help-23

[devout shale]

@sacred vale actually let me stop you completely

do not keep trying

we are not allowed to do partial fractions in this way

it will not result in a proper decomposition

e^x is not a linear factor

even if it composes a polynomial

[sacred vale]

ok

[devout shale]

wait

[sacred vale]

this is how i was taught

?

[devout shale]

I made an error in my math, one second

forget I said that last part for now lol

Okay yes, I completely take that back

you can decompose this way, it will work

however your A and B are still incorrect

so just retry finding those

@sacred vale is decomposing it taking you some time to do all the algebra?

[sacred vale]

b=5

no i wrote plus sign by accident

[devout shale]

b=-5, A=-6 is the correct

[sacred vale]

yea

i got that

[devout shale]

ok good

I am getting off of discord now, so here are the steps to the rest of it

[sacred vale]

this is what i go for the first integral

[devout shale]

Sure, and then it looks like some more partial fractions fun

and the second integral will be the same methods

so looks like you will get this one easy! nice

[devout shale]

feel free to DM me if you have some more questions in the future

[sacred vale]

now do we

seperate?

[devout shale]

yes looks like some more partial fractions

[devout shale]

well that is true, but it does not help you

perform another partial fraction decomposition

[sacred vale]

we have t odo this again?

wtf

[devout shale]

with linear factors v and v-3

yeah but this time it is much easier!

lots of good practice

[devout shale]

👍

You don't need me to watch over you step-by-step @sacred vale you got this. Feel free to DM me if you have more questions in the future though, I will be getting off for tonight.

[sacred vale]

alright\

[sacred vale]

.close

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[deep oak]

can someone walk me through the process of this problem

[deep oak]

just part a

i know tan 3pi/4 is -1

which means i would have tan inverse of -1

and then from there my brain goes brrrr

[sacred vale]

wdym

[worn lark]

remember that $\arctan$ outputs values within the interval $(-\pi/4,\pi/4)$

[deep oak]

my brain doesnt work past that point lol

[deep oak]

as in the graph right

[flat frigate]

🤓

[deep oak]

cuz it oscillates through -pi/4

right?

i know that tan ^-1 (-1) can be rewritten into

tan theta = -1

[worn lark]

yes

[deep oak]

ok cool

[worn lark]

these are the angles arctan outputs

[deep oak]

so isnt tan of 3pi/4 or 7pi/4

the answert then

[deep oak]

right

[worn lark]

the blue line is the line passing through the origin with slope -1

[deep oak]

yes

[worn lark]

the arctan of -1 is the angle of the intersect of the blue line and the red curve

[deep oak]

right

wait but

[worn lark]

what is it

[deep oak]

im kinda lost cuz tan theta = -1 and we know 3pi/4

is -sqrt2/=,2, sqrt2,2

[worn lark]

tan is not injective, so arctan isn't a true inverse of tan

[deep oak]

wait but like i said if we know that at 3pi/4 the coordinates are (-sqrt2/2 , sqrt2,2)

isnt tan just the later over the first

[worn lark]

yes

[deep oak]

giving us -1

[worn lark]

yes

[deep oak]

so wouldnt the answer to theta be

3pi/4 or 7pi/4

[worn lark]

no

[deep oak]

.-.

[worn lark]

.

[deep oak]

right?

[worn lark]

3pi/4 is not the answer

and 7pi/4 isn't either

[deep oak]

oh

ok shoot

[worn lark]

they are not in the interval (-pi/4,pi/4)

[deep oak]

ok i see

[worn lark]

you must find an angle within (-pi/4,pi/4) such that the tan of it is -1

[deep oak]

ok so

pi/3 is the answer

but im not sure why it is

cuz isnt tan of pi/3

sqrt 1

[worn lark]

tan(pi/3) is sqrt(3)

[deep oak]

shoot i was looking at the wrong answer

lol

sorry

sqrt3 / 2 is the answer

I MEAN

-pi/4

sorry

its been a long day

wait really?

[worn lark]

yes

[deep oak]

uhhh ok im kinda lost im sorry

-pi/4 + k ?

[worn lark]

wait

sorry

i confused pi with tau

mb

-pi/4 is the answer

[deep oak]

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[covert merlin]

hi could someone please help with how i would find t1 and d in this? thank you

[lapis shadow]

havent u obtained two equations in two unknowns

3.75 = t1 + 3d
9.75 = 2t1 + 9d

solve them as a system

[covert merlin]

uhh could you please explain how i'd go about that?

OH WAIT I GOT IT NVM

i had a lil dum brain moment

tysm

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[lusty nova]

how to prove this

i am dying inside

i dont even get the answers

this is part c btw

[final halo]

Do you get the very first line

[lusty nova]

yea

that makes cents

[final halo]

And the second? Same idea

[lusty nova]

no not really

[final halo]

Well how did you convince yourself of the first line? Using (c)

[lusty nova]

wait

they just subbed it in

ok

yea i get that

now

[final halo]

Yeah replace x with cuberoot( 1/x^2)

For x being a,b,c

[lusty nova]

mhmm

so how does it move to the next step

actually

now i get it

its just algebraic manipulation after that

[final halo]

Yeah then you just bound each piece

And they cancel

[lusty nova]

mhmm

thankz

i would not have been able to see this first try

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[lusty nova]

3rd times the charm

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[empty gyro]

test

[empty gyro]

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[idle parrot]

[final halo]

[empty gyro]

.coose

[lapis shadow]

lol

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[stuck jackal]

hello, im just looking for an explanation on why 6 wasnt inculded when it was flipped to change the fractional index to positive

[sand minnow]

because the negative exponent is not over the 6

[brave wolf]

The 6 is completely unrelated

[sand minnow]

if it said $$\frac{(6(3x+8))^{-\frac{2}{9}}}{7}$$then the 6 would also be flipped

[flat frigate]

Duh Hello

[brave wolf]

The power doesn't apply to the coeffecient. It applies only to what's inside the parenthesis

Exponentiation is also the first operation that has to be done (except for parenthesis). Even if there are no parenthesis, exponentiation has to be done before multiplication

[stuck jackal]

thank you

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[formal dawn]

can someone explain where 1/x^n comes from?

[safe radish]

@formal dawn Has your question been resolved?

[peak estuary]

from the derivatives of lnx

[snow robin]

so all the terms in the taylor polynomial

have to be less than that term with M

or are like that

and so they use the fact that they are less than or equal to 1

for x > 1

to set M

is how i understand what is written

[formal dawn]

but if f(x) = ln(x), |f^n(x)| maximized on the interval [1, 1.3] increases as n increases, so how does 1/x^n relate to that?

for my attempt at the problem, I found a pattern to generalize increasing derivatives of f(x) = ln(x), then maximized that generalization on the interval [1, 1.3], as given by Taylors inequality where 1 is the center of the polynomial expansion and 1.3 is the x value to determine the error bound. The magnitude of the maximized expression however, increases with increasing values of n so I'm not sure where to go from here since idk how I'm supposed to bound it

[safe radish]

@formal dawn Has your question been resolved?

[formal dawn]

<@&286206848099549185>

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@formal dawn Has your question been resolved?

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[onyx wagon]

all of the formulas i have

only work when k=1

when the value is more how do i solve?

[old drift]

Try shifting the index. So start the summation at k=1 and fix the expression, so that it is still the same.

Also mind the upper limit of the summation

[icy lance]

do the (sum from 1 to 30) - (sum from 1 to 5)

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@onyx wagon Has your question been resolved?

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[scenic valley]

Question is what area is the red area?

Here, the R is 0,5 cm?

[worthy gorge]

what is the formula for the circumference

@scenic valley

[scenic valley]

To clarify, is this correct?

[worthy gorge]

u dont need to square radius

circumference is 2 X pi X r

[scenic valley]

When do we square radius?

[scenic valley]

Isn’t it Pi * R^2?

[worthy gorge]

thats for area

[scenic valley]

Yeah and this is for area

[scenic valley]

I don’t understand

[worthy gorge]

cant u just find the circumference of the first circle 2.5 - circum of 2

[scenic valley]

Pi x 0,5^2?

[worthy gorge]

no

[scenic valley]

But this told me otherwise.::

[worthy gorge]

isnt that for a different question

[scenic valley]

So how to do this correctly other than I did?

[worthy gorge]

yes ig

u could put pi as full tho

[scenic valley]

This is correct? @vocal rampart

<@&286206848099549185>

[worthy gorge]

pretty sure

[zenith quartz]

<@&286206848099549185>

[scenic valley]

Go away

[worthy gorge]

u have to claim your own channel

it is correct

[scenic valley]

<@&268886789983436800> this person is disturbing us

[scenic valley]

ye I know I’m so smart

[worthy gorge]

ye u are

[wild cape]

@zenith quartz channel occupied, see #❓how-to-get-help

[zenith quartz]

Bro I am just asking my dbts

[scenic valley]

The first one is also correct? (Find area of the whole thing)

[worthy gorge]

do u have to find area of the shape?

if so then it is correct

[scenic valley]

Yes.

[worthy gorge]

sorry bro but i gtg ask someone else

[scenic valley]

Alr

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[granite aspen]

x^2 + ax + 25 = (x + b)^2
trying to complete the square, i think it's called
i am kind of stuck

[empty gyro]

What are you trying to find?

[vestal sedge]

do you have to find the value of a that you can write it as (x + b)² or what exactly

[granite aspen]

the value of a and b

sorry i forgot to mention that

[empty gyro]

Is this the start of your problem or are you halfway through the problem right now

[granite aspen]

no, still the start
i've watched tutorials and such but i still don't get it

[empty gyro]

Could you share your original problem with full context please

[granite aspen]

this is the full context,
"what numbers should a and b represent"
then the
x^2 + ax + 25 = (x + b)^2

[vestal sedge]

well try expanding rhs

[granite aspen]

heyy, omori ost, nice
and what's the rhs?
english is not my main language

[vestal sedge]

right hand side

the square thinf

also ty

[granite aspen]

well, that's easy
x^2 + ax + 25 = x^2 + 2bx + b^2

[vestal sedge]

now you can subtract x² right?

from both sides

[granite aspen]

yeahh

but then what?

[vestal sedge]

and youre left with ax + 25 = 2bx + b²

move the x terms to the left and the constants to the right to obtain ax - 2bx = b²-25

wait a sec

do you have to solve this mathematically or can you just imput what the result with not so rigurous explanation as to why

because you can just say a is whatever its needed to complete the square for the rhs and be done without further explanation

[granite aspen]

hmmm, what do you mean?
so i've checked the like correction sheet and
a = 10 b = 5
a = -10 b = -5

[vestal sedge]

yeah thats what it would be if it was what i just said

well you know the formula for (a+b)²=a²+2ab+b²

now think what a and b are for your case

[granite aspen]

i've honestly got no idea,

that's kind of where i'm stuck

square root of 25 is atleast 5,

but

[vestal sedge]

well x is already squared so you know a = x in the formula

[vestal sedge]

it is 5

precisely 5

[granite aspen]

i meant like, at least

or

gahhh translation

[vestal sedge]

so you know 5² = 25

well thats your b right?

in the formula

[granite aspen]

mhmmm

[vestal sedge]

but remember it can also be negative

because (-5)²=25 too

so, from the formula, the middle term is 2ab and you take it by each case
if b = 5 that means the middle term is 2*5x = 10x so a is 10
if b = -5 that means 2*(-5x) = -10x is the middle term, therefore a is -10

[granite aspen]

oh wait sorry

huhh i suppose

it's all a bit confusing for me but i'm going to try and solve some other... things now,
thanks for the help

but how would you even draw up this..

is this correct @vestal sedge ?

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[molten elm]

2x + 7y = -4

[safe radish]

Please don't occupy multiple help channels.

[molten elm]

Determine the slope and y intercept from an equation in standard form

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[novel magnet]

what is the purpose of doing the highlited step in the picture

[final halo]

you eventually want your pivot entries to be 1

doesnt matter when you do it

[novel magnet]

wdym by pivot?

[final halo]

If a matrix is in row-echelon form, then the first nonzero entry of each row is called a pivot, and the columns in which pivots appear are called pivot columns.

[novel magnet]

so in this case the pivot is the top left number right?

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@novel magnet Has your question been resolved?

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[vital vortex]

I need help

how does it have a imaginary number

from that step

[compact ferry]

$\sqrt{-20}$ is imaginaryu

[vital vortex]

should it just be -sqrt 5

[flat frigate]

heavy

[lofty temple]

that is what happens when u root a negative num

sqrt(-1) = i

[vital vortex]

so negative root doesnt work

[wheat cave]

$\sqrt{-20}=\sqrt{-4\cdot 5}=2i\cdot \sqrt{5}$

[compact ferry]

+?

[worthy hemlock]

You mistyped this

[flat frigate]

Jigglyproff

[vital vortex]

how did that do that

why is it 2

i

2i

why is it turned 2i but the 5 didnt turn into anything

[compact ferry]

$\sqrt{-20} = \sqrt{-1 \cdot 20} = i \cdot \sqrt{20} = i \cdot 2\sqrt{5}$

[flat frigate]

heavy

[vital vortex]

ok so if it egative root, then it just an i?

so if -4i, how would that be 2i

i mean -4 root

[wheat cave]

$\sqrt{-(a)^2}=i\cdot a$

[flat frigate]

Jigglyproff

[vital vortex]

is the A the number inside

[wheat cave]

a is a number that gets squared, then multiplied by minus one

[lofty temple]

i would say it is better to think of it as sqrt(-1 * 20) = i * 2sqrt5 = 2i*sqrt5

[vital vortex]

why does it get squared, is it because it needs to because it turns into a imaginary number

[wheat cave]

because I was lazy, so the result becomes just i*a

like, its the same as $\sqrt{-(a)^2}=\sqrt{-1}\cdot \sqrt{a^2}=i\cdot a$

[flat frigate]

Jigglyproff

[wheat cave]

just took sqaure to show that you still have to actually take a root of a afterwards, the i just resolves the - in the root

[vital vortex]

so should i do it like that or another way

[wheat cave]

like where specifically

its just a rule that $\sqrt{-1}=i$

the rest is your usual math

[flat frigate]

Jigglyproff

[wheat cave]

or

,w i^2

[flat frigate]

Results provided by WolframAlpha

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Requested by Jigglyproff#4511

[vital vortex]

Ok

Sorry I was transitioning in class

<@&286206848099549185>

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[lost nexus]

I feel like I am overcomplicating this question but at the same time I am unsure

[lost nexus]

isnt it just the double derivitive of both of the parametric functions for both

and plug in 314 as t

and find which is faster?

[frank glen]

Hmm

From your past knowledge

What represents speed: first derivative or second derivative?

[lost nexus]

first

no derivative = position
first = velocity/speed with direction
second = accel

@frank glen

[frank glen]

Okay

Which one is a measure of how fast or slow: velocity or acceleration?

[lost nexus]

oh shoot

its the first derivative

that is the speed

[frank glen]

Yep

So can you find the first derivative and plug in t=314 and see which one has a greater magnitude?

[lost nexus]

i guess 🤷‍♂️

im assuming it is that

but it also seems too easy?

[frank glen]

Well

[lost nexus]

i know you have to sqrt((x^2)+(y^2))

[frank glen]

Sometimes really easy questions are the most easily mistaken questions because of how we are used to the tough stuff

[lost nexus]

for the overall motion

[frank glen]

Use that but for the derivatives at t=314

[lost nexus]

yes

got it

lemme do that now

thanks!

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[lean cedar]

Honestly just looking for help for my daughter in 4th grade. I know this isn't hard but can't figure out the formula. Can anyone help?

[vague reef]

If $1000 is invested for 4 years and the interest earned was $200. What was the interest rate?

i need help with this please

[karmic zenith]

bruh

get ur own channel

[vague reef]

oh is that how it works

[karmic zenith]

they have available next to them

yah

[vague reef]

oh ok

[karmic zenith]

@lean cedar im not a certified helper or anything but im pretty sure this is just 2880

I got that from multiplying all of the numbers together

[lean cedar]

@karmic zenith ah ok. Thank you

[karmic zenith]

once again, not like a pro or anything but that should be right unless im overlooking something

[lean cedar]

I thought the same Honestly but wanted to make sure.

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[craggy trout]

For part a, the probability of no smartphones would be 95/100 or 0.95. But I don't understand how the sample of 3 phones plays into it. How do I take the probability and apply it to the sample?

[safe radish]

@craggy trout Has your question been resolved?

[craggy trout]

<@&286206848099549185>

[frank glen]

Hmm

Have you considered the fact that you have a dependent probability?

Like when you pick a phone, you reduce the total number of phones

Also have you considered using binomial distribution?

[craggy trout]

Maybe something like

,w 95/100 * 94/99 * 93/98

[flat frigate]

Results provided by WolframAlpha

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Requested by dopediscorduser#4906

[craggy trout]

?

[craggy trout]

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[signal dome]

binomial expansion using Pascal's triangle, can somewone please explain this?

[worthy hemlock]

https://www.youtube.com/watch?v=s19dWIHficY

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[meager jungle]

I'm really confused as to how to do this problem:

red marbles and three blue marbles. A marble is selected at random from box I and
placed in box II. A marble is then selected at random from box II. If this marble is
observed to be red, what is the probability that the marble selected from box I and
placed into box II was also red?```
Does anyone have any idea?

[lapis shadow]

draw a tree diagram

[meager jungle]

that helped thanks

.close

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[dreamy finch]

could someone please help me with e?

[dreamy finch]

<@&286206848099549185>

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[dreamy finch]

oh sorry

[stoic dune]

That's just "someone has a cat, someone has a dog, someone has a ferret" but weirdly worded

[dreamy finch]

That is worded very weirdly lol. So would it be "Ex(C(X)) and Ex(D(X)) and Ex(F(x))

[stoic dune]

Yeah I like it. I suggest using x,y,z instead of x,x,x for clarity.

But I think that's right

[dreamy finch]

oh okay. Thank you so much for your help!

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[main ice]

what’s up

[devout shale]

can I ask why you can't figure it out using the try feature?

seems like that would help you out a lot

[terse current]

i dont understand it, i dont want to just randomly guess until i get it right,

I want to actually understnad the material

[devout shale]

sure, but won't the guessing help you understand the material?

if you don't know what a reflection is, why dont you click try and see what it is

I think it would be more helpful for you to see it rather than me explain in the chat

[main ice]

you could sketch it and try rotating it translating it and reflecting it

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[lean otter]

i need help with this

[glass carbon]

can you determine radius from provided info?

[lean otter]

ig

is it 10?

[glass carbon]

yes

now imagine cross section of a cone

[lean otter]

yeah half

[glass carbon]

there you have a right triangle

[lean otter]

ok continue

[glass carbon]

with legs 10 and 8

[lean otter]

ok

[glass carbon]

determine hypotenuse (l)

[lean otter]

10^2 + 8^2 = c^2?

[glass carbon]

yep

[lean otter]

100 + 64 = 164

square root of 164 is

?

i dont know

its not a natural square

more than 12 less than 13

[glass carbon]

just sqrt(164), it might be rewritten as 2sqrt(41)

now just use formula for the surface area of a cone

[flat frigate]

Modus

[lean otter]

10* 10+ (sqrt)64?

[glass carbon]

or just add pi * r * l to 100pi (because area of the base has been given)

[lean otter]

yeah but what is l

without l its gonna be too complicated

i just put it in a sqrt calculator

[glass carbon]

l is sqrt(164)

(your c)

sometimes denoted as s

,calc pi * 10 * (10 + sqrt(164))

[flat frigate]

Result:

716.47942664582

[glass carbon]

or:

[glass carbon]

,calc 100 * pi + pi * 10 * sqrt(164)

[flat frigate]

Result:

716.47942664582

[lean otter]

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[sharp wharf]

Why is this a linear map? Where have I gone wrong?

[obtuse widget]

Doesnt look like youre wrong

Unless T(3t+1) = 0

[cosmic grove]

Welp
T(p(t)) + T(q(t)) = T(3t+1) + T(3t+1) = 2T(3t+1)
🤔

[obtuse widget]

Whats the full excerpt of the example?

[sharp wharf]

The question is to find eigenvalues and eigenvectors of this Linear map

[cosmic grove]

can you give the whole exercise tho ?

[cosmic grove]

not the solution

[sharp wharf]

The exercise of checking that it’s a linear map?

[cosmic grove]

yes

[sharp wharf]

They haven’t given a solution for that

[cosmic grove]

im not asking for the solution

im asking for the whole exercise

[sharp wharf]

That is the entire exercise

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[plucky elk]

Very careful chain rule

[rugged hound]

after applying chain rule you get: 1/sqrt(1-((sqrt(1+x)+sqrt(1-x))^2)*d/dx((sqrt(1+x)+sqrt(1-x))/2)

what follows are pretty much normal differentiation

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[hushed furnace]

triangles abc and mnp have the same length of the radius of the circumscribed circle, bc = np, but angles A and M of different measures. Calculate the sum of the angle measures. Any ideas?

[hushed furnace]

I tried the sinus formula but it doesn't end anywhere for me

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[tired pond]

Can someone please clarify something about ${n \choose k}$ for me? If a formula says $f(n) = {n \choose k} a^{n-k} b^k$ for $f(3)$ does that evaluate to $a^{3-k} b^k$ or is the $k$ there as a multiplier in place of n choose k? Like $(k)(a^{n-k} b^k$

[flat frigate]

math

[lean otter]

yes, n choose k is a number https://en.wikipedia.org/wiki/Binomial_coefficient

[tired pond]

yep, that clarifies things lmao

thanks!

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[stray cypress]

how do i integrate this using complex analysis

[stray cypress]

im assuming its either a rectangular or a keyhole contour

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@stray cypress Has your question been resolved?

[stray cypress]

<@&286206848099549185>

[waxen cove]

yes what

[stray cypress]

oh it says i can ping helprs if i dont get a response

[vestal sedge]

do you have to do this for all p or do you have some kind of restriction for that

[stray cypress]

im working with gamma(p), so i think it must be positive (is that correct)?

[vestal sedge]

well you can also have negative inputs using a bit of manipulation but if gamma p is given in some way then yes

[plucky elk]

the integral doesn't converge for any real p-1>0

[stray cypress]

apparently i should be getting pi/sin(p*pi)

so it definitely does

[plucky elk]

show this

screenshot or picture of the information

[vestal sedge]

a quick wolfram lookup also reveals this to be true

[stray cypress]

im trying to not use a keyhole if possible tho

[plucky elk]

try #real-complex-analysis

#get-advanced-access if you don't have access

[plucky elk]

the condition for p is Re(p) in (0,1)

https://web.mst.edu/lmhall/SPFNS/sfch2.pdf

[stray cypress]

ty

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[nocturne blaze]

I know 7 factorial is 1 x 2 x3 x4 x5 x6 x7 but this question I dont understand it

[solid timber]

Try prime factorising it first

Now for total number of divisors…

The divisor may have a 2 or a 2^2 or 2^3 or 2^4 or no 2 at all… 5 possibilities

Similarly for 3, there r 3 possibilities

For 5: 2

And for 7 also its 2

Thus total number of divisors is the product of 5,3,2 and 2

For odd divisors we don’t have any powers of 2… so the no of odd divisors is the product of 3,2 and 2

Taking the fraction, its 1/5

@nocturne blaze

If you want a general formula for such questions, prime factorise it first. The answer is 1/(1+exponent of 2)

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@nocturne blaze Has your question been resolved?

[nocturne blaze]

sorry i was afk

i will see now

thanks for answering btw

[nocturne blaze]

why u deleted 1

[nocturne blaze]

do you know whats the name of this lessoon

A or D

[nocturne blaze]

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[vital vortex]

i dont get this part

is sqrt -1 = i?

[ancient escarp]

yes

[vital vortex]

what does -i stand for

turn to

[ancient escarp]

what

[empty gyro]

i is called the imaginary unit

[vital vortex]

is -1 just 1

[ancient escarp]

no

[vital vortex]

i mean -i is 1

[empty gyro]

It does not represent any real number (hence why they call it imaginary)

i represents a value that does not exist on the real number line, but that has the property $i^2=-1$

[flat frigate]

SWR

[vital vortex]

ok

so what is $-i^2

what would -1^2 be

i think it represents something because my teacher showed it to represent something

[empty gyro]

Hold on. Let me switch off of mobile

Okay. PC power activate

So first important thing to remember is that a negative multiplied by another negative will give you a positive

So $-2\times-3=6$

[flat frigate]

SWR

[empty gyro]

This is important when you remember what squaring a number does. It just multiplies that number by itself.

For example $5^2=5\times5=25$

[flat frigate]

SWR

[empty gyro]

So, when you square a negative number by itself, you get a positive

$(-5)^2=-5\times-5=25$

[flat frigate]

SWR

[empty gyro]

This creates a problem, because we now arrive at the fact the no number, when you square it, will give you a negative number

This is where the imaginary unit comes in.

The imaginary unit, is denoted as $i$. It exists outside the real number line, and it has the property $i^2=-1$

[flat frigate]

SWR

[empty gyro]

Now, you were asking "What would $(-i)^2$ be?"

[flat frigate]

SWR

[empty gyro]

To understand this, we need to refine the words when I say "a negative times a negative returns a positive"

This doesn't work for the imaginary unit, i, because it exists outside the line of reals, so it is neither a negative nor a positive

This is where it's important to remember some properties of algebra

First, multiplication is commutative. That is, multiplying $a\times b$ will be the same as multiplying $b\times a$

[flat frigate]

SWR

[empty gyro]

You still around @vital vortex?

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@vital vortex Has your question been resolved?

[plain ingot]

can they help me

(x+1)^2-(x^2+2x+1) factor this expression

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[red lintel]

i havent learnt this in school, so can someone teach me how to work it out?

[bold ferry]

do you know how to find circumference of a circle?

[red lintel]

pi x r?

[bold ferry]

2 * pi * r

or

pi * d

sound familiar?

[red lintel]

yeah

does the * stand for + ?

[bold ferry]

no, times

[red lintel]

👍

[bold ferry]

so basically if you know the wheel made 9 rotations, you can multiply the circle's circumference by 9

and you know how far she travelled

[red lintel]

OHH

okay

so 1. figure out circumference
2. times it by 9?

[bold ferry]

yep

[red lintel]

alright ill try answer it

[bold ferry]

you can also use this to find B

[red lintel]

alriught

whats the distance travelled though

it doesnt say

wait

no

[plain ingot]

(x+1)^2-(x^2+2x+1) factor this expression

[red lintel]

4.4

[bold ferry]

er

that's the circumference yeah

[bold ferry]

that's why A is before B and you can solve for the distance :)

[red lintel]

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[dense token]

I used tan(A+B)

[dense token]

but i cant get a quadratic from that

[glass carbon]

hint:
tan(theta + 45) = -1/tan(theta - 45)

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[fickle abyss]

i dont get it

should |x| always be positive?

so should it not be something like this? ⬇️

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[lean otter]

im confused for a)

I thought that since the order of the flags didnt matter you would use the divider method

where you add 9 dividers and see how many ways to arrange them

so it would be 34!/(9! * 25!)

dividing by 9! because of the added dividers and by 25! because the ordering of the flags doesnt matter

i also got C wrong

no combinatorics fans in this discord 😭

[buoyant shadow]

it's because the flags are different

divider method only lets you distinguish the amount of flags

however it should work for b

34!/(9! * 25!) * 25!

[lean otter]

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[calm fern]

Consider a polynomial function with real coefficients and a zero at 4+7i what is another zero

[calm fern]

trying to decipher this

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[arctic stone]

how did the numerator get 10???

[plucky elk]

this screenshot has a lot. where exactly are you lost

[arctic stone]

Right where I highlighted

1+2=10

[plucky elk]

the 10 is part of the problem. they didn't calculate anything

[arctic stone]

ohhhh I was just looking it over it again

and realized im a dumbass ha

thank dude

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[prisma garden]

Hey im having trouble with a calc hw question. In 3a, for $\phi_r$ and $\phi_z$, why is the result divided by rho?

[flat frigate]

punchster2

[prisma garden]

Im not sure if im missing some nuance of the problem, but im losing my mind here

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@prisma garden Has your question been resolved?

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[dreamy finch]

is there a way to make this equation in the form of y=? I'm just curious. So the logistic equation xn+1 = a*xn(1-xn) is an equation where the nth input is the output of the (n-1)th input. Is there a way to put this in the form of y= so i can graph the equation on desmos?

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[dreamy finch]

<@&286206848099549185>

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[safe radish]

Please don't occupy multiple help channels.

[violet veldt]

someone pls help with this problem

its due in 10 min

9 now

@whole acorn

<@&286206848099549185>

@vocal rampart

[wild cape]

pls dont ping mods for math help

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@violet veldt Has your question been resolved?

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[tawdry plover]

Let $f:[0,\pi] \to \mathbb R$ be the function $f(x) = x$. Find its even periodic extension. Find the corresponding Fourier cosine series.

[flat frigate]

blanket

[tawdry plover]

i dont know what i did wrong, i found the coefficients to be

$$A_n=\frac{2}{\pi n^2}((-1)^n + 1)$$

[flat frigate]

blanket

[tawdry plover]

did i integrate wrong?

never mind yes i did

i just realized that i had the wrong signs when i plugged in my bounds

.close

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[tawdry plover]

Given coefficients $A_0, A_n, B_n$ for some Fourier series of a periodic function $f(x)$, consider the function $f(ax), a \in \mathbb R$. Write the coefficients of the Fourier Series of $f(ax)$ in terms of $A_0, A_n, B_n$.

[flat frigate]

blanket

[tawdry fox]

just rewrite the fourier series with ax instead of x

and then use some trig identities

[safe radish]

@tawdry plover Has your question been resolved?

[tawdry plover]

i just got every coefficient divided by a?

im not sure if that's right

[tawdry fox]

not really

show your work maybe i can see where you made a mistake

[tawdry plover]

wait so far

so i have f(ax), right?

[tawdry fox]

ok

[tawdry plover]

if f(x) had a period p, then f(ax) has period p/a right?

so

$\alpha_0 = \int_{-p/a}^{p/a}f(ax)dx$

[flat frigate]

blanket

[tawdry plover]

im using alpha 0 alpha n and beta n to denote new coefficients

is this right so far?

[tawdry fox]

ok

[tawdry plover]

wait so am i on the right track

because the rest of the coefficients are basically the same thing

[tawdry fox]

so alpha0 is A0 right?

[tawdry plover]

or what im connecting it to, yes

wait are they all just going to be the same?

because

oh i just realized i forgot to divide by the period

[tawdry fox]

but A0 should be the average of the function so its unchanged by scalar

[tawdry plover]

oh i see

so

$\alpha_0 = \frac ap \int_{-p/a}^{p/a}f(ax)dx$

[flat frigate]

blanket

[tawdry plover]

$\alpha_n = \frac{2a}{p}\int_{-p/a}^{p/a}f(ax)\cos\left(\frac{2a\pi n x}{p}\right)dx$

[flat frigate]

blanket

[tawdry plover]

$\beta_n = \frac{2a}{p}\int_{-p/a}^{p/a}f(ax)\sin\left(\frac{2a\pi n x}{p}\right)dx$

[flat frigate]

blanket

[tawdry plover]

this is what i ended up obtaining

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@tawdry plover Has your question been resolved?

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@tawdry plover Has your question been resolved?

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[regal hatch]

Any complex number can be written as re^iθ
I want to ask the θ angle here is the argument or the angle made by z verctor with x axis ??

[safe radish]

Please don't occupy multiple help channels.

[regal hatch]

Any complex number can be written as re^iθ where r is the distance from origin
I want to ask the θ angle here is the argument of z or the angle made by z verctor with x axis ??

[tawdry plover]

the angle made w the real axis but yeah it is

[regal hatch]

So if the complex no. Is e ^i4π/3 can we write it as e^-2iπ/3 ???

[tawdry plover]

id believe so, yes

[regal hatch]

I.e if the angle with real axis is ex 190 degree can we write it has -170 ?

[tawdry plover]

yes

[regal hatch]

Thanks blanket 🙂

[tawdry plover]

np

[regal hatch]

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[lean otter]

Can someone help me

I don’t understand why pi is positive not negative

Also, am I on the right track?

$z^6 + 1 = 0 \implies z^6 = -1 \implies z^6 = e^{i(\pi +k2\pi)}$

[flat frigate]

Ryuzaki

[lean otter]

$z = e^{\frac{i\pi(2k+1)}{6}}$ where $k = 0, 1, \hdots, 5$

[flat frigate]

Ryuzaki

[lean otter]

then convert this to polar form

i.e. cis(theta)

[royal skiff]

For me, I'll just find the principle solution and add 2pi/6 every time (as roots of unity have equal magnitude but equally spaced argument)

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[lean otter]

option

d

[pseudo scroll]

No option D is incorrect

@winged willow Are you aware of Newton's First Law?

[lean otter]

oh sorry

[winged willow]

yes

[pseudo scroll]

Can you state it for me?

[lean otter]

sum of forces will be 0

[winged willow]

unless you force a body it will not move

[pseudo scroll]

Partially right

A body retains its current state of motion unless compelled to change it by an external force

[winged willow]

okay...

[pseudo scroll]

If a body is moving with constant velocity or is at rest what can you say about the forced acting on it

[pseudo scroll]

With respect to this law

[winged willow]

so option c ?

[royal skiff]

Has to be c, others are definitely incorrect

[pseudo scroll]

Yup

I hope you know why

[winged willow]

thanks

Yes got it

[pseudo scroll]

Alright cool

[royal skiff]

I don't know whether you learned this or not, but the gravitational force in the direction of the inclined plane is mgsin(θ),,and norma force is mgcos(θ)

[winged willow]

.close

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[zealous nest]

Can someone help me get C

[brave wolf]

Not +4x-12, rather use -4x12

-9+3-4x12

[zealous nest]

How would I get a?

Is there a bracket in a?

That’s only thing messing me up

[brave wolf]

Sorry, what I said was wrong

[zealous nest]

Would it be 6 x 5 divided by (9)

[brave wolf]

except for 1 thing. The (- ) part

[zealous nest]

(-9)?

[brave wolf]

Nope

30/9 isn't integer

[zealous nest]

I’m confused how would I do it?

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@lean otter Has your question been resolved?

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[shy void]

Hi is it correct how you can check if the function is in sobolev space?

[opal oak]

Is this not undergrad? If it is, you'd be better getting the "undergrad" role and asking there, this is more meant for high school help

[shy void]

ah ok thx

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[timid fossil]

How do you find the shortest possible distances between two trigonomic graphs? So far, I tried deriving them like how you would with a normal quadratic function and another quadratic, but the derivative produced didn't give a conclusive result as to the solutions of the functions. The two trigonomic functions are:

• Tan(x-45deg)
• -2cos(x)

[pseudo scroll]

Are you sure these graphs don't intersect

[timid fossil]

Oop

my bad, its in a specific range 0<x<pi

[pseudo scroll]

I see

[timid fossil]

they will intersect when at the full range 0<x<2pi

[timid fossil]

also yes ive graphed them out and im sure they dont intersect within this range

so im just wondering if theres a conclusive proof for it

[sand minnow]

take the derivative of the difference between them to find the minima

if there is no minima then it will be one of the endpoints

[timid fossil]

yeah

i did

[pseudo scroll]

Would that work

[timid fossil]

it works for normal quadratics dunno about trigs

[pseudo scroll]

For points (x1, y1) and (x2, y2) on both graphs

[timid fossil]

i tried it and the result wasnt exactly what i was looking for

[sand minnow]

i dont see why it wouldnt

[pseudo scroll]

The minimum difference utilizes the derivative of y1 - y2?

[timid fossil]

yeah

[pseudo scroll]

[timid fossil]

which is the difference between the two functions

[pseudo scroll]

I was thinking of assuming parametric coordinates and differentiating that with the distance formula

[timid fossil]

()x1, tanx1-45) (x2,-2cosx)

maybe ill redo the math again soon

[timid fossil]

deeeefine parametric coordinates?

[pseudo scroll]

Instead of (x1, y1) you have (f(t), g(t)) basically cuts your variables in half

[timid fossil]

ah

hmm,

[timid fossil]

in the end though i did this

and i still ended up with only x as a variable

will be back if my maths prove incorrect once more

.close

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[little kettle]

@icy jolt I have a question for the induction problem, if you don't mind

[little kettle]

It's in help 20, for reference.
I was wondering. Isn't the Inductive Hypothesis that n = k? If so, how does that prove that a list that's n/2 size will be sorted correctly?

Like, which part of the logic says that if a list of n size can be sorted, a smaller list can too?

I mean, we proved that "a smaller list" can be sorted with n = 1, but that doesn't necessitate that ALL smaller lists can be sorted, does it?

Also; Logically, if it can merge two sorted lists properly and we know that for a list with only one element, it is always sorted, shouldn't we try to prove that mergesort will split every list/array up until it's only 1 element big, from whence it can merge everything properly?

[old drift]

When proving something by induction, you can always assume your statement is true for all i <= k, for some k. This is sometimes called "strong induction hypothesis" and it can be shown that this is equivalent to "normal induction" where the hypothesis applies only to n = k.

https://en.wikipedia.org/wiki/Mathematical_induction#Complete_(strong)_induction

[little kettle]

is normal induction the same as weak induction?

[old drift]

yes

[little kettle]

okay, thank you so much. I'm not too comfortable with induction yet, so this was very helpful

[subtle heron]

It's also good to think of it building itself up. So if you prove it's true for the smallest list 1. And then show that it's true for any n+1 given that it's true for n. You can now build. when n+1=2 you know it's true because n=1 is. Which means 3 is proven because 2 is. And so on til infinity. So you've proven that ANY positive integer length is possible

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[stray axle]

,rotate

[stray axle]

how does the integral from 1 to b dx/x^p =1/(1-p)x^(1-p)

i dont understand the transformation

what type of integration is it

[opal oak]

It's the same as 1/(x^p) dx

Or x^(-p) dx

Try using standard integration power rules from there :)

[stray axle]

so substiution?

[opal oak]

Nope, just standard power rule

The integral of x^n = (x^n+1)/(n+1)

[stray axle]

oh didnt know that! is that just a basic fact?