help-49

[tidal turret]

lowkey onfused

<@&286206848099549185> 😨 😨 😨

[tribal temple]

Well, you want to change the expression
[
(\frac{3n + 2}{3n + 1}) \ln(a_n) = (\frac{3 + \frac2n}{3 + \frac1n}) \ln(a_n)
]
so that inside it, you'll see a $\frac1n \ln(a_n) = \ln(\sqrt[n]{a_n})$
what can you multiply and divide by to make that happen?

[grand pond]

@tribal temple

[tidal turret]

Multiply by n both sides

whats next

[tribal temple]

Well multiply and divide by n, sure-

[tidal turret]

then wat

[tribal temple]

That then gets you
[
n \cdot {\color{green} \qty( \frac{3 + \frac2n}{3 + \frac1n}) } \cdot {\color{red} \frac1n \ln(a_n) }
]

[grand pond]

@tribal temple

[tribal temple]

And you said the green goes to 1, and the red goes to ln(4), right?

[tidal turret]

True

nln(4) is what that thing approaches

so?

[tribal temple]

but n is blowing up to infinity, and ln(4) is strictly greater than 0, soooooo...?

[tidal turret]

so bn diverges to infinity

i guess

[tribal temple]

Yep, left hand side of here diverges to infinity, so b_n must do too

[tidal turret]

how do i check my answer doe, is it possible

[tribal temple]

I mean, other than arguing through logic, without an explicit sequence it's pretty hard to in this case

[tidal turret]

aaaaaaaaaaa

okay

I am closing then, thx!

ty for the help

.close

[midnight plank]

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[brittle spade]

how is the derivative of g(x) 2,4,6,etc when the first difference is 3,5,7,9,11?

<@&286206848099549185>

[surreal moon]

!15m

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[brittle spade]

ok sorry but can you help me pls

[celest harness]

whats the derivative of g(x) ?

[brittle spade]

i know the first difference is 3, then 5, then 7, then 9, then 11

so i just don't understand how then a derivative of this function ends up being 1 less then each of those first differences

like for a linier function, i see how the derivative is the first differenc3e

[celest harness]

calc the derivative it will be clear

[brittle spade]

so is each of those derivatives for the quadratic the tangent line? like the slope of each of those points?

[celest harness]

yes

[brittle spade]

ok tahnk you

i think i get it now

[celest harness]

but g(x) describes a quadratic functions whilst g'(x) describes a linear function

[brittle spade]

right. the linear function for each of those points

?

[celest harness]

so when u calculate the values given one will go up faster than the other

[brittle spade]

ok igu

[celest harness]

,w derive x^2-9

[grand pond]

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Requested by spookyspaghetti_

[celest harness]

see

thats why

[brittle spade]

ok

[midnight plank]

@brittle spade Has your question been resolved?

[midnight plank]

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[austere lodge]

so i think i know the steps

the lesson didn't cover much

[celest harness]

what steps do u think u know ?

[austere lodge]

i substitude y in for f(x)

2. switch the variables (x and y)

3. solve for y

[celest harness]

thats for a normal function yes

[austere lodge]

oh

[celest harness]

here ist just an equation

u want to have F in one part and then all the rest in the other part

give that a try

[austere lodge]

thx

[celest harness]

show what u got when ur done

[austere lodge]

so im isolating f?

[celest harness]

yeah

[austere lodge]

-5/9F=-c-17 and 7/9

[celest harness]

huh

can you write it between $

[austere lodge]

$-5/9f=-c-17 7/9$

[grand pond]

chilly

[austere lodge]

it should be 17 7/9

[celest harness]

$-\frac{5}{9}F = -C-17\cdot\frac{7}{9}$

this ?

[austere lodge]

$17 7/9$

[grand pond]

chilly

[austere lodge]

wait

no

the 7/9 is a fraction of the whole (17)

[celest harness]

wdym ?

[austere lodge]

the 7/9 is with the -17

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[grand pond]

spookyspaghetti

[celest harness]

like this

[celest harness]

@austere lodge

[austere lodge]

sending a pic

[celest harness]

ait

[celest harness]

yeah so multiply

[austere lodge]

oh

[celest harness]

welp doesnt rly matter cus its not completely the right answer

[austere lodge]

how should I do it?

[celest harness]

you start with $C = \frac{5}{9}(F-32)$

[grand pond]

spookyspaghetti

[austere lodge]

yes

[celest harness]

the 5/9 has to go to the other side first

how would you do that

[austere lodge]

the recipricol

9/5 multiplied?

[celest harness]

yeah

on each side

[austere lodge]

oh

[celest harness]

that gives $\frac{9}{5}C = F-32$

[grand pond]

spookyspaghetti

[celest harness]

correct ? @austere lodge

[austere lodge]

correct

[celest harness]

alright then whats the last step to only get F on the right side ?

[austere lodge]

add 32

[celest harness]

add it where ?

[austere lodge]

the left side

[celest harness]

so $\frac{9}{5}C+32 = F-32$ ?

[grand pond]

spookyspaghetti

[austere lodge]

no

the 32 on the right cancels out

[celest harness]

so u add it on both sides then

[austere lodge]

yes

[celest harness]

important to specify that

[austere lodge]

oh

[celest harness]

then ur left with $\frac{9}{5}C+32 = F$

[grand pond]

spookyspaghetti

[austere lodge]

yes?

[celest harness]

now u have a function of $F$ with respect to $C$

[grand pond]

spookyspaghetti

[celest harness]

thats the solution that they are asking for

[austere lodge]

ohhh

thank you

[celest harness]

np

u solved it not me, remember that

[austere lodge]

i will

[celest harness]

.close

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[runic sage]

yo

[runic sage]

what is this rule lol?

is there a another way to solve it or just this way?

[runic sage]

<@&286206848099549185>

[spare forum]

which rule?

[midnight plank]

@runic sage Has your question been resolved?

[runic sage]

the rule where they change it to lim

idk what or how they did that

[midnight plank]

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[obtuse totem]

for a continous random variable, is it strictly defined to be (-inf, inf) or a disjoint closed union? (as an example, [0,10] U [70,90]

[obtuse totem]

so if it was [0, inf) it would be discrete random variable? This is a stats question by the way

wait what

why is that

[queen ermine]

How can it be discrete if it's from a continuous set?

[obtuse totem]

oh so its continous if it consists of all numbers in an interval?

oh so like what we learned in analysis

its the same thing

ooh im stupid

[queen ermine]

Doesn't discrete mean countable

can you count the set of [0, inf)

[tribal temple]

Also e.g. the exponential distribution is continuous and has [0, infty) as where it's defined

[obtuse totem]

Its about time I realize,...

that all branches of maths are connected😮

[queen ermine]

Except for stats

stats is the black sheep

[twilit field]

Stats too

[queen ermine]

we hate stats

no

I refuse to accept stats

as a member of the core family

[tribal temple]

It should be connected to the bin

[queen ermine]

it can take a seat

but I don't grant it rank of master

[obtuse totem]

honestly, Im sort of agreeing

for now

its not very concrete..?

stats is so.... random😂 thats the way I'd describe it rn

[tribal temple]

Random variables

[obtuse totem]

https://tenor.com/view/reboot-cgi-binome-badum-tss-drum-gif-24916102

[tiny drum]

Stats roolz

[midnight plank]

@obtuse totem Has your question been resolved?

[obtuse totem]

ive got an interesting question

a,b

So at first, I thought 0 could be a possible value for X right? Because X denotes the number of moves BEFORE returning to 0. So if Claudius never makes any move he's never returning to 0

but 0 is not in the set of X in the solution

and one other value I thought of was 1, because the first move must always be B1,B2,B3,orB4. and then the 2nd move theres a chance of returning to 0

so my set for X is {0,1,2,..} but thats wrong?

[tiny drum]

Well it says that he makes an initial move

So 0 isn't an option

[obtuse totem]

ohh i see

X is saying nunber of moves, and after moving to either B1,B2,B3,B4 Claudius has to make another move after to see whether he goes to 0 or not

so that would be 2 moves

not counting 0, the set would be {2,4,6,8..}

[tiny drum]

I think so

[obtuse totem]

ye the answer says {2,4,6,..}

I think I have the right interpretation of why that is

[midnight plank]

@obtuse totem Has your question been resolved?

[obtuse totem]

.close

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[elder prawn]

How does one go about evaluating the following limit to find the derivative of sin(sqrt(x)) using first principles?
I have tried using the sum-to-product formulae but didn't know what I could do to simplify the expression further.
I was wondering about using small-angle approximation but that doesn't make much sense.... x may not be small and I don't think it is safe to make the assumption.
(I explicitly don't want to use L'Hopital's rule here, ruins the fun for me if I use these "shortcuts" 😂)

[midnight plank]

@elder prawn Has your question been resolved?

[robust flint]

$sin(A) - sin(B) = 2sin((A-B)/2)cos((A+B)/2)$

[grand pond]

Diborane

[elder prawn]

I have tried using the sum-to-product formulae

[robust flint]

rationalise it so now it becomes

$2sin(h/2(sqrt(x+h) - sqrt(x)))cos(sqrt(x+h) + sqrt(x))$

[grand pond]

Diborane

[robust flint]

then evaluate

[robust flint]

sorry is sqrt(x+h) + sqrt(x)

[elder prawn]

$2\cos{\frac{\sqrt{x + h} + \sqrt{x}}{2}}\sin{\frac{\sqrt{x + h} - \sqrt{x}}{2}}$

[grand pond]

HTGAzureX1212.

[robust flint]

yeah now rationalise the argument of sin

then take limit

[elder prawn]

now I have this

[robust flint]

you missed multiplying the entire thing by 2

but yes its correct otherwise

[elder prawn]

ah yeah

[robust flint]

now multiply and divide 2(sqrt(x+h) + sqrt(x)) with the h in the denominator

then the limit would be of the form lim x -> 0 of sinx/x =1

[elder prawn]

oh sin x / x

[robust flint]

yeah

[elder prawn]

thanks

It makes sense now

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[last slate]

It says as the angle gets smaller the adjacent size gets larger

How is this possible if just the hypotenuse is moving

Oh no I understand

My bad

.close

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[median ridge]

i'm marking my physics practice test

[median ridge]

and the question was

"a small aluminium ball is statically charged with - x C. The aluminium ball is suspended in an external uniform vertical electric field."

there was mass and charge but its not important

"Draw the external electric frield, indicated at least 4 field lines"

i have no idea what this looks like, and the answers only say "draws electric field direction and shape" as a marking guide, it doesnt show me what the field looks like

can someone please help

[sharp coral]

well we know the electric field is uniform

in order to keep the ball suspended in the air, it should be in a particular direction to counteract the force of gravity

[midnight plank]

@median ridge Has your question been resolved?

[median ridge]

yes i know

but given that the ball is in the middle and its statically charged negatively, does the electric field interact at all with this?

[sharp coral]

well the force from an electric field is $\va F_E = q\va E$ (since the field is uniform, the same force applies to the entire ball)

[grand pond]

cloud

[median ridge]

so is it just like electric field lines through the ball? i get mixed up with magnetic field lines because they curve due to the ball

[sharp coral]

it just says "draw the external electric field", not the field of the ball itself

[median ridge]

so its a bit of a trick question? its just a normal electric field? not impacted by the charge of the ball at all?

just like that?

[sharp coral]

well if the ball was positive vs negative that would change which direction the field would need to be

[median ridge]

because its suspended, the positive plate would be at the top yes? because the negative ball is attracted to it and repelled from negative

[sharp coral]

you can think of it that way

[median ridge]

ok cool thank you

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[light cliff]

hey]

[light cliff]

got a terms question

state the four first terms

Tn = 4n - 3

how am I meant to know the term when the question doesnt state t1 nor n

[summer remnant]

Just substitute n=1,2,3,4

[light cliff]

ok thx

.close

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[vagrant storm]

Proof by contradiction question, how would i start this off?

[vagrant storm]

i meaan i know the assumption but how would i contradict the assumption?

[odd solar]

What would be your assumption

[midnight plank]

@vagrant storm Has your question been resolved?

[hollow oyster]

assume that the function has stationary points

then differentiate the function

[odd solar]

Okay, now break down the assumption. How are stationary points defined?

[hollow oyster]

no u assume that the function has stationary points

then differntiate

[odd solar]

Right, and what are you planning on showing?

After differentiating

[hollow oyster]

what do u get?

a stationary point means that after differentiating its value will be 0

[midnight plank]

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[light cliff]

hello

[hollow oyster]

hello

[light cliff]

the sequence is tn = 2700 (n-1)d

d being -150

[hollow oyster]

k

[light cliff]

So when im finding a rule for the nth term

[light cliff]

how do i go beyond this point

[hollow oyster]

wdym?

[light cliff]

after ive found out tn=2700+(n-1)x150

what do I do

because I thought you just leave it at that

[hollow oyster]

is tn the sum of an ap?

[light cliff]

yes

[hollow oyster]

do you want to find the nth term of ap?

[light cliff]

no im just trynna figure out why question c

does =2700-150n+150

are you always meant to simplify an explicit rule

[hollow oyster]

not alwys depend on the question

[light cliff]

i just dont get the simplifying in this case

[hollow oyster]

well its more simple in terms of n than n-1

[cerulean glade]

They're both different ways of writing the same expression

we just simplify because it looks nicer and simpler

[light cliff]

yes i know but how does the simplyfing work in this equation

[hollow oyster]

its just more elegant to look at

[cerulean glade]

the distributive law?

[light cliff]

bruh

[cerulean glade]

I really don't get your question

[light cliff]

okay so how does it go from tn= 2700 + (n-1)x-150 to 2700 - 150n + 150

[hollow oyster]

distributive law

[cerulean glade]

well its by the distributive law, you distribute -150 over the bracket

ie, you multiply each term in the brackets by -150

so it becomes n*(-150)+(-1)(-150)

-1 * -150 is just 150

so it becomes -150n+150

[light cliff]

oh wait

i thought you could only do that if the number was infront of the bracket

e.g. -150(n-1)

[cerulean glade]

multiplication is commutative, it doesnt matter whether the number is before or after

you can rearrange the bracket and the number in any order you'd like

[light cliff]

I had no clue

[cerulean glade]

$ab = ba$

[grand pond]

proofAd

[light cliff]

okay that makes more sense lol

cheers

[cerulean glade]

alright, dont forget to do .close

[light cliff]

.close

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[cobalt swan]

do u have idea on limit of summation?

do not ping until no one responds atleast for 15min

[flat spire]

<@&268886789983436800>

[hollow oyster]

well lim as n approach infinit(summation (k/n0)^4*1/n is = integral 0 to 1 x^4

[cobalt swan]

lol bro got hacked ig

[olive matrix]

bit odd that

.close

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[severe glade]

im following a problem on quadratics and in the first step he factors the problem without really explaining how he did it. can someone break it down how he did it?

[still nacelle]

firstly, observe that (or test a couple numbers until you spot that) x = -2 is a root. So (x+2) is a factor of x^3+8.

Then use algebraic/long division

[severe glade]

i can understand how he got the x + 2, because he did square root of 3 on both sides i think

but how does he get (x^2 - 2x + 4) from the (x+4)

[still nacelle]

cube root*

[severe glade]

like can i get a step by step, im strugglign in math here

of how he got the 2nd part

[still nacelle]

the method is algebraic/long division

if you don't know what that is, I advide looking on yt

[severe glade]

what would i look up specifically to figure out how he got that?

[still nacelle]

it's hard to explain how that works without writing on a piece of paper and going through it step by step

[worthy wing]

Is the sum of cubes

[still nacelle]

oh yh that ^ also

[worthy wing]

8 = 2^3

[still nacelle]

either look up sum of cubes, or just google "algebraic/long division" and learn the general method, then apply it to the question. No shortcut

[severe glade]

ohh ok

i found the exact problem

he just used that formula

A^2 - AB + B^2

for the 2nd part

[still nacelle]

yh

[severe glade]

ty

.close

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[severe glade]

.reopen

[midnight plank]

✅

[severe glade]

Hi i had a quick question while doing my homework

When i did the problem: x^3 - 64 = 0

When i factored it out

I got ( x - 4 ) ( x^2 - 4x + 16 )

but in google it says
( x - 4 ) ( x^2 + 4x + 16 )

so the difference is that google is 4x while i have -4x

but why would it be positive?

[torn compass]

$$a^3 - b^3 = (a-b)(a^2 + ab + b^2)$$

[grand pond]

JustToPro

[burnt flame]

well do the foiling/multiplication out, or use the formula :3

[rain wasp]

:3

[severe glade]

based on @torn compass 's formula,

a = x
b = -4

so (a - b) would be (x +4) then?

but the issue is, in the video i watched it said the formula is (a + b)

[torn compass]

a = x and b = 4

both formulas are same

in ur video , plug in a = x and b = -4

[burnt flame]

you can use that formula too, with b=-4

[torn compass]

u will get same answer

[severe glade]

I noticed that in his problem, the A^3 + B^3 are added together

but in my problem it's subtracted

does that make a difference?

can i sitll use the same formula he used

[torn compass]

u can

but u need to use a = x and b = -4

$$x^3 - 4^3 = x^3 + (-4^3)$$

[grand pond]

JustToPro

[torn compass]

a = x
b = -4

[severe glade]

OHH i see

so i just want to confirm

i can use the formula in the video i sent regardless of if the problem is:

x^3 - 64 = 0 / x^3 + 64 = 0?

[torn compass]

yes

[severe glade]

so the addition or subtraction doesnt matter since if its subtracting b is negative instead

[torn compass]

yes

ALL the b's will be negative btw , make sure u plug in correctly

[severe glade]

i see

thank you for the help

🙂

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[slender cobalt]

isn't this outright wrong. like if epsilon_n = 1 for all n then its just a geometric series that evaluates to 1.5?

[hard umbra]

,w sum n=0 to infinity of 1/3^n

[grand pond]

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[hard umbra]

index should start at 1

[midnight plank]

@slender cobalt Has your question been resolved?

[slender cobalt]

ah so its a book error, thought so thnx for the help

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[last slate]

So for this I end up getting [
\6\FF\omega = \412\6\delta\omega - \sum_{\substack{n=2k+1\ \ k \in \Z \ n\ne 0}} \4j{n\pi}\6\delta{\omega - n}
]

[last slate]

I am not sure if thats correct though. i decomposed the sequence into even and odd parts and got that

[hard umbra]

,w fourier transform of HeavisideTheta[x] - HeavisideTheta[x-pi]

[grand pond]

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[hard umbra]

that doesnt look very delta to me

[last slate]

well like

what i did is like

,, c_n = \41T\int_0^T\6fte^{-jn\omega_0t}\dd t =\4j{2\pi n}\8{e^{-jn\pi}-1}

[last slate]

so like

,, [2\textwidth] c_n = \41T\int_0^T\6fte^{-jn\omega_0t}\dd t =\4j{2\pi n}\8{e^{-jn\pi}-1} =\4j{2\pi n}\8{(-1)^n-1} = \e{dcases*}{0, &if $n = 2k\q k \in \Z$ \ \5{-j}{n\pi}, &if $n=2k+1\q k \in \Z, n \ne 0$}

[hard umbra]

what is this integral for

[hard umbra]

this looks like fourier series

[last slate]

oh yea woops

first part of the question

[hard umbra]

ew

[last slate]

i dont get why the sigmas are so fat

why did they do this

[hard umbra]

well this is quite disgusting

[last slate]

Quite.

[hard umbra]

god what even are the scaling factors

well

its probably somewhere in the vicinity of correct

but im too lazy to work out the fourier series

actually i literally have the formula i was doing it yesterday

,, a_k = \f 1 {2\pi} \int_0^\pi e^{-ik\theta} \dd\theta = \f 1 {2\pi} \f 1 {-ik} (e^{-ik\pi} - 1)

[hard umbra]

,, a_k = -\f 1 {2\pi ik} ((-1)^k -1)

[hard umbra]

,, a_{2k + 1} = \f 1 {2\pi i(2k + 1)}

[hard umbra]

,, a_0 = \f 1 {2\pi} \int_0^\pi \dd\theta = \f12

[hard umbra]

,, f(t) = \f12 + \f 1 {\pi i} \sum_{k \in \Z} \f {e^{i(2k + 1)t}} {(2k + 1)}

[hard umbra]

ok

@last slate i think its fine

[last slate]

oki ty

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[pulsar sun]

why can we define the set of complex numbers equals to the set of 2x2 matrix such as $M =
\begin{array}{cc}
a & -b \
b & a\
\end{array}$ with $(a,b) \in R²$ and how to prove that ?

[grand pond]

phoestaclies

[midnight plank]

@pulsar sun Has your question been resolved?

[quiet hinge]

Because they possess the same algebraic structure under addition and multiplication

If you want to be morqe formal,
The ring of complex numbers is isomorphic to the ring of those matrices

[midnight plank]

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[misty marlin]

Hi, is this overall correct? The question: if 3sinx+4cosx=5 find sinx, cosx

[last slate]

u forgot to square the 3 in hte denominator

[misty marlin]

Oh yeah

I just see it

[last slate]

but yea otherwise i think its ok

[misty marlin]

Okay i will just send how i did it just in case

[last slate]

aight

[last slate]

what happened here

[misty marlin]

Ou

...well that's one stupid mistake

9/15=sinx

That's it

Lol

Thank you anyway

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[ivory saddle]

could i get some help please

[keen seal]

send the questions you need help with

[ivory saddle]

.reopen

[keen seal]

its already open lol

[ivory saddle]

how do i reopen it

oh ok

i dont actually know if it is possible

@keen seal do you know this

[keen seal]

you have no angles no measurements nothing?

[ivory saddle]

no i dont think so

[keen seal]

are angle ABC the same?

[ivory saddle]

we are given that they are integers

[ivory saddle]

they cant be

but i guess it could be possible that 2 of them are

<@&286206848099549185>

OH WAIT @keen seal i messed up and found the question

[naive dove]

m=3,n=4,p=12...idk i think this is it

[ivory saddle]

however that isn't P at its largest

[naive dove]

soo is it the minimum?i found it idk

[ivory saddle]

it could be the minimum

[naive dove]

is it 1/2=1/m+1/n+1/p

[ivory saddle]

where did u get that from

[naive dove]

and i give values fpr m,n,p but there has to be a better way

[naive dove]

soo u get the interior angles in a k-gon

==>180-360/k=thetha K
soo thetaK+thetaM+thethaM=360
(180-360/m)+(180-360/n)+(180-360/p)=360
540-(360/m+360/n+360/p)=360
subtract 360
180=360/m+360/n+360/p
divide by 360

1/2=1/m+1/n+1/p

[ivory saddle]

and then how do you get the maximum

[midnight plank]

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[last slate]

Can someone please explain the 2nd step. I didn't understand how they got that value in the numerator. I understood the denominator part.

[last slate]

,rotate

[fresh sparrow]

how did they get the denominator part?

[last slate]

According to me

!!!

[plain kestrel]

you cant do that

[last slate]

Can someone please explain how to solve this?

[plain kestrel]

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

[fickle oriole]

looks like they mutiplied the top and bottom by the numerator

[plain kestrel]

try to see pattern and set x, y for something

and remind yourself that ( sqrt(a+2b) )^2 = a + 2b

[last slate]

Oh so rationalizing?

[fickle oriole]

ye

[last slate]

Alright

Thanks you all

[fickle oriole]

i would honestly

try to set

u=a+2b

and other constant

like z=a-2b

to help you

rationalize

@last slate

[last slate]

Yeah else it is very confusing

Im solving it one sec

[fickle oriole]

ok i can show u my work if u want

when ur done

or i can check urs

[last slate]

Alright ill solve once and ill show

Thank you so much everyone.

@fickle oriole thanks

[fickle oriole]

ye thats what i did

np

[last slate]

Can i dm you if i have doubts? Im a lil self conscious to post my doubts here. I understand if im not allowed to do that.

[fickle oriole]

ok np

idrc

no need to be self conscious tho

its a help server

[last slate]

Thanks a lot.

[midnight plank]

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[twin ridge]

Is anyone able to proofread this for me rq? I haven't done a set proof in about a year, so I just wanted to make sure it sounded ok.

[twin ridge]

I didn't grab the entire thing.

[plain kestrel]

you need to chope it up to pieces.

[grim vector]

where black orange and red are the same or B is greater than A

hoping no error made

[plain kestrel]

A interset B = A then A is subset of B.
Proof,
Assume A intercept B = A.
Let x be some element in A.
Then if we can prove that x is also in B then we will have proven that A is subset of B. because it can only happens if A is subset of B
Since x is in A then it means that x is in A intercept B. which means x is in both A and B.
So x is also in B.
||We proved it.||

[twin ridge]

Is that not what I put, or are you saying I need to write it differently?

Just trying to understand the thought process because in my mind, we wrote the same thing.

Am I able to say that If $x\in A, x\in A\cap B$, by the def of an intersection?

[grand pond]

Narutoes

[twin ridge]

Or do I have to prove more than that

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[keen seal]

using either ratio or root test

im left with (x-2)/2

and theres no n so is this still fine?

[sharp coral]

well hopefully you are able to find the limit of a constant

[keen seal]

ah okay so if i continue then i get x< 0

is it the possibility 1? that i have here

[sharp coral]

using the root or ratio test you should be left with $\abs{\frac{x-2}{2}} < 1$ (since that is the limit you calculated)

[grand pond]

cloud

[tribal temple]

(in particular, where are your absolute values )

[keen seal]

i absolutely forgot them

okay so -1 < x < 3

right? O_O

[worthy wing]

[keen seal]

me frl

i meant 0 to 4 sorry

tysm!

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[tribal temple]

.(also this one does happen to come under the geometric case [which is why you can automatically exclude the endpoints!])

[keen seal]

ah so whenever geometric you exclude endpoints?

wait oh yeah i can see what you mean

[tribal temple]

.Yep, though not exclusive to geometric ofc

[keen seal]

wait just to make sure

lets say x = 0

[tribal temple]

.reopen

[midnight plank]

✅

[keen seal]

(-2/2)^n

geometric

soooo a = 1

r is -2/2

[tribal temple]

Yep, -1, and of course that series doesn't converge

[keen seal]

ah okay tysm!

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[limber shell]

How can I integrate this?

[plain kestrel]

from inside out.

integral of (your func) from 0 to 2 with dn first.
cos( theta) is just a constant then.
and then integral of (your func) from 0 to pi/2 with d theta.

[limber shell]

My bad, it's dr, but how can I integrate r(sqrt(4-r^2cos^2(theta))

Variable change or something else?

[plain kestrel]

its r * sqrt(4-r^2) * sqrt(cos^2(theta))= is constant.
try u v = u dv - v du.

[midnight plank]

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[last slate]

Greetings

[last slate]

Is
$\
\sqrt{\frac{x}{y}}=\frac{\sqrt{x}}{\sqrt{y}}$

[grand pond]

GinNBread

[last slate]

True for complex numbers

[worthy wing]

What is i/i?

[last slate]

1

Is it not?

[worthy wing]

Yes it is

[last slate]

I don't know. Someone told me some rules of algebra that we apply to real numbers are not applicable to complex numbers

[summer remnant]

You can take two numbers
x= a+ib and
y= a'+ib' to verify

[worthy wing]

Some does not mean all

[last slate]

That's why I asked

[summer remnant]

Actually nvm
The polar form will be an easier approach

[dreamy lichen]

hmm

what about sqrt(-i/i)

that doesnt equal sqrt(-i)/sqrt(i), does it?

[last slate]

wait. √(e^(ix)) is e^(ix/2) right?

[summer remnant]

Yes

[worthy wing]

It doesnt

[last slate]

Or is that operation even valid 😭

[worthy wing]

And it does

[last slate]

It does?

[dreamy lichen]

sqrt(e^(2pi*i)) = 1, e^(pi*i) = -1

[summer remnant]

sqrt(-i)= 1/sqrt(2) - i/sqrt(2)
sqrt(i)= 1/sqrt(2) + i/sqrt(2)

Divide em both, I think you should get the same on both sides

[dreamy lichen]

You get -i here

[worthy wing]

U get i and -i, dont u?

[dreamy lichen]

and i with sqrt(-1)

[dreamy lichen]

yes, you do

square roots are pain

[summer remnant]

Hmm

[worthy wing]

The original question is true for complex set but real numbers

[dreamy lichen]

?

[worthy wing]

In this case x, y must be reals

[summer remnant]

Nvm then

[dreamy lichen]

What?

[dreamy lichen]

What does this mean?

[worthy wing]

I mean if the numbers inside the root are reals

Then the equality is true

[dreamy lichen]

sqrt(-1/-1) = sqrt(-1) / sqrt(-1)?

[worthy wing]

Yes

[dreamy lichen]

oh, if we are working with the real numbers but with complex operation, then yes

[worthy wing]

Exactly

[dreamy lichen]

I'll expand on the answer a bit

it somewhat works in complex numbers too

[last slate]

Bro 😭

[dreamy lichen]

the problem is that the principal square root only takes the solution with smallest argument. So e.g x^2 = -1 has solutions i and -i, but sqrt(-1) = i, because it has smaller argument

The reason why sqrt(-i / i) = sqrt(-i) / sqrt(i) didnt work is that the first one took the solution with smaller argument, while the second one took the solution with larger argument

[last slate]

What do we mean by larger argument?

[dreamy lichen]

but essentially, they are both solutions to one quadratic equation

[dreamy lichen]

do you know polar coordinates?

[last slate]

Yes

[dreamy lichen]

argument is the angle basically

sqrt(z) returns a solution of z^2 = x, but there are always 2 solutions (except z = 0), so it returns the one solution that has smaller argument

or equivalently stated, that has positive coefficient of i

[last slate]

But larger as in?

[dreamy lichen]

Sorry i meant smaller

it means that the angle in polar coordinates is smaller

[last slate]

Smaller "argument"?

[dreamy lichen]

out of the 2 solutions, one has smaller angle in polar coordinates

[last slate]

Ok, that makes sense

[dreamy lichen]

and sqrt(z) returns that one

[summer remnant]

Hang on so while finding sqrt(-i), if we take the arg as 3pi/2 instead of -pi/2, it comes out to be a diff value

[dreamy lichen]

Indeed it does

[summer remnant]

hmm

[dreamy lichen]

As I said, there are always 2 solutions to z^2 = x, except for z = 0

this is an example of 2 solutions of z^2 = x

note that they are always symmetric about the origin

[last slate]

I never knew square root returned the one with lesser argument, that clears so much up.

[dreamy lichen]

and what square root does is it only returns the one in upper half of the plane, that is the one with smaller argument

[dreamy lichen]

If we could just choose any argument, then sqrt wouldnt be a function anymore

[summer remnant]

Right

[dreamy lichen]

because it would return 2 values for 1 z

[dreamy lichen]

And this is exactly the reason why some rules dont work in complex numbers

[summer remnant]

So we neglect the one with the larger arg?
Like sqrt(+4)= 2 and not -2?

[dreamy lichen]

because we can sometimes manipulate it, such that one returns the 1st solution and the other the 2nd solution

[dreamy lichen]

yep, this was also done with reals

sqrt(4) = 2

not -2

[summer remnant]

Makes sense now

[dreamy lichen]

To generalize, we can say that either
sqrt(a/b) = sqrt(a)/sqrt(b)
Or
sqrt(a/b) = -sqrt(a)/sqrt(b)

so they are either equal, or one is equal to the negative of the other

[summer remnant]

Idts not necessarily upper half

Coz for example sqrt(-i), both solutions are in 3rd and 4th quadrants

Oh wait nvm

I miscalculated

[dreamy lichen]

there are always 2 solutions of form: z, -z. So they always lie in opposite quadrants. It makes sense because z^2 = (-z)^2

[summer remnant]

Yeah mb

[dreamy lichen]

and there cant be more than 2 solutions for quadratic equation

[midnight plank]

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[midnight plank]

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[last slate]

Show that if :

[last slate]

i need help with my words

How do i end this

[midnight plank]

@last slate Has your question been resolved?

[last slate]

<@&286206848099549185>

[tiny drum]

You need to justify going from line 2 to 3

[last slate]

No

I need help after the last step

[tiny drum]

Yes, you do have to justify it

[last slate]

I'm sorry?

[tiny drum]

You need to justify your steps if they are not based purely on definitions

[last slate]

I proved x belongs to C-A. I need to show that it is the proper superset of C-B

[last slate]

You're telling me I missed something?

[tiny drum]

Yes. You're missing the justification for going from line 2 to line 3

[last slate]

I have, no? On the right side I wrote since a is the proper subset of b

[tiny drum]

Have you proven that?

[last slate]

I'm asking this because other friends of mine simply concluded with

C-B proper subset of C-A

[last slate]

It is given A is the proper subset of B, I'm confused.

Could you give a proof for this question, please?

[tiny drum]

Yes, but have you proven (or has your book or teacher) that "x not in B" implies "x not in A"?

[last slate]

Yes

And do we have to prove that too?

[tiny drum]

Not if it has been proven for you

[last slate]

Ok

How do I conclude tho

[tiny drum]

It's pretty much done IMO

You could let x be an element of B not in A. Then it's in C - A, but not C - B. So the sets can't be equal

[last slate]

I feel it's incomplete

[tiny drum]

If you wanted

[last slate]

No I mean I could yap and finish

But isn't there a simpler way

Like the other steps

[tiny drum]

Simpler than what

[last slate]

You know complete it in a single line

[tiny drum]

That should not be the goal and is not going to be possible in general

[last slate]

Ok, thank you for you patience

.close

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[sullen aspen]

second one has no solutions correct

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[sullen aspen]

no exact solution

[sullen aspen]

For the given problem

Since in essence the transversal lines in the problem do absolutely nothing

as the only given information is in the interior triangle and the information we want to find is in the interior triangle

with 1 angle it should be impossible

[surreal moon]

Which second one? Where am I looking?

[sullen aspen]

The second triangle

the second question

in the picture

Ignore all extra info

Pretend that just x and 46 are in the diagram

[midnight plank]

@sullen aspen Has your question been resolved?

[sullen aspen]

<@&286206848099549185>

[wary cypress]

whats the question?

[sullen aspen]

Second question

Solve for x

Only given is 46

Isn’t this question absurd?

impossible

[wary cypress]

is 46 really the only given? no parallels or anything like that?

well, x+46=b, that might be helpful

[midnight plank]

@sullen aspen Has your question been resolved?

[sullen aspen]

Well the parallel line is outside the triangle with all the info

it’s like “noise”

provides nothing

to the problem

I can imagine u can just isolate the triangle itself and that’d be the same exact problem as what’s going on here

[wary cypress]

Well, the task isn't to find x, but find the calculations needed to get to x

[midnight plank]

@sullen aspen Has your question been resolved?

[midnight plank]

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[sturdy tendon]

I need help with this kinematics as I tried my best wit this

<@&286206848099549185>

[last slate]

!15mins

[midnight plank]

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.

[sturdy tendon]

Sorry

[sullen aspen]

Since the initial vertical velocity upwards will be the same (but with opposite sign) as the vertical velocity downwards when the diver is at a height of 2.00 m, u can 1) find the velocity at the halfway distance in question in the problem using the time given, a distance of 1, and -g (this is basically like finding v0); and 2) find the downwards vertical velocity at a height of 2.0 m using a distance of 1, an acceleration of -g, and a final velocity of the velocity u just calculated

the answer you obtain for this downwards velocity will be the same in magnitude as the initial velocity upwards

giving u the answer to the question

[sturdy tendon]

Need help with the equations

[sullen aspen]

For 1), use -1 = v_halfway (0.12) - 0.5 ( 10) (0.12)^2

For 2), use v^2 = (v_halfway)^2 - 2(10)(1)

v_halfway will be calculated from 1

1)*

[sullen aspen]

Yep your initial equation looks good

I’ll assume the arithmetic is right past that

Continue

that is your halfway velocity

[sullen aspen]

Yes but you should probably be consistent with your value of g

I used 10

you used 9.8 and 10

[sturdy tendon]

Agreed

I switch it back to 0

[sullen aspen]

Alright that should be the answer

[sturdy tendon]

Us that good

[sullen aspen]

Yep

[sturdy tendon]

Thank you

[sullen aspen]

No problem

[sturdy tendon]

Have a good one and thank you as I will keep asking for help her

Sit I got another one

[sullen aspen]

You too. Also kinematics?

[sturdy tendon]

It is about circuits as no idea if it can be solved here

[sullen aspen]

Ah I haven’t covered this yet I’ve only taken mechanics

But currently taking electromagnetism so this should come up soon

[sturdy tendon]

That is okay as thank you for your help

[sullen aspen]

@ the helpers

[sturdy tendon]

Hope to get help like this <@&286206848099549185>

[sullen aspen]

👍🏻

[sturdy tendon]

Are you a physics major

[red mesa]

bruh

2 times

sus

[sullen aspen]

No I’m a highschool student

Junior

[sturdy tendon]

Been told to do this

I need to show significant figures as is this good then

[sullen aspen]

I think you should write it as 6.3 for sig figs and scientific notation

[sullen aspen]

Except it should be 6.3 not 6.30 (least number of sig figs in question is 2)

[sullen aspen]

Also for your 7.73, you might want to underline the second 7 to show that quantity is accurate to two significant figures

And -20 should be -20.0 (because of the 1.00 as a distance)

[sturdy tendon]

Is that good

[sullen aspen]

Yeah pretty much (but u could still underline the 7 in the next line as well and change 20 to 20.0)

[sturdy tendon]

I’d di underline the 7

[sullen aspen]

yes on one line

But on the next line too

[sturdy tendon]

Okay

[sullen aspen]

👍🏻

[sturdy tendon]

Thank you so much,icy for your help once more

I will close this channel now

.close

[midnight plank]

Channel closed

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[sturdy tendon]

.reopen

[midnight plank]

✅

[sturdy tendon]

.close

[midnight plank]

Channel closed

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[sturdy tendon]

Sorry

[midnight plank]

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[obtuse totem]

would it also be right if I did (8C3)(3!)

choose 3 bottles from 8 and they can all be ordered 3! ways

[drifting cloak]

oh god combinatronics

😭

[fathom knoll]

yes that works too

and gives the same answer

[obtuse totem]

ok

thank you

[drifting cloak]

(8C3)(3!) = 8P3

which is 8 x 7 x 6

P is permutation btw

[obtuse totem]

ok

[drifting cloak]

but yea u got it

[fathom knoll]

mhmm

[obtuse totem]

d is so trippy

cause if you choose Sony from Reciever, then its 4x4x3

but if you choose sony from disc player, its 5x3x3

oh noo

what am i doing

nononononon

[fathom knoll]

3 slots one for receiver, one for compact, one for speaker

[obtuse totem]

please ignore me sometimes💀

lol yea

[fathom knoll]

now for speaker neither is sony

so 3 possibilities

now either say receiver is sony

so we have 4 options for compact

or compact iis sony and 5 options for receiver

so 3x(4+5)=27

[obtuse totem]

24 ways, for 1 sony we either have it for receiver or compact, thats 9+12 ways
then for 2 sony we have 3 ways

[fathom knoll]

sure that works too

[obtuse totem]

holyyy im actually getting it?

like actually understanding stuff now

im reviewing the counting and comb and permutations stuff rn

whatttt?

nahhhh

i swear ii was so confused a few days ago

oop

nvm mind

i just messed up on a question😭

[midnight plank]

@obtuse totem Has your question been resolved?

[midnight plank]

@obtuse totem Has your question been resolved?

[obtuse totem]

i dont know why they split it into two def

they could've just said A and B are independent if ".." or "...."

[obtuse totem]

alright

i think its sleep time

Ive reviewed everything, just wish I had some time to do and redo some questions

if I can do that I'd be more confident

oh welll

thank you all, byeeee

.close

[midnight plank]

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[midnight plank]

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[midnight plank]

Please don't occupy multiple help channels.

[eager anchor]

what

[hushed herald]

hey u already opened a channel right

[eager anchor]

idk i did but its gone

[hushed herald]

its still there

[eager anchor]

...

i pressed the collapse button...

someone help me close this how do i close this

.close

[midnight plank]

Channel closed

Closed by @eager anchor

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[eager anchor]

ok good

[midnight plank]

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[sharp coral]

what have you tried so far?

[minor obsidian]

im not sure what the question is asking me to do haha

is this what the question is asking me to do?
or are there proper way of answering this question?

i dont have the answer for this question unfortunately

[midnight plank]

@minor obsidian Has your question been resolved?

[minor obsidian]

.close

[midnight plank]

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[minor obsidian]

.reopen

[midnight plank]

✅

[minor obsidian]

.close

[midnight plank]

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[midnight plank]

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[sage helm]

I can't seem to get a picture of this in my head, surely it can't mean for all x in those sets right

[runic hamlet]

why not

intuitively the condition means that A is on the left and B is on the right of the real number line. so c is exactly the point where they "meet"

[sage helm]

Okay I made this diagram from reference

[sage helm]

Oh

I did consider that

[sage helm]

Yeah