC||

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[gentle coyote]

<@&286206848099549185>

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[gentle coyote]

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[torn jolt]

guys

[torn jolt]

in a mirror

the reflection

of a hooman

is the inverse function?

0:

the mirror is 2D rightt

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[vivid berry]

is this right

[vivid berry]

OMG WTF IS THIS

[thick hedge]

Can you shade the region common to T,R, C?

[vivid berry]

🙅‍♀️

[thick hedge]

ok, what's the region common to T and R

[vivid berry]

oh

r is the rectange

[thick hedge]

Think about why this is the answer

[granite torrent]

do you know what the ∩ symbol means

[vivid berry]

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[granite torrent]

great answer

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[idle shale]

How many roots in X² = 0

[stuck glacier]

Just one root, x=0

[idle shale]

Even in imaginary numbers

I think that's right

[stuck glacier]

but it has multiplicity of 2

[stuck glacier]

yep

[idle shale]

0 × 0 ?

[stuck glacier]

(x - 0)(x - 0) = 0

[idle shale]

Yeah

K

Thanks

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[stuck glacier]

all good :)

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[timber juniper]

hi... How did this turn into this

[undone vector]

a^2-b^2=(a-b)(a+b)

[timber juniper]

oh that rule

thank you

[undone vector]

np

[timber juniper]

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[tropic yarrow]

How do I factorise this

[neon geyser]

you mean factorize 2x^2 + 9x + 9?

(x+3)(2x+3) lol

[mortal kraken]

no

look at the previous line

it's 2x^2+8x+9

[neon geyser]

then u don't factorize

¯_(ツ)_/¯

[mortal kraken]

what is the original function

[random shoal]

I think its 18x^2 not 16x^2

[mortal kraken]

my eyes are blind

[tropic yarrow]

How’d u get that tho

[neon geyser]

it's just the regular factorization gimmick?

,, (2x + ?_1)(x + ?_2) = 2x^2 + (2(?_2) + (?_1))x + (?_1 \cdot ?_2)

[glossy valve]

nyxie9151

[neon geyser]

?_1 * ?_2 has to multiply to 9

and 2(?_2) + (?_1) has to add to 9

[mortal kraken]

the

[tropic yarrow]

this lowkey confusing tf outta me

[mortal kraken]

=2x^2+9x+9
=2x^2+3x+6x+9
=2x(x+3)+3(x+3)
=(2x+3)(x+3)

[tropic yarrow]

ohh oaky

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[limpid monolith]

what is the area of the square? (cm)

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[limpid monolith]

I closed the other one so it should be fine.

[blazing dome]

Why are all the commas there

For example 17,5

[vast fossil]

Decimal points

[blazing dome]

Or is that

Yea

Okay

[vast fossil]

Start by noticing that the smaller triangles are similar, that should allow to solve for their sides

[blazing dome]

This is the are of the biggest possible triangle:

$\frac{1}{2}14(10.5)$

[glossy valve]

the numbers mason50_50…

[blazing dome]

Then

$\frac{14}{2}(10.5)$

[glossy valve]

the numbers mason50_50…

[blazing dome]

Then

$7(10.5)$

[glossy valve]

the numbers mason50_50…

[blazing dome]

Which equates to 73.5

cm

Then subtract the area of the two inner triangles from that area of 73.5

[limpid monolith]

I'm confused by what you did sorry

by the way, the square is NOT in the middle of the hypothenuse line.

[blazing dome]

I see

[limpid monolith]

I got this on a 10th grade maths test.

[random shoal]

Did u get it?

[limpid monolith]

no sorry

[random shoal]

The area of all three equals total area of triangle

The area of square is (root x)(root x) equals x

[limpid monolith]

yea ik

that's why I wrote sqrt x

[random shoal]

And area of bottom triangle is 1/2 x height x base

Height is root x

[limpid monolith]

we don't know any values though.

[random shoal]

Base is 14- root x

[limpid monolith]

OH

[random shoal]

Same for upper triangle

[limpid monolith]

yeah

that makes so much sense now

on the test I literally measured the length haha

I didn't have any other choice

[random shoal]

0.5 x (root x) x (10.5- root x)

[limpid monolith]

answer was 36 cm^2

[random shoal]

What the

That is not possible if the diagram is not to scale

[limpid monolith]

well my professor said that it doesn't matter how I do it, just that I show how I do it.

[random shoal]

Don't tell me you drew the triangle?

[limpid monolith]

I made it to scale on papper.

[limpid monolith]

haha yes

but I believe I got every other question right

[random shoal]

Well, it isn't wrong

[limpid monolith]

so probably still an A.

[random shoal]

Just that it takes time

[random shoal]

That's good

[limpid monolith]

yeah I know

I used laws of cosines when I just had to use the pythagorean theorem 😂

[random shoal]

But for me, they'll reduce marks if I do your way

[limpid monolith]

yeah I know that

[random shoal]

Bruh

[limpid monolith]

but hey at least it's not a 0 score.

[random shoal]

[limpid monolith]

thanks for your help

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[quiet tundra]

im confused are they asking for the concavity here or what

[quiet tundra]

i did concavity and it was wrong it must be asking for something else

[hot herald]

show work

[quiet tundra]

ok wait

@hot herald

[hot herald]

for local min/max, you'd want to determine location of stationary points using the first derivative
and first derivative also tells you when a fucntion is increasing/decreasing

[quiet tundra]

x=0 and x=2

[hot herald]

and then you can plug those values into the second derivative to determine what kind of stationary points they are

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[loud ferry]

I am wondering if someone can help me with some Discrete math questions! Thanks so much

[thick hedge]

ask your questiosn

if we can help, we will

[loud ferry]

Thanks, im unsure how this is wrong

[thick hedge]

I haven't studied this yet, so I can't help, sorry

[ancient folio]

Its just logic?

[ashen crescent]

Assume the statement “No one loves spinach” to be true.
Would that mean “Someone doesn’t love spinach” is false?

[ancient folio]

If at least 1 person loves spinach then it must be false (negation) to conclude "No one loves spinach"

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[rain briar]

can someone, slide me a hint to calculate for y ?

[torn jolt]

hint: collect like terms, separate constants from variables (sqrt(ab) = sqrt(a) * sqrt(b) is good here)

[rain briar]

so i get something like this: 11\sqrt{2}\sqrt{y} = 2(8\sqrt{2}\sqrt{y}) - 5

[torn jolt]

yes

indead

[rain briar]

well nvm i figured it out thanks

[torn jolt]

now continue

oh okay great!

good job

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[rain briar]

So i have a problem in this question. I'm attempting it like this,

1. first squaring both side and got the equation this like this: \frac{100}{9} < h < \frac{575}{25} (since +ve numbers, the equality doesn't changes).
2. then what im doing is to calculate the integers values satifying this problem is: \frac{576}{25} - \frac{100}{9} - 2 ( This will give all the numbers between these 2 fractions). But it seems it wrong. My approch is not the correct way of doing it.

[torn jolt]

for your information, there is an integrated texit bot here and u can surround ur math with dollar signs to produce an output

[hot herald]

whats your reasoning for doing

\frac{576}{25} - \frac{100}{9} - 2

[rain briar]

i thaught when we need to find how many numbers are between 2 numbers like 1 and 10, so there are 8 numbers in between. We subtract 1st and last nnumber and then subtract it by 2 to get all the values in between ??

[hot herald]

subtracting by 2 doesn't even work for integers

[rain briar]

yea i just realised it.

[hot herald]

and also you want for first consider either the floor and/or ceiling of your fractions depending on personal preference

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[queen jetty]

How to deal with my numbers being squared

It's very close to format @-@

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[livid herald]

There's got to be an easier way of solving this, any ideas?

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[rapid siren]

I've been stuck on this question for a decent while.
I tried making a venn diagram but it was useless.

[atomic venture]

can you show the venn diagram you tried?

[rapid siren]

ok one sec

I didn’t know where to go from here

[craggy verge]

The outer three are 0

All did >1 activity

[atomic venture]

you can let some variable be the total amount of people, then solve for k

[rapid siren]

x = total
k% of x = ??

[atomic venture]

uh

:3

[rapid siren]

oh like 0.6x + 0.5x + k%(x) + 10 = x or smt?

[sharp thorn]

yeah

[rapid siren]

then what can i do

<@&286206848099549185>

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[rapid siren]

<@&286206848099549185>

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[rapid siren]

<@&286206848099549185>

[static cipher]

your 50%, 60%, and k% are wrong

[polar valve]

doesnt "50% participated in at least hiking and canoeing" include the 10 students participating in all three activities? so my venn diagramm would look a little bit different.

[static cipher]

yeah, that's what i mean

[polar valve]

as you have "at least" thsi woukdnt be an equality

a+b+c+10=x
a+10>0.5x
and so on.
and you know that a,b,c,x must be natural numbers.

@rapid siren

[rapid siren]

I see

I think I understand now

Thanks

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[long bloom]

My notes are unclear what to do next

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[nocturne garden]

why is tan (360 - theta) = tan theta

why isn't tan theta negative even though tan is negative in QIV

[hot wadi]

Where'd you get that ss from

tan(360-x) is -tanx

cos(360-x) is cosx

Sin is correct

[nocturne garden]

prob a textbook error

okay thx

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[livid herald]

That's the question and I know I need a system of equations, however I can only come up with these but they give me a negative answer which I know it cannot be, any tips?

[lusty socket]

A nice way to find more equations is noting that the sum of each probability*Tokens awarded = 11

and the sum of each of the probabilites = 1

aka a ball will be pullled out

[livid herald]

Hmm, lemme try that and get back to you

Same problem, it's still negative

We know that m has to be positive since that's also the number of red balls so the unknowns are all positive

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@livid herald Has your question been resolved?

[livid herald]

Anyone who can help?

[random shoal]

Hey

The second equation is wrong?

First the probability of white will be $\frac {15}{4m}$

[glossy valve]

Monarch of Eternal Night

[random shoal]

So token will be $\frac{(15)(m)}{4m}$

[glossy valve]

Monarch of Eternal Night

[random shoal]

And probability of yellow toke will be $\frac {(5n)(23)}{8m}$

[glossy valve]

Monarch of Eternal Night

[random shoal]

If you add all these you'll get relation between n and m

@livid herald

[livid herald]

I'm sorry can you explain how you got this? I'm on a serious mental block

I get the next steps you've done

[random shoal]

The probability for red ball

It's $\frac {m}{6+m+n}$

[glossy valve]

Monarch of Eternal Night

[livid herald]

Yea

[random shoal]

Which equals to 5/8

[livid herald]

Yes

[random shoal]

So $6+m+n =\frac{8}{5m}$

[glossy valve]

Monarch of Eternal Night

[random shoal]

And for probability of white is $\frac {6}{6+m+n}$ just substitute 6+m+n

[glossy valve]

Monarch of Eternal Night

[random shoal]

@livid herald

[livid herald]

Ahhhh

Okay got it, thanks a lot

[random shoal]

Sure np

[livid herald]

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[terse dove]

Hey is this correct?

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[craggy tapir]

Let there be $f(x) = ax^4+bx^2+c$ $(a, b, c \in\mathbb{R},a>0)$ such that $f(x)$ has 3 extrema at 3 points A, B and C$\left(1,-\frac35\right)$. Let there be $g(x)$, a quadratic function having the graph passing though A, B and C. The region confined by $f(x)$, $g(x)$, $x=0$ and $x=1$ has an area of $\frac25$. Evaluate $\int_0^1 f(x)\dd x$

[craggy tapir]

rough diagram

[low echo]

Wow

[craggy tapir]

not sure where to go from here

[low echo]

What is the range beside C

[craggy tapir]

what do you mean?

[low echo]

Is the point c goven ?

Given ?

[craggy tapir]

yes, (1, -3/5)

and from this i can deduce that A coordinate is (-1, -3/5)

[low echo]

Yeah

[craggy tapir]

the problem is i dont know B

[low echo]

Hm

[craggy tapir]

(0, c)

[low echo]

4ax³ + 2bx

= 0

Is satisfied by c

And by a also

[craggy tapir]

what do you mean

isnt that the deriv

[low echo]

yeah

[craggy tapir]

oh wait

[low echo]

If there is max or min it should give 0

[craggy tapir]

yes

[granite torrent]

this feels like a problem you posted a long time ago

[low echo]

satisfy it with A and C to get a, b