help-0

[lime tangle]

do you go to school?

or you are self-taught

[silver dew]

then www.youtube.com is your teacher

[lime tangle]

thats sad

is this in the US?

[raw shard]

that’s basically not school lol

[mighty grotto]

hmm how do you guys get good at math :0

[raw shard]

@mighty grotto practice

[lime tangle]

practice

and asking for help when you need it

[mighty grotto]

does it take alot of time

[lime tangle]

yes

[raw shard]

yes

[silver dew]

i lowkey suck. my undergrad was physics and masters was math but i forgot so much of the advanced stuff.

[mighty grotto]

ooo thanks :DD

[raw shard]

oof

[silver dew]

the key was that i enjoyed it

[raw shard]

honestly if you enjoyed it idk how you forgot it

[lime tangle]

but if you are dedicated and you love it and you ask questions it will take less time or it will feel like it was less time

[lime tangle]

congrats btw

[raw shard]

i guess you just never used a lot of it

[silver dew]

mostly because i stopped using it

[lime tangle]

congrats on your degrees @silver dew

[silver dew]

i do data science now so i remember applied linear algebra and multivariate calc, but no way can i do real analysis or abstract algebra without thorough review

[mighty grotto]

alr ;)

[lime tangle]

nice

hard work pays off

always

[silver dew]

@lime tangle thanks. im a phd dropout lol

[lime tangle]

thats fine masters is amazing

i dont think ill even get to an undergrad tbh

but ill try

[silver dew]

are you in high school

[lime tangle]

well

last year

pretty much

last year of high school then going to college

[silver dew]

what do you plan on majoring in

[lime tangle]

i like computers

but

its hard

like the math involved is rlly hard

and then there is physics and stuff its not easy

i like math i enjoy it a lot but it just gets harder and harder

especially that this year i am doing cal 1

its unusual that i do cal in high school but i am doing it cuz my school offers it

[silver dew]

are you in the united states

[lime tangle]

canada

[silver dew]

oh. it's pretty typical for cal to be taught in high school.

i think now they even go up to differential equations

[lime tangle]

wow

i do lims now

limits is fine but like its not easy

we are getting on to derivatives soon

but its very difficult

[silver dew]

i would just say that "easy" is obviously a relative term

if you enjoy it, continue to work hard

[lime tangle]

i will

[silver dew]

but never compare yourself to how others are doing

[lime tangle]

i find it hard not to but that is very good advice

its so competitive as well

every person is to themselves

nobody helps, nothing

its very tense

but ill try

thanks

[silver dew]

the good news is that you're not only in the internet generation, but a generation in which resources are seemingly unlimited

[lime tangle]

thats true

i agree

[silver dew]

take your time in deciding a major, especially if you think you'd like to go into academia.

[lime tangle]

ok i will

[silver dew]

if you like computers and just want a job, i'm just gonna say that computer science is the way to go

[lime tangle]

yeah thats what i heard

comp sci is good

i might go in that

but if i go in math what can i do with that?

[silver dew]

i definitely am not interested in romanticizing college. if your plan is to get a good paying job in STEM, do CS or engineering

if you go math, complement it with something marketable like a CS minor

[lime tangle]

yeah

[lime tangle]

hmm i might do that

thats a good option

[silver dew]

i will say that this discord is very useful to get a scope of what to expect from a math major i.e. browse the early university channels.

[lime tangle]

yeah i do that sometimes

it seems super hard

like another language

the beginning of cal is a new language to me with limits and stuff but at least there is some factoring and algebra but when i like at university channels damn it aint easy its like a problem of 2 lines with barely any info

[silver dew]

i mean there's no way i would expect you to grasp the difficulty, i just think it's good exposure to show you that upper division mathematics isn't necessarily computation like you're used to

linear algebra will probably be the first course to help you decide if you like math, or if you'd rather do physics/engr.

[open idol]

3cosx -3cos^3x

what is that

[umbral kindle]

need help

how to write in this form

[gritty dune]

recall the definition of tangent

have you tried that?

[umbral kindle]

i think i forget it

[wary stream]

Does SOH CAH TOA ring any bells?

[umbral kindle]

yea

oic

[wary stream]

So how would you set up the tangent ratio?

[umbral kindle]

2+sr3 /4+sr3?

[wary stream]

Add some parentheses

[umbral kindle]

oh okok

thx

i think i know already

[wary stream]

Then rationalize the denominator

And simplify and that's it

[warped holly]

Can someone help me distribute this?

1 [1 - (-3)^n]

[inner sentinel]

how do i do this

[radiant barn]

do you know the conditions for EVT?

[bleak basalt]

"PROVE OR DISPROVE THE FOLLOWING STATEMENT"

for hashtag #discrete-math

my issue is that i am so sick that i dont even understand the question. Could there even be anyone out there that could help me with this homework? if so then thank thank you so much

?can anyone help

[inner sentinel]

i solved it but thx

how do i go about solving this

[alpine sable]

for every single value on [6,9] the derivative is greater than 9

what does that say that value has to be — at a minimum — at r(7), r(8), and r(9)?

[inner sentinel]

im confsued

[fresh parcel]

derivative must be larger than 9

wait

[raw shard]

basically the slope is at least 9 for every x value in that interval

notice how the start of the interval is 6, and you’re given r(6)

so r(7) should be at least r(6)+9

and so on

@inner sentinel

it took me a while to come up with something too

i think i just thought of it as an actual graph

you weren’t supposed to say that

they aren’t here so delete that @fresh parcel

[fresh parcel]

okay

mb

[sick torrent]

How do you do a question like this?

[mighty sinew]

you go in the physics server lol

[fresh parcel]

tf is that

[small cypress]

how do u find the line of regression?

[fresh parcel]

that could be googled

[small cypress]

bruh

[raw shard]

half the questions here could be lol

[daring prawn]

elements with the same number of valence electrons have similar physical and chemical properties, # of valence electrons is # of electrons in highest level which is represented by the number coefficient before the letter, and generally s and p sub levels are the ones that contain valence electrons, so 4s2 4p6 is similar to 3s2 3p6 (the exponent represents # electrons in that sub level and the total valence is the sum) and the 3s2 3p3 is similar to 2s2 2p3

[brisk lantern]

The original cords was a typo. But I figured it out already

[wary badge]

Excuse I have a doubt

[signal forge]

'bout what?

[lapis valley]

For integration by parts I use $$ uv = \int u' v dx + \int uv' dx$$
Then to solve for $$\int \sec^3 \left(x\right)$$
$$ u = sec x, v = sec^2 x$$ and Idk how to proceed with this

[ocean seal]

hduxueiwkwnxudi

[narrow meadow]

can't seem to figure out the steps in between this

kinda confused about how it happened

[quartz oxide]

sin(2x)= -sin(2x+pi)

Similar for the other sin()

[narrow meadow]

ohhhh right yeah

me dumb nvm

thx <3

[night narwhal]

Is anybody using this channel

Am wondering why they integrate this way as wondering if i am able to get the right answer or not

I integrate my to be 2pie((9/28)y^7/3 -(4/15)(y/6)^5/2). If somebody can help me with this

<@&286206848099549185>

[pulsar aspen]

You mean $2\pi({9\over28}y^{7\over3}-{4\over15}({y\over6})^{5\over2})$

[ocean seal]

Xwtek

[night narwhal]

Yes

[pulsar aspen]

Sounds to be incorrect to me

[night narwhal]

Okay

Why

[pulsar aspen]

Try to derive it

It should result in the original equation

[night narwhal]

I am doing on calc and so am being told that it suppose to be (3^(2/3)*y^(4/3))/7

for the first part

and second part should be ((6)^(1/2)*y^(5/2))/15

Is that correct

[pulsar aspen]

Before integration or after integration?

[night narwhal]

After

[pulsar aspen]

The first part is wrong.

When you are given that type of equation, try to convert that into a form of $$c_{1}x^{b_{1}}+c_{2}x^{b_{2}}+...$$ before integrating

[night narwhal]

Oksy

[ocean seal]

Xwtek

[pulsar aspen]

Then the result of integration will be $${c_1\over b_1+1} x^{b_1+1}+{c_2\over b_2+1} x^{b_2+1}+...$$

[ocean seal]

Xwtek

[night narwhal]

But with the integration why is there the (9)^(1/3)/3

then

Is it because you cannot have square root on the bottom even though many equations allow for it

to be that way

[pulsar aspen]

Your teacher probably wants to simplify the equation further, but it's not necessary

[night narwhal]

Okay

[pulsar aspen]

You just need to extract the y inside the roots outside

[night narwhal]

But how would I get the first part done then as you describe the entire integration is wrong as a way to reduce it not to be like that

[pulsar aspen]

I mean your answer

What your teacher says at the image is correct

[night narwhal]

Yes I got orignally integrate (y/3)^(4/3) - (y/6)^(3/2)

When trying to integrate it I would bring the 3/7(y/3)^(7/3) - (2/5)(y/6)^(2/5)

[pulsar aspen]

The y has to taken outside the power, so you get c^something y^something

[night narwhal]

Csn you be able to write it down looks like

Like should not be taking out y with integration

[pulsar aspen]

$$({1\over3})^{4\over3} y^{4\over3}+({1\over6})^{3\over2} y^{3\over2}$$

[ocean seal]

Xwtek

[night narwhal]

Okay

[pulsar aspen]

Then you can do the integration

[night narwhal]

GIves to me 8.50

Wait

Gives to me 10.599

(1/3)^(4/3)(3/7)y^(7/3)+(1/6)^(3/2)(2/5)y^(5/2)

Is what I got

Is that corect

[pulsar aspen]

The result of indefinite integration is correct (up to constant of integration). I didn't check the resulting definite integration yet

[night narwhal]

Okay so my answer could be correct as this is the right asnswer

Still need some help in why it is the way it is expect it is sitll not the right answer

[pulsar aspen]

I calculated the answer, but I got 10.744

[night narwhal]

Yes but for some reason still not right as they reduced it down

For me if not reduced down it is not right answer

[pulsar aspen]

What is your teacher's answer?

[night narwhal]

8.241

And you see it reduced to the integral with the photo above

Still no idea why as feels like with integration have to reduce it down not to be the form you shown earlier

This is ruining my life

@pulsar aspen feel stupid and all

Hate thios

[alpine sable]

so i got gemetry homework and iforgot what a = sign with ~ above it means, its like triangle ABC =~ triangle DEF

[pulsar aspen]

My bad, my mistake

[night narwhal]

Okasy so what is it

[pulsar aspen]

Other channel, please?

[night narwhal]

@pulsar aspen will you still answer my question

[alpine sable]

sry

[pulsar aspen]

The equation before integration should be $$({1\over3})^{1\over3} y^{4\over3}+({1\over6})^{1\over2} y^{3\over2}$$

[ocean seal]

Xwtek

[night narwhal]

Okay whay about after then

@pulsar aspen

[pulsar aspen]

When integrated, it should be $$({3\over7})({1\over3})^{1\over3} y^{7\over3}+({2\over5})({1\over6})^{1\over2} y^{5\over2}$$

[ocean seal]

Xwtek

[night narwhal]

Got the right answer then

@pulsar aspen thanks for the help as could see that now and with the y(√y/√6) I can see it now having to reduce it to be y^(3/2)(1/6)^(1/2)

Still thanks for the elp and all

It does make sense now

[pulsar aspen]

Me first

[alpine sable]

oh sorry i mistaken you for john

[night narwhal]

Sorry got to got to sleep

[pulsar aspen]

I mean, I answered john answer, so I could ask a question

[night narwhal]

@pulsar aspen thanks again for your help your the best

[alpine sable]

oops

my bad

[night narwhal]

GOt my daily dose of math thanks for ythe help and good night

[pulsar aspen]

Is there any algorithm like Elo for this scenario:
We have m players and n puzzles, we like to assign a skill on each player and a difficulty on each puzzle so that a harder puzzle requires a more skilled player. The puzzle and the player may be added later, and player may improve or degrade over time

[alpine sable]

Can someone help me with this?

Idk my teacher gave us this questions as a homework and I can't even understand them

[pulsar aspen]

Q1. find a vertex (intersection between a parabola and the line of symmetry) then find the roots

[alpine sable]

Ugh

How

[pulsar aspen]

Find the maximum value of a parabola, the find what x that makes the function returns that maximum value

[alpine sable]

What's the meaning of parabola

[pulsar aspen]

That f(x) is a parabola

[alpine sable]

o

I think I'll fail at preparing this hw

[pulsar aspen]

wdym?

[merry robin]

If x and y are two rational numbers such
that
(x + y) + (x – 2y) √2 = 2x – y + (x – y – 1)
√6, then:
(A) x = 1, y = 1
(B) x = 2, y = 1
(C) x = 5, y = 1
(D) x and y can take infinitely many values

how i solve this equation?

[quaint mantle]

Try pluging in the numbers in each option

If left hand side equals to right hand side, then you got correct answer

Else if more than one option satisfied both left hand side and right hand side, then (D) is the answer

[merry robin]

consider the equation x^2 - x - 1/x-1 = 1 - x/x-1
A) pi - 22/7
B) pi - 3.14
C) cos 90°
D) cos 0°

[ocean seal]

Tuk Tuk Driver

[quaint mantle]

????

[ocean seal]

Tuk Tuk Driver

[quaint mantle]

???

[barren sinew]

anyone?

[merry robin]

this one

@quaint mantle

[quaint mantle]

1. Apply LCM to cancel x-1

[merry robin]

so which one is correct?

x = 0

@quaint mantle

[alpine sable]

hello can someone help me

[white swallow]

FYI there is a physics discord server just like this one

[rapid hazel]

can't I transfer 2x outside the root?

[vale wigeon]

what do you mean?

[merry robin]

u can do 1/2x + 2 i think

[vale wigeon]

are you trying to turn $\sqrt{2x+2}$ into $2x + \sqrt{2}$?

[ocean seal]

Ann

[vale wigeon]

because that's illegal.

[merry robin]

can we do this?

[vale wigeon]

no

$\sqrt{2x+2} \neq \frac{1}{2}x+2$

[ocean seal]

Ann

[vale wigeon]

(and it's not equal to $\frac{1}{2x+2}$ either, in case that's what you meant)

[ocean seal]

Ann

[merry robin]

if we want to remove root from that, then what we need to do

[alpine sable]

fyi thats also mathematics

[white swallow]

well good for u

[alpine sable]

[merry robin]

3+√2 = 3+√6

need to make it LHS = RHS

[golden bridge]

Hello can someone explain to me how 2 multiplied by -1 = 4

[ocean seal]

Dev Shah

[craggy salmon]

@golden bridge ^

[vague coral]

4 x 2 = 8 = 1 in Z_7

[whole zinc]

ok so I have this sequence of numbers that are awkwardly defined like this:

S_1 = a+b
S_2 = (a+b)(j+l(d+kc(-a+(a+b)...)
S_3 = (a+b)(j+l(d+kc(-a+(a+b)(j+l(d+kc(-a+(a+b)...)
S_4 = (a+b)(j+l(d+kc(-a+(a+b)(j+l(d+kc(-a+(a+b)(j+l(d+kc(-a+(a+b)...)
etc.

is there a way to rewrite this so that it won't require increasingly more characters?

or to simplify it in some way so that it isnt a mess like that

[lapis sluice]

how to show that $$\sum _{i = 0}^{n} \binom{n}{i} \frac{1}{n^{i}} = 2 + \sum _{i = 2}^{n} \frac{1}{i!} ( 1 - \frac{1}{n} ) ( 1 - \frac{2}{n} ) ... ( 1 - \frac{i-1}{n} )$$

[quaint trout]

👀

[lapis sluice]

oups

what is wrong???

ffs

[ocean seal]

Danajax

[lapis sluice]

there...

[whole zinc]

how to find if this converges?:

(j+l(d+kc(-a+(a+b))))

j+l(d+kc(-a+(a+b)j+l(d+kc(-a+(a+b)))))
j+l(d+kc(-a+(a+b)j+l(d+kc(-a+(a+b)j+l(d+kc-a+(a+b)...)))

[lapis sluice]

use $\epsilon - n - \text{definiton of converging}$

[tawny fable]

Is it true that Every real number can be written as a sum of two irrational numbers?

[ocean seal]

Danajax

[lapis sluice]

try it for yourself

$\forall n \in \bR, \text{you proove that:} \ n = q + r, q,r \in \bQ$

[ocean seal]

Danajax

[tawny fable]

Do you have any suggestion how to start working with that?

[lapis sluice]

perhaps you should think of $\bQ \subset \bR$

[ocean seal]

Danajax

[lapis sluice]

thought you meant rational

sorry

[tawny fable]

Well, I know that Real numbers conatins all irrational and rational numbers.

I previously proved wrong that Every real number can be written as a sum of two rational numbers. (I proved it by saying that 2 rational numbers's sum will be always rational and that means that if a number is real and irrational it cannot be a sum of 2 rational numbers).

Yeah... and now I need to prove of prove wrong the other way. Im pretty stuck.

[lapis sluice]

yeah, i see your dilemma

[tawny fable]

It might also be true, but I don't know about that

[lapis sluice]

I believe that in some cases the sum of two irrational numbers will be irrational

[ocean seal]

Dev Shah

[craggy salmon]

@tawny fable ^

[tawny fable]

Why does it mean that it is true for all other irrational numbers? what if they are different then those you wrote.

[lapis sluice]

https://math.stackexchange.com/questions/2743302/prove-that-any-real-number-can-be-expressed-as-the-sum-of-two-irrational-numbers

[craggy salmon]

I'm talking about this question

[tawny fable]

why did you delete it?

I see @craggy salmon , but what you wrote is correct only for one situation isn't it?

[craggy salmon]

which one situation?

[golden bridge]

@vague coral @craggy salmon i didnt understand can you explain more 😅

[craggy salmon]

n can be any finite real number

[golden bridge]

Actually do you have any videos explaining the topic

[craggy salmon]

look up modulo function i guess, don't have any particular videos tho

[vague coral]

8 = 1 mod(7)

[craggy salmon]

^

[craggy salmon]

it's correct for any finite real number n and any finite irrational instead of root2

the link danajax posted has a bit more formal proof

[lapis sluice]

I am asked to proove that the solutions are on a circle in the complex plane: $z^6 + 4z^3 + 16 =. 0$

[ocean seal]

Danajax

[lapis sluice]

i have found the solutions and they have a common modulus but different argument

do you think i could argue with that statement or should i find an equation for the circle, and show that it works for every point

[craggy salmon]

that is the proof

[lapis sluice]

that is enough?

[craggy salmon]

imo yeah

might wanna explain why, i.e. with the definition of a circle

[lapis sluice]

yeah, perhaps i should ask my professor?

[craggy salmon]

that'd be better 😅

[lapis sluice]

what do you mean by that

[craggy salmon]

like just saying that a circle is the locus of a point having fixed distance from a specified point, and all the roots have the same distance from origin, hence it's a circle

[lapis sluice]

oh ok, thx then i get what you meant

I also have another dilemma

show that $$| cos z | ^2 + | sin z | ^2 = 1$$, if and only if $z \in \bR$

[ocean seal]

Danajax

[lapis sluice]

I have shown the first part of the problem, and i am stuck on showing that: $$z \in \bC \not \bR \implies \left | \frac{e^{iz} + e^{-iz}}{2} \right | ^2 + \left | \frac{e^{iz} - e^{-iz}}{2i} \right | ^2 \not = 1$$

[ocean seal]

Danajax

[lapis sluice]

and i know that $z \cdot z^* = |z|^2$, but I don't see how that could help

[ocean seal]

Danajax

[lapis sluice]

omg I think i got it this time

yeah, i solved it

*know how to solve it

[tawny fable]

If x isn't a part of the rational numbers, does it mean that it has to be a part of the irrational numbers?

Or can it be a part of any other group

[lapis sluice]

i just figured out that it is the same as cos(z-z*) = 1

[quaint trout]

@craggy salmon n = (sqrt(2) +n) + (-sqrt(2)) is not sufficient. If n = -sqrt(2) for example. Better to do cases for irrational n and rational n.@tawny fable

[craggy salmon]

even if n = -sqrt(2), that equality still holds

doesn't it?

[craggy salmon]

if it's a part of real numbers but not rational, then yeah

[quaint trout]

Yes, but then n is no longer written as a sum of irrationals @craggy salmon

[craggy salmon]

ah right

fair enough, for irrationals though, n/3 + 2n/3 also works as both will be irrational

[quaint trout]

👍

[tawny fable]

Ok here is a hard one for me:

Let x not in Q (Rationals) such that x^3 is in Q . Then x^2 + x + 1 is not in Q

Do you have any clue where do I start this?

I dont think I get it..
Does every x that is not rational has to be irrational?

Do you mean that is i say XXX is rational it means that X has to be rational? since a rational*rational has to be rational?

I don't get it. they say that X is not rational. is that a prove wrong then?

Im so confused...

[cobalt lichen]

suppose we have 40 videos in our dataset. Each video falls in at least one of the two categories, comedy videos and music videos. It is known that there are 27 comedy videos and 22 music videos in the dataset. How many videos fall into both categories?

[craggy salmon]

take x as (t)^(1/3) where t is rational

[craggy salmon]

total videos has to be 40, so, in 27 + 22 we double counted some videos. If we remove the double counted videos, the answer has to be 40

[tawny fable]

I tried to play with t^(1/3), well I do get the answer that the second polynomials is irrational but it doesnt mean that it is correct for all situations doesnt it?

I think I dont get the write way to prove this one..

[lusty garnet]

how do i do this?

idk where to ask

[craggy salmon]

liks' answer is right, factorise x^3-1

[tawny fable]

What do you mean? how to I continue with that?

[craggy salmon]

try to factorise x^3-1 in terms of x^2+x+1

not in terms, but like f(x)(x^2+x+1)=x^3-1

[tawny fable]

But how do you do this equation? no one told you they are equal to each other

[craggy salmon]

i'm not doing the equation, that's why f(x)

also an update*

[tawny fable]

I don't get it.
If someone could lead me to the solution it would be great.
Thank you so much for you help!\

[ocean seal]

Dev Shah

[tawny fable]

Damn

I wouldnt ever think about it...

Ughhhhhh

I wonna die

[lapis sluice]

$z-\overline{z}$ is always imaginary right?

[ocean seal]

Danajax

[craggy salmon]

if z is non real complex yeah

[lapis sluice]

if z is complex

[craggy salmon]

edited

[lapis sluice]

ok, thx

can i use that in a proof? Such as writing z-z* is always imaginary, therefore we can conclude that z = z*

i have prooved that $z-\overline{z} = 2k\pi$ for k is an integer

[ocean seal]

Danajax

[lapis sluice]

that means that z = z* right?

in other words z is real?

[craggy salmon]

no, that's just never true

for real z, z-z* = 0

for complex numbers, it's purely imaginary

[lapis sluice]

yes z is complex

[craggy salmon]

^ it can never be that

[lapis sluice]

yeh but just assume

it is part of my own proof

[craggy salmon]

your proof must be wrong then, i think

cause let z = a+ib, z* =a-ib z-z* can only be 2bi, where b is real

[lapis sluice]

i have written previously $e^{iw} = 1$ which means that $ iw = 2k\pi \cdot i$

[ocean seal]

Danajax

[lapis sluice]

so $w = 2kpi$

[ocean seal]

Danajax

[craggy salmon]

sure

[lapis sluice]

and previously i have written w as z-z*

i have defined that $w = z-\overline{z}$

[ocean seal]

Danajax

[fierce frigate]

is this question used?

[lapis sluice]

yes it is occupied here

[fierce frigate]

once ur finished i need help

if x = -4
what is -x^2?

[craggy salmon]

i dunno where it's wrong, but your conclusion is wrong so something's off

oh right

[lapis sluice]

hmm

[craggy salmon]

w has to be real in e^iw i think

yeah, the argument has to be real

[lapis sluice]

yes so that means that $z = \overline{z}$

[ocean seal]

Danajax

[lapis sluice]

?

[craggy salmon]

it means that this step is wrong

[craggy salmon]

since you need to assume this for that to satisfy

[lapis sluice]

that $iw \not = i2k\pi$ ?

[craggy salmon]

if and only if w is real

[ocean seal]

Danajax

[lapis sluice]

yes but that is what i want

i want to proove for w is real

[craggy salmon]

but you assumed its real first

so it's not a valid proof

[lapis sluice]

fuuuuuck

bad language sorry

[craggy salmon]

nw from me lol

[lapis sluice]

lemme just search a quick sec on web

@craggy salmon this is literally the exact same as i have done

so this is not valid?

[craggy salmon]

huh

I still don't think it is, but I've been wrong before soo 🤷‍♂️

[lapis sluice]

do you agree with the image above?

yah, perhaps i was not so specific?

but is the bottom part true or false?

I don't see how it is false

wait what does that mean

Ok, thx so much for spending time on helping

haha yes

but one sec

but how does $z-\overline{z} = 2k\pi \implies z = \overline{z}$

[ocean seal]

Danajax

[lapis sluice]

I honestly dont understand that part

[craggy salmon]

because z-z* has to be either 0 or imaginary

so it has to be

0

[lapis sluice]

I still don't get it though if z-z* is imaginary that it is equal to 2kpi

[craggy salmon]

if it's imaginary, it can't be equal to a real number

[lapis sluice]

ok so that implies that z has to be real then

= -2ib

yes

and that means that z is real then

which ofc implies that z = z*

now i get it!!!

thank you

[rugged latch]

Can someone teach me Pythagoras theorem

[lapis sluice]

a^2+b^2 = c^2

you're welcome

[rugged latch]

I'm also confused on Soh CAH Toa

[placid zinc]

If you know any two sides of a triangle, you know the third

(right triangle, that is)

[rugged latch]

Yea

[placid zinc]

Sin, cos, tan

If you know any two sides, you know all of the angles (of a right triangle)

[lapis sluice]

you mrsn pythagorean identity or pythagorean theorem?

[rugged latch]

So if ur trying to find sin, cos, tan what are the rules for them?

[placid zinc]

That's a bit general. Have an example?

[lapis sluice]

you're welcome

There are hundreds of way of finding $\sin \theta, \cos \theta \text{and} \tan \theta$ but that is the geometric way

[ocean seal]

Danajax

[lapis sluice]

you could integrate, derivate, use the sin theorem, cos theorem, use area formula etc.

[rugged latch]

Thank you

[lapis sluice]

xD

The standard rule of clicking on a link: If something seems to good to be true, then it probably is

[next zephyr]

How do i find the derivative y(dy)

[lapis sluice]

yes that does not make any sense

[distant scarab]

How would i go about plotting a vector like this to a point on a graph?
[3,0,5,7,2,11,0]

[lapis sluice]

with python?

c++?

[distant scarab]

c#

[lapis sluice]

geogebra?

idk c

[distant scarab]

well, just looking for a way to calculate it

a hidden formula

[lapis sluice]

yeh

[distant scarab]

i realise i can calculate the distance between two vectors like
[3,0,5,7,2,11,0]
[4,2,5,0,1,3,6]
with this formula, but how do you actually plot them into a graph

like on a cartesian coordinate graph

[lapis sluice]

i think that would be pretty complicated

I think MatLab is more suited for that sort of stuff

nvm MATLAB only allows for plotting 3 dimensions

[clever tapir]

can i use the channel

[lapis sluice]

yeh i suppose so

[clever tapir]

Idk where to begin bro

[lapis sluice]

"The sum of the cubes of three consecutive integers is 405"

it says what you need to do and how to begin

define and integer x

[hushed pasture]

$(a-1)^3 + a^3 + (a+1)^3 = 405$ and continue

[ocean seal]

Salah

[lapis sluice]

yeh that is one way to do it

[clever tapir]

But see (a-1)^3 isnt a sum of cubes

[hushed pasture]

no it is

[lapis sluice]

where $a \in \bZ$

[hushed pasture]

no a in Z

[clever tapir]

I thought sum a cubes where (x^3 + 1^3)

[ocean seal]

Danajax

[hushed pasture]

no it's not

[lapis sluice]

cube means to the power of 3

[clever tapir]

ok my mistake

because when I set it up like that its all just positive terms dude LOL

i was getting unreal solutions

[lapis sluice]

you should note the difference between (a^n + b^n) and (a+b)^n

[clever tapir]

The one on the left is a sum of cubes right

[lapis sluice]

I don't think u understand the definiton of a cube and three consecutive numbers

[clever tapir]

I MEAN SQURES

[lapis sluice]

3 cube = 3^3

consecutive numbers, could be 1, 2, 3

[clever tapir]

alright i got it ty

[lapis sluice]

just pretty much use this equation

[clever tapir]

the wording was confusing bro

[lapis sluice]

and you have the answer giving directly to you

[formal spoke]

Why did it become negative?

[distant scarab]

So apparently what i need to do is use some magic called T-SNE

But my feeble mind cannot comprehend the math behind it enough to utilize it in c#

So to sum up, i need to be able to display an n-dimensional vector ([3,0,5,7,2,11,0]) on a scatterplot, in c#. I can use T-SNE but I just have no idea where or how to even begin.

[lapis sluice]

jesus christ that seems complicated, good luck with that

[formal spoke]

<@&286206848099549185> Hope you don't mind but I don't understand why 2y-3 turned into 3-2y

[raw shard]

don’t immediately ping helpers

it probably got multiplied by -1

[wispy urchin]

Is this a free channel?

[raw shard]

not sure, most of the other channels are free though

[formal spoke]

What I don't understand is why the position of 2y-3 swapped

[raw shard]

multiply the whole thing by -1

[formal spoke]

Why -1? And why isn't 2y+5 isn't swapped. Is it made like that so the number with the variable is in front

[raw shard]

2y-3 = 2y+-3, -(2y+-3) = -2y+3

[tame sorrel]

What do I do next?

[clever tapir]

I solved it wanna know the answer

[formal spoke]

I'm just gonna guess the program made a miscalculation or mistake

[raw shard]

ok

[rare ledge]

how is that possible?

[bold panther]

can someone plz help me?

[lofty dirge]

can't u use calculator?

[raw shard]

probably not

[bold panther]

i dont think they want me to use a calculator

[lofty dirge]

oh okay

[raw shard]

considering that it says exact value, i’m not sure what to do

[lofty dirge]

yo what year r u in?

like skl

cause

[bold panther]

sophomore

[lofty dirge]

that big maths

um, is that like year 10?

[bold panther]

yh

[nimble wyvern]

um

you can use

[bold panther]

i can change it into degree right?

[nimble wyvern]

cos(x)=sin(pi/2-x)

[devout summit]

Try and start with cos^2(x)=(cos(2x)+1)/2

[bold panther]

kk

[nimble wyvern]

usin mine is much easier bcs you will immediatly get the value but you can use what you like

[bold panther]

kk

[raw shard]

wow that’s smart

didn’t think of that

[lofty dirge]

bro what is this, i haven't seen these equations in school in my life

[devout summit]

cos(2x)=cos^2(x)-sin^2(x)=cos^2(x)-(1-cos^2(x))=2cos^2(x)-1

[devout summit]

Rearrange a bit and you get this

[raw shard]

no that way is harder

[lofty dirge]

what year is u learning this??

[raw shard]

i tried the way @nimble wyvern suggested and it was easy

[flint badger]

After you apply: $\cos \left(x\right)=\sin \left(\frac{\pi }{2}-x\right)$

[ocean seal]

eggdog

[flint badger]

$\cos ^2\left(\frac{\pi }{12}\right)-\sin ^2\left(\frac{\pi }{2}-\frac{5\pi }{12}\right)$

[ocean seal]

eggdog

[flint badger]

You should get this

[lofty dirge]

ayoo chillll

[flint badger]

Pretty simple from there, I think

[raw shard]

yeah

[devout summit]

Umm 11th std…

[bold panther]

tyyy

my dumbass tried to switch both to sin 💀

[flint badger]

lmao

[distant scarab]

<@&286206848099549185> any of you that can give me a pointer on how to achieve this?

[raw shard]

@flint badger wait this is a dumb question, but i assume you need to apply cos^2(x)+sin^2(x) = 1, but how would that work?

unless i’m thinking wrong

[flint badger]

ok so after you apply the identity you get this right

[raw shard]

yeah

but that’s subtraction

[flint badger]

yeah then just solve for cos(x) and sin(x)

and then square that whole value that you get

[raw shard]

oh

[flint badger]

$\left(\frac{\sqrt{2+\sqrt{3}}}{2}\right)^2-\left(\frac{\sqrt{2-\sqrt{3}}}{2}\right)^2$

[ocean seal]

eggdog

[flint badger]

because

$cos\left(\frac{\pi }{12}\right)=\frac{\sqrt{2+\sqrt{3}}}{2}$

[ocean seal]

eggdog

[flint badger]

and similarly

$sin\left(\frac{\pi }{12}\right)=\frac{\sqrt{2-\sqrt{3}}}{2}$

[ocean seal]

eggdog

[tight locust]

holy shit really?

wow

why is there a square root inside a square root what the fuck

that's gotta be like a 4th degree polynomial or something

,w (x-sin(pi/12))(x-cos(pi/12))(x+sin(pi/2))(x+cos(pi/12))

[ocean seal]

Results provided by WolframAlpha

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

Requested by EndTimes#8053

[crystal narwhal]

i got 49.9 but i fail to see why there can be 2 anwsers can someone explaine pls?

oohhh

nvr mind i got it now

[rugged dust]

Does this suffice for a proof of surjectivity for f(x) = ax + b; a != 0?

Since f: R -> R f(x) = ax + b; a != 0 is a linear function with constant slope a != 0, and the domain is unlimited, ∀y ∈ ℝ ∃x ∈ ℝ : f(x) = y.

I'm doing proofs and the likes for the first time now, so I don't know how deep I have to go.

[small bear]

I don't see a proof of the most crucial part, which is f(x) = y. You talked about the domain, and it has a constant slope, but what does that say for the range.

And you'd have to argue more things, but there is a simpler way of course

You are trying to prove that f(x) = y for some x (and all y). Why don't you try to find an expression for x (in terms of y) that guarantees to give f(x) = y

Ex. If I have f(x) = x^2, I can always plug in sqrt(y) to get y in the end.

(Well, x^2 isn't surjective, perhaps a better example would be x^3, but I hope you get the point)

[rugged dust]

I see, thanks for the hint

[devout summit]

Or better yet cos^2(x)-sin^2(x)=cos(2x) so the answer is cos(pi/6)=sqrt(3)/2

[bold panther]

can someone help me with this too. I tried doing (12C8 * 7C4) + (12C7 * 8C5) and i got the number in the answer box but its evidently wrong

[bleak rose]

can someone help me with this

[craggy salmon]

You gotta make cases, like 8 in train 1 and 4 in train 2, 7 in train one and 5 in train 2 etc

[nimble wyvern]

you can simply multiply 2 and 10

because they have the same power

[soft crag]

Guys can u actually get a paying job with no gcses?

Or after failing em

[bold panther]

but how do i make the cases?@-@ like for case 1 would i do 12C8 *4! or 12C8 *4?

[craggy salmon]

No, for case 1 it'll just be 12C8, as they didn't specify any position in each trolley that matters, i think

If that doesn't work, try 12C8*8!*4! and so on

[glass lichen]

If you know one solution... what do you need help with?

[mighty grotto]

can i know the way to solve that :0

I'm interested

alr

that's all :0?

Coooool

[mild plover]

are angles between vectors unique? For example if an angle between 2 vectors is π, would 2kπ + π be all the angles where K is an integer

[bleak cobalt]

Occupied?

[frigid gorge]

help

[glass lichen]

Yes there is 1 angle, however of course any extra rotation looks the same. But we just say the acute one is the angle

[lime tangle]

RS is ur slant height

do you know how to calculate surface area of pyramid

[frigid gorge]

no

[lime tangle]

so learn how to do it

here ill give it to you

[feral lodge]

can someone solve me step by step and explain this to me

[lime tangle]

$A_s = A_b + 0.5(P_b \times s)$ \ where $A_b$ is area of base, \ $P_b$ is perimeter of base and \ $s$ is slant height

[feral lodge]

0,9(200-100(1+r))+0,1(-cl)⩾60

[ocean seal]

IAMTHEFARMER

[alpine sable]

Hi

How to do part b?

Please explain with diagram

[lime tangle]

do you know ur kinematic equations

@alpine sable

[alpine sable]

yes

[lime tangle]

this seems like physics to me

[alpine sable]

this isn't

it is "mechanics 1 "

[lime tangle]

thats physics

basically

mechanics 1 is physics

anyway

[alpine sable]

We do it in maths but anyways

How do we do it?

[lime tangle]

ok so i always write down information

first

[lime tangle]

we know the particle is decelerating which means that your displacement-time graph will look like a negative parabola

[alpine sable]

I could do part a

got 2 seconds

But can't do part b

[mellow raft]

Hey I was wondering where would I put my question for BioMathematics?

[lime tangle]

and your VT graph will be positive

[alpine sable]

And?

How do you use the SUVAT formulae to do part b?

[lime tangle]

as it is moving in the + direction

[lime tangle]

never heard of suvat i just memorize the equations

[alpine sable]

My teacher shows us

But I don't get how she got the 4

For part b

Any clue?

[lime tangle]

well she used formula 4 i think

[alpine sable]

it gave her 24.5

[lime tangle]

yup she used equation 4

[alpine sable]

So how did she get 4?

[lime tangle]

she used this equation

[alpine sable]

v^2 = u^2 + 2as ?

[lime tangle]

wait

lemme write it

[lime tangle]

$v_f^2 = v_i^2 +2a\delta d$

[ocean seal]

IAMTHEFARMER

[alpine sable]

@lime tangle Exactly what i said

[lime tangle]

yeah

[alpine sable]

The equation gave her 24.5

How did she acquire 4 from that?

[lime tangle]

the question is asking for total distance

which is delta d

in letter a you got?

[alpine sable]

a is 2 seconds

[lime tangle]

ok so

[alpine sable]

s = 24.5

How did teacher get 4 m ?

[lime tangle]

why does ur teacher say decerating when it has a positve accel and positive velocity

its moving in the + direction with a positive accel and velocity

why does the problem say decel

anyway

[lime tangle]

@alpine sable v initial is 14

[alpine sable]

negative acceleration

[fading patrol]

15 pls

[lime tangle]

@alpine sable i am trying to figure the problem out

i am drawing a vt graph

but im pretty shit at physics tbh

at least this isnt variable accel phew

[late parcel]

I have a sciency/ math definition question. If a population has a high morality rate does that mean more people die or live that a population with a low morality rate

[lime tangle]

high mortality means a lot of people die

low mortality means not many die

[wintry prawn]

mortality just means death

So death rate

[lime tangle]

looking at ur teacher's solutions, she but v final as 0 and v initial as 14

[late parcel]

Thx

[lime tangle]

then you just plug in the accel (-4) and find delta d

[keen star]

Anyone know AP Calc?

[lime tangle]

im also confused as to why she got 4m

[late parcel]

Is this channel open

[lime tangle]

yes

[late parcel]

Am I doing number 5 correctly

[lime tangle]

hmm

well country A starts better

but at the end it reaches the same

its a rapid decline in life expectancy for country a between 3-5

but the fact that country b always has the lower survivorship for the most part i would say country A as well

so yeah i would put country A

[late parcel]

Thx

[lime tangle]

however

[late parcel]

?

[lime tangle]

do keep in mind that at age group 5 they are the same

both countries are the same at age group 5

so that was the tricky part of this question

but u did it correctly

on average, country A is better

[late parcel]

So A is better because

It has a higher morality rate?

[lime tangle]

its better because its average mortality rate is lower than country B

in other words, country As avg life expectancy is higher than country Bs avg life expectancy

therefore country A is a better place to live

[final tartan]

this channel free now?

is the answer just n(B)= 12 n i dont have to calculate anything?

[crystal narwhal]

can anybody explain to me why y = sin 3x results 3 periods within the orginal period of sin x i know how (i did the table and graphed it) but i dont get why

[daring prawn]

you are inputting a larger number (3 times x) so the output oscillates faster/frequency is higher and so it looks horizontally compressed

[past ivy]

because its as if the graph is being compressed by a scale factor of 3. every value of x is graphed as if it is triple the value

[placid venture]

Hi guys, I'm struggling a lot with this question from my econometrics problem set and I was hoping one of you could help me out.

To answer it I essentially follow this proof from the textbook

using X_i^2 instead of just X_i
naturally that (XiX')^-1 doesn't cancel out with X_iX_i'
so the end result is something other than zero
so the statement is false
I think that's correct but I can't figure out the "intuition" for the last part that asks me to explain why this is the case

[cyan basalt]

x(x+1)+(x+1)(x+2)+···+(x+n−1)(x+n) = =1·2+2·3+···+(n−1)n (n ∈ N, n>2016).

How to solve it

[alpine sable]

I got it

[placid zinc]

x = 0 seems like a good idea

[alpine sable]

Your method is wrong

It should be 2 x ( 24.5 - 22.5)

@lime tangle

[hollow tide]

can anyone help with the very last part

[upper escarp]

not sure if this channel is still in use but ill post my q here

how can i solve this q?

[alpine sable]

Im not fully sure but you could start by using the formula for sphere, substituting the values and solve it

[dawn plaza]

how would i use that formula in implicit differentiation?

[buoyant kayak]

F_x is taking the derivative treating y as a constant, vice versa for F_y

first get it in the form it tells you though, F(x,y)=constant

[winter wing]

can someone explain why the numbers go where they go

how do you solve this step by step

[sturdy nova]

You multiply like terms @winter wing

[mellow linden]

y=-f(x+1)-3

the - negative symbol in front of f is reflection across the x axis?

so it would be (-x-1, y-3) for mapping notation?

[oak chasm]

No, y is negated by it. It is the same as -y = f(x + 1) + 3 if you negate both sides, right?

@mellow linden

[mellow linden]

what does negate mean

[oak chasm]

Change the sign of.

Multiply by -1.

[mellow linden]

i’m kind of confused

[oak chasm]

Well, you are allowed to do the same thing to both sides, right?

[mellow linden]

i’m not sure

[oak chasm]

When you solve an equation, like 3x = 6, what do you do?

[mellow linden]

divide

[oak chasm]

<@&268886789983436800>

[mellow linden]

but i’m not solving for y though

[oak chasm]

Right, you divide both sides by 3, right?

[mellow linden]

yea

[sly mantle]

b&

[oak chasm]

So doing the same thing to both sides is allowed.

[mellow linden]

with the negative sign?

[oak chasm]

If we multiply both sides of your equation by -1, we see that y has a negative sign, so it is really y that is flipped.

[mellow linden]

oh wow

but let’s say

y=-3f(x+1)-3

would it still be relfection across the y axis?

[oak chasm]

No, it is across the x axis for both.

Changing the sign of a y coordinate gives you a point on the other side of the x axis.

[mellow linden]

ok

thank you for clarifying

for vertical reflections

[mellow linden]

like the ones in the bracket

[oak chasm]

(2, 5) is above the x axis. (2, -5) is below the x axis.

[mellow linden]

they will always be vertical?

[oak chasm]

If you change x itself, that will affect the x coordinate.

[mellow linden]

Yes

[oak chasm]

y = f(x) and y = f(-x) will change the x coordinate since you changed x itself.

y = f(x) and y = -f(x) changes y, as seen before.

[mellow linden]

👍👍👍

ty

[oak chasm]

No problem.

[misty mortar]

Can someone help me with a problem?

[oak chasm]

@misty mortar Please read the rules and tips for getting help in #❓how-to-get-help.

[misty mortar]

How do I solve this problem?

<@&286206848099549185>

[alpine sable]

Didn’t even look at the work but why is that your answer, based on what you got from your work?

@misty mortar

[misty mortar]

The answer I got is that if x is less than or equal to 0 then it converges

[alpine sable]

Why?

[misty mortar]

Can someone help me with this problem?

[charred flint]

@misty mortar negative numbers don't work either because it'll go like -1 +2 -3 +4 -...., going back and forth between bigger and bigger numbers

[misty mortar]

What will go back and forth?

[glass lichen]

the partial sums will alternate b/w +-1

[misty mortar]

Why will it alternate between 1 and -1?

[glass lichen]

check it for yourself

im not gonna do simple sums for you

[vagrant rover]

you can't write a vertical line in slope-intercept or point-slope form, right?

[quartz osprey]

how do i sovle for 0s for the first derivitive

arent the zeros only 0?

[glass lichen]

yes

[quartz osprey]

oh wait

here

so is this viable

2xe^x^2-4=0
divide by 2x and ln it

[glass lichen]

no

[quartz osprey]

oh

alright

[glass lichen]

you're solving $2xe^{x^2-4}=0$

[ocean seal]

Mosh

[glass lichen]

so either 2=0, x=0 or e^(x^2-4)=0

2 of which are clearly impossible

[quartz osprey]

im ngl i dont get it but ill stick to 0 lol

[glass lichen]

how would you solve a quadratic like (x-2)(x+1)=0?

[quartz osprey]

set them = to 0

both of them

[glass lichen]

yep

you do the exact same thing

[quartz osprey]

oh i see

yea its impossible for e^x whatever to = 0

[glass lichen]

yes

since e^x>0 for all x in R

[quartz osprey]

yes

[harsh steeple]

is there a symbol that can replace "or"

[sly mantle]

we can say $\lor$ but "or" is very clear

[ocean seal]

RokabeJintaro

[harsh steeple]

ya i was just wondering, the symbol just seems to me as more universal

[quartz osprey]

confused on how to find the absolute minimum

i found the zeros

which is 0

only i think

0^2-4 is -4 right

[sly mantle]

most math ppl say "or" bc it's very clear

[amber depot]

@quartz osprey Abs min on a graph is the lowest point

Abs max is the highest

[quartz osprey]

yea

[amber depot]

FIrst, graph it. In your case you are given an interval

graph between the interval, see where the lowest point lies

[quartz osprey]

hmm is there a way to solve without plugging into a calculator

i kinda wnat to do everything without a calculator

[amber depot]

Do ti without a calculator xd.

I would do it without a calculator

bruh

[quartz osprey]

wait what

oh

graph it without

xd

[amber depot]

https://www.youtube.com/watch?v=qLEtZ8L61rI

[quartz osprey]

well is ther a way to solve it without graphing is what i mean

[amber depot]

If you don't know how to do it

hm

Let me think

[quartz osprey]

11min fuck :pepeosadcrfy:

[amber depot]

it is helpful

Grapch and then you should be able to see it or use points

to grapch the function

[quartz osprey]

ok well let me try that then

but i really rather do without a calculator cuz idk how to fucking use one

[glass lichen]

By Fermat's Theorem, extrema occur at endpoints, when f'(x)=0, or non-differentiable points

[amber depot]

ignore what I said

[glass lichen]

you also just have the 0 and 1/e^4 in the wrong spots

at x=0, f(0)=1/e^4

[quartz osprey]

wait what

wait

[glass lichen]

you have it right, just swapped.

[quartz osprey]

im so mad

alright htakns

i didnt need the second derivie for this one

it seem s

[amber depot]

Im sorry, I don't know this topic well. Just know that to find the abs min/max you can grapch the equation.

[quartz osprey]

yea

alright thanks guys

[upbeat heath]

mmm

what's the other point of 0,3 with a slope of 1/2

it's my last question and I'm too lazy to do it

🥲

[trim folio]

A pot of soup is one-fourth full. When 20 ounces of broth is added to the
soup and mixed thoroughly, the pot is two-thirds full. How many ounces
of soup would there be in a full pot?

[placid zinc]

Any thoughts on the problem so far?

[trim folio]

me?

ii have slight progress.

THIS IS THE ANSWER THEY GAVE;The 20 ounces of broth added takes up 2/3 - 1/4, or 8/12 - 3/12, or 5/12
of the pot. One-twelfth of the pot would be 4 ounces. A full pot,
therefore, would be (4) (12), or 48 ounces.

I got everything else

just not how they got 1/12 = 4oz

[placid zinc]

5/12 of the pot is 20 oz

So 1/12 is 4 oz

By dividing both sides by 5

[trim folio]

wait had a brain fart

how did I not do that

ty

[placid zinc]

Haha np. Feel free to ask if you have anything else

[stiff idol]

Im so confused

[wary stream]

You have two choices, R and L, what are the chances you get R in branch b?

[stiff idol]

idk like 5 or something lmfao

Ive got no idea

[wary stream]

Without replacement

Meaning it doesn't get put back in

[stiff idol]

hmm

is it 6

[wary stream]

No

[charred flint]

is that question even well-posed

[wary stream]

The percent is even for both

[stiff idol]

oh

so 4

[wary stream]

Yes

[stiff idol]

so D and E is 4/8

oh crap my bad is B and C 4/8

[wary stream]

Because initially, you have 9 cards, labeled R and L, the chance of getting R the first time is 5/9, then you don't replace so you have 8 cards remaining, so you have 8 cards

[stiff idol]

OHHHH

[wary stream]

Hopefully that's enough for you to solve it all

[stiff idol]

I got it

I think

[wary stream]

You add

[trim folio]

What is the greatest number of consecutive positive whole numbers whose
sum is 85 ?

[alpine sable]

10

4+5+6+7+8+9+10+11+12+13

[glass lichen]

dont give answers.

[alpine sable]

It doesn’t exactly say that in the rules

[glass lichen]

It actually does say so.

"We are not here to do your homework for you, but we are here to help you learn how to do your homework."

Direct from #rules

[solar pebble]

(I did break that rule in #help-4 but oh well)

[glass lichen]

Yeah, you did their homework/question for them... which goes against this rule/bit of common sense.

In regards to King's deleted message.

[alpine sable]

I deleted it because I realised you typed it

[glass lichen]

Ye

[placid zinc]

Note that 85/10 is 8.5, which is the average of those added numbers. Gives a nice way to find the answer

[solar pebble]

In my defense, they already had the answer and did have the "why not what" mindset

[quartz osprey]

i dont get this question B isnt it just plugging into the derivitive

and if so

why do i keep getting a negative number

[glass lichen]

cause it's decreasing