help-10

[empty maple]

Yes but I don't know how can I rewrite y in terms of x when $y = ax^2 + bx$

[warm shale]

MetuMortis

[marsh geyser]

First of you have to put some restrictions

If not, that function does not have inverse

[empty maple]

Like a and b is not equal to 0?

[marsh geyser]

It is about x

x^2 does not have inverse

But x^2 for x>=0 does have inverse

[empty maple]

Sorry for not giving details, x is time and is positive

[obtuse pebble]

@empty maple Has your question been resolved?

[marsh geyser]

You can complete the square

Or use quadratic formula

[empty maple]

This makes sense

[marsh geyser]

$y=ax^2+bx$

[warm shale]

Samuel

[marsh geyser]

$x=ay^2+by$

[warm shale]

Samuel

[marsh geyser]

$ay^2+by-x=0$

[warm shale]

Samuel

[marsh geyser]

Take x as c

And solve for y

[empty maple]

Can I complete it to a square no matter what are a and b?

I guess I can: $y^2 + \frac{by}{a} - \frac{x}{a}$

[warm shale]

MetuMortis

[empty maple]

Which is equal to $(y + \frac{b}{2a})^2 - \frac{b^2}{4a^2} - \frac{x}{a}$

[warm shale]

MetuMortis

[empty maple]

Thanks

.solved

[obtuse pebble]

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[obtuse pebble]

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[dreamy fjord]

How can I factorise x^3 - 1

[tender tusk]

like difference of 2 cubes?

(x-1)(x^2-x+1)

[dreamy fjord]

yes, but how do you get to that

I was sick when we went through this at school

[tender tusk]

theres a more general formula for different powers

[dreamy fjord]

oh, I see

thanks a lot!

.close

[obtuse pebble]

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[timid silo]

how many 50 x 50 grids of white and black cells are there such that

1. no two white cells touch
2. every black cell touches a white cell
   (diagonal touching by corners counts as touching)

[deep haven]

alright lets go

@zenith raft

[zenith raft]

hello

[deep haven]

for 1x1 theirs 1 for 2x2 theirs 4

now for 3x3

4 rotations of BBW/WBB/BBW

2 rotations of BWB/BBB/BWB

[dusk widget]

round 2, eh?

[deep haven]

and BBB/BWB/BBB

any others?

ah wait

[timid silo]

guys, if we made it 1 d it maybe easier

[deep haven]

WBW/BBB/WBW

[dusk widget]

someone ping the original asker

I don’t remember their username

[deep haven]

that's layla

alr pinged

oh wait

im stupid

[dusk widget]

no

[deep haven]

lol

ok that's 8 for 3x3 are there any others?

[zenith raft]

let them start it over it will be more fun

8 is correct

[deep haven]

i'm gonna periodically update this:
1x1: 1
2x2: 4
3x3: 8

[timid silo]

a black cell an touch 2 white ones right

[zenith raft]

yep

[deep haven]

did the helpful ping thing get accepted yet

2^(n - 1)

WBBW/BBBB/BBBB/WBBW

[oak escarp]

[deep haven]

well first theirs the square packings

so

WBWB/BBBB/WBWB/BBBB (4 rots)

which has 4 rotations

now the diamond packings

WBBB/BBBW/WBBB/BBBW (4 rots)

4 rots

also this asymetrical one

WBBW/BBBB/BWBB/BBBW (4 rots, 2 refs)

[timid silo]

how about we start at 1d

and then we go into higher dimentions

[deep haven]

for 1d it seems trivial...

yeah

for an n long thing

lemme cook rq

[timid silo]

array of n points

[deep haven]

ok densest possible packing is WBWB....

[timid silo]

how many ways to arrange black and white points in that array

[deep haven]

least dense is WBBWBBWBB...

at each one theirs a choice

1 or 2 B's before next W

[abstract beacon]

for the asymmetrical one you dont jsut have 4 rotations but also 2 reflections of each

[deep haven]

ah yeah

[timid silo]

so for 4 points array, how many possibilities are there?

[deep haven]

wait im cooking

ok lets say you want m W's to fill up an n long array

if 3m < n then there are 2^m ways

if 2m = n then there is 1 way

now for the weird cases

ok for n=1 its 1

for n=2 its 2

for n=3 its also 2

ok lets say that 2a + 3b = m

so for each solution in naturals of that equation, you get 2^{min(a, b)} possible arrangements

ok i think i've solved it for nx1 arrays

@timid silo

how many solutions in naturals for given m?

[timid silo]

what

i was looking at my paper

wym

[deep haven]

ok so you want to fill a nx1 array with up to m white ones right

[timid silo]

yes

[deep haven]

now you can have a gap of up to 2

so repeating things of either WB or WBB

let a be the number of WB's and b be the number of WBB's

we have 2a + 3b = n

for any a and b there are 2^{min(a, b)} possible solutions resulting from it

times 2 due to reflection

[timid silo]

why do you assume it starts with W

why can't it start with BW

[deep haven]

...times 2 due to reflection

tf is an E

so we have

wait

[timid silo]

BW works

but not BBW

[deep haven]

ahh yeah

ok

so you solve it for the limited case for n and n+1 simple

[timid silo]

can we forget about the rules laid by the problem for a second

[deep haven]

now for any natural n, how many natural numbers a and b exist such that 2a + 3b = n

[timid silo]

for m white points how many ways to fill an n array?

without any rules

[deep haven]

easy

yeah just combinations

[timid silo]

how

[deep haven]

wait

uh

why are we doing this

do you know

how to find

how many solutions of 2a + 3b = n

this is diophantine equation

n - 2a must be a multiple of 3 so for any a there is a 1/3 chance it works

so n/2/3 = n/6?

if im right

then for an nx1 array there are (2n + 1)/6 solutions

[timid silo]

for 2 array you will know youa re not right

[deep haven]

except for 1 and 2

only works for 3 and more i think

[timid silo]

2n + 1 is always odd, it is not divisible by 6

[deep haven]

shit

wait

how do we find

possible (a, b) where 2a + 3b = n

[timid silo]

can we just do my easier version

[deep haven]

ok sure

expand that

wait

for m = 1 its n

for m = 2 its n*(n-1)

works up to m = n/2

[zenith raft]

hmmm

[deep haven]

wait

did you see my thing

[deep haven]

this stuff

[deep haven]

so then double that

wait

[zenith raft]

i haven't been reading everything. i know why you would care about the number of solutions to 2a + 3b = n but not sure what you're doing with it

[deep haven]

well i also care about the specific form

the specific (a, b)

[zenith raft]

right, because you can permute the 2s and 3s

[deep haven]

because if you have a list of a, b pairs for n, the restricted solution for nx1 is the sum of 2^(min(a, b)) for all (a, b) pairs, the full solution for nx1 is the sum of the restricted solutions for n and n+1 (because you can have a B at the start which restricted doesn't account for)

i think

[timid silo]

no

in m = 2 and n =3 the only possibilities are

WBW WWB BWW

[deep haven]

wdym

what are your rules lol

oh

no spacing between white ones

uhh

i don't get it

what are your rules

[deep haven]

can someone verify this

idk if its right

well i mean

verify the reasoning

as to why its right

[timid silo]

no rules

m white points in an n array

organize them as you want

[deep haven]

ok

what about WBB

lol

wait...

its different

well

my 2a + 3b = n thing is wrong i forgot about smth

[zenith raft]

for n = 9 the only solution is (0,3), but there is more than 1 grid

[timid silo]

m=2

[deep haven]

that's the restricted solution

[zenith raft]

what restriction?

[deep haven]

the full solution can have a B before or have a B after

ah wait

restricted thing is incorrect

no its not

[deep haven]

composed of WB and WBB only

anyway so

wait

the ending one is different

a solution matches regex

B?(WB|WBB)*

incorrect but its progress

oh wait

can't you just add an ending

either W or WB

so then

restricted(n-1) + restricted(n-2)

but you can also have an optional B at the beginning

so

restricted(n-1) + 2*restricted(n-2) + restricted(n-3)

where restricted(n) is the sum of 2^{min(a, b)} for all natural (a, b) where 2a + 3b = n

yeah, i think this works

do you guys understand?

ok bye

sorry

[zenith raft]

huh

it is an idea

[obtuse pebble]

@timid silo Has your question been resolved?

[timid silo]

BINARY NUMBERS

[obtuse pebble]

@timid silo Has your question been resolved?

[deep haven]

ok im back guys

@timid silo

wdym binary numbers

oh yeah

that makes it much simpler LOL

wait

if we can come up with a valid(x) function that returns 1 if the binary expansion of x, treating 1 as white and 0 as black, is valid, and 0 if it isn't

then it's just

$\sum_{k = 1}^{2^n} valid(k)$

[warm shale]

speedydelete

[deep haven]

ok

now

actually i'll wait for someone else and yap about 2a + 3b = n for now

ok

n = 1: nothing

n = 2: (1, 0)

3: (0, 1)

4: (1, 0)

5: (1, 1)

[timid silo]

f(1) = 1
f(2) = 2
f(n>2) = f(n-2) +1

[deep haven]

1, 2, 2, 3, 4, 5, ...

wait

the a and b don't matter

its just their min

6: (3, 0), (0, 2)

7: (2, 1)

ok lets call it sol(x)

and sol2(x) is only the minimum of the pair

wait

if n = 2a + 3b

then b <= n/3

and a = (n - 3b)/2

so

half of possible b values work on average

so floor(n/6)

now which is bigger, a or b

which is bigger, (n - 3b)/2 or b

,w graph (n - 3x)/2 < x

,w graph (n - 3b)/2 < b

[warm shale]

Results provided by WolframAlpha

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

Requested by speedydelete

[timid silo]

did you check this?

[deep haven]

that produces 1, 2, 2, 3, 4, 5, 6, ...

for n=5 i can only find 4 whats the 5th

10101, 01010, 10010, 01001

[timid silo]

exactly

f(5)=4

[deep haven]

oh

for n=6

101010, 010101, 100101, 101001

aren't those the only ones

010010

ah yeah

ok whats your proof

[deep haven]

so if b < n/5 then a is smaller

so it's

wait

[timid silo]

no proof

trial and error

[deep haven]

well can you help me with my thing

what is the sum of 2^(the smaller value in the pair) for all ordered pairs (a, b) where 2a + 3b = n

[timid silo]

i would if I understood it

[deep haven]

do you understand this?

[zenith raft]

i can chime in and say this is wrong, f(1,7) = 7

[timid silo]

why how

[deep haven]

ok i am gonna do a computer simulation

i'll get back here in about an hour and a half ish

bye

[timid silo]

oh wtf

[obtuse pebble]

@round scroll Has your question been resolved?

[deep haven]

@round scroll I FIGURED IT OUT

well chatgpt helped

f(n) = f(n - 1) + f(n - 5)

and ik why

well i reverse engineered the reasoning from the formula

[zenith raft]

what values do you get?

[deep haven]

1, 2, 2, 3, 4, 5, 7, 9, 12, 16, 21, 28, 37, 49, 65, 86

here's why

there are 2 ways to extend a sequence

if it ends in B, you can extend it with W or WBBWB

if it ends in W, you can extend it with B or BWWBW

@zenith raft

[zenith raft]

yes that looks right

[deep haven]

ok

now if i have a solution for nxm, then all the rows, columns, and diagonals must work independently

now is the converse true?

[zenith raft]

good question

[deep haven]

i think it is

if it is, then it's much simpler

[zenith raft]

:worry:

WBBW works, but

WBBW
WBBW

does not

[deep haven]

the columns don't work, even tho the rows do

oh wait

no

[deep haven]

this is not true actually

counterexample:
WBB
BBW

all the rows, columns, and diagonals being good is stronger than it being good

[zenith raft]

that was my point but maybe i misunderstood what you were saying

[zenith raft]

whatever, never mind that

[deep haven]

if the rows, columns, and diagonals all work, then it works

however the converse isn't true

[zenith raft]

right

[deep haven]

ok how to extend a row downwards

wait

what about nx2

so

[obtuse pebble]

Channel closed

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[deep haven]

.reopen

[zenith raft]

you may just want to start a new channel

[deep haven]

yeah ig

[zenith raft]

since abstract algebra simp left the server

[obtuse pebble]

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[obtuse pebble]

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[steady yew]

Hi. Is there a book that talk and explain these topics in a simple way.

[tardy epoch]

#book-recommendations or #calculus

[steady yew]

ohh, sorry I don't know about this channel

.close

[obtuse pebble]

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[wanton wharf]

need help on this equation

[wanton wharf]

I need to use, by factoring, completing the square, and quadratic formula.

need explanation cuz, my teacher told me to create an inforgraphic brochure

[high lily]

is that supposed to be 6x?

[wanton wharf]

yup

[solar meadow]

OK, What can you factor from the whole equation?

[warm shale]

ren

[last pilot]

for later reference

[last pilot]

note that you can easily factor out a two here

[wanton wharf]

yup, 2x8

=16

[last pilot]

...?

where did-

that come from

[wanton wharf]

Am I wrong?

[last pilot]

no i just don't get where and why you got 2x8 = 16

it's correct

i just don't see how it fits in

[wanton wharf]

I completely forgot on how to haha

[last pilot]

okay notice

[wanton wharf]

then find the sum of the second term right?

[last pilot]

$2x^2 + 6x + 8$ is $2(x^2 + 3x + 4)$

[warm shale]

ren

[last pilot]

oh you're attempting to find the roots; i suggest you don't do that

they're irrational here

mb i mean complex

by the way are you absolutely sure this is the equation

[obtuse pebble]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[zealous shell]

use the quadratic formula

[last pilot]

apparently they need an explanation too

[zealous shell]

oops wrong person

[last pilot]

here it is fyi

[last pilot]

no right person just late

[zealous shell]

i need a tutor for my math also

[last pilot]

i don't think this is the place to ask; tbf i'm not sure where you'd ask

[zealous shell]

i have an exam soon

[last pilot]

@wanton wharf fam you alive

[zealous shell]

and my tutor flak on me

[wanton wharf]

yup

[zealous shell]

last min

[wanton wharf]

just checking on my notes, if I am right

[obtuse pebble]

Please read the channel description before posting, and stay on topic.

[wanton wharf]

um

[last pilot]

alr

just show me the original equation will you

[zealous shell]

is pemdas with the quadratic equation?

[wanton wharf]

nevermind, I got it now haha

[last pilot]

what?

!topic

[obtuse pebble]

Please read the channel description before posting, and stay on topic.

[wanton wharf]

thanks for all the help though, I was having a mental block

[last pilot]

nw

[wanton wharf]

@last pilot

[last pilot]

where tf did that come from

[last pilot]

great

[wanton wharf]

should I close now?

[last pilot]

sure

[wanton wharf]

!close

[last pilot]

.close

[obtuse pebble]

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[opal totem]

I need to apply a eulers totient function to a non-whole number, how would you do this?

[red ice]

you probably did one of the earlier questions wrong then

[opal totem]

i mean maybe but is it possible?

or like "technically" doable

[zenith raft]

[red ice]

no, Euler's totient function is defined for natural numbers only

[opal totem]

i see

oki lemme do the first one here then

ok so

its (13 +x)/(13*19 + x) right

for f_1(x)?

[opal totem]

1/this to get f_1(x)

,help

[warm shale]

A brief description and guide on how to use me was sent to your DMs!
Please use ,list to see a list of all my commands, and ,help cmd to get detailed help on a command!

[opal totem]

,list

[surreal forge]

there appears to be a mistake

[opal totem]

wdym?

,tex f_1(x)=\frac{1}{\frac{13 + x}{(13)(19) + x}}

[warm shale]

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

[obtuse pebble]

@opal totem Has your question been resolved?

[obtuse pebble]

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[obtuse pebble]

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[round pewter]

So I understand range is -inf < x < inf

What do I do because it stops in the middle

Like the line

Is it still -inf<x< inf

I lowk think it's like -inf < x < 1 and then there's another one like 1 < x < inf

[restive acorn]

Do you mean domain or range when you say that

[round pewter]

Domain sorry

I also noticed it's neither not discreet

[restive acorn]

Yea should be cont.

Ok so

You see how

f(1) is defined

[round pewter]

No our teacher said it's niether

[opal gyro]

it's not continuous

i think

[restive acorn]

I'm sorry yes my bad

I was thinking in regions not overall

[opal gyro]

It's definitely not discrete.

[round pewter]

Yay nay?

[restive acorn]

Consider this

You agree that something is defined at x = 1

[round pewter]

Yes

The end of line

[restive acorn]

Then the domain includes 1

[round pewter]

Yes

[restive acorn]

You dont need to skip.over it or do a weird construction

Not in this case

[round pewter]

😔

Okay

[opal gyro]

it is -∞<x<∞. Why do you think that might be?

[round pewter]

So it's just -inf < x < inf

[restive acorn]

Because although there is a strange behavior occurring at x = 1, there is nonetheless something defined there

[round pewter]

Because it goes on forever it's just when the line ends it confuses me 😔

[restive acorn]

I can point to x = 1 and say yes that location is part of the graph

[round pewter]

Yeah so it's just part of the x

[restive acorn]

If it was 2 open circles with a similar graph

Then you'd say

Hold on

[round pewter]

Okay thanks, wish me luck on my quiz in literally 5 minutes 😔

[opal gyro]

It ends, but it doesn't have two output values at that point. The reason it ends is because it's not continuous.

[restive acorn]

X = 1 isn't defined

[restive acorn]

Good luck!

[round pewter]

Okay bye bye

[opal gyro]

x = 1 is defined.

[round pewter]

.close

[obtuse pebble]

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[restive acorn]

Sorry was talking about another example

[opal gyro]

oh ok

[restive acorn]

Where it would be similar with 2 open circles

[obtuse pebble]

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[vocal hatch]

is this even possible with these units i got

[vocal hatch]

@sonic rapids

<@&286206848099549185>

this might be more physics than mathematics

.close

[obtuse pebble]

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[tiny kelp]

<@&268886789983436800>

[obtuse pebble]

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[oak escarp]

yeah shut the fuck up

[tiny kelp]

I saw it

[oak escarp]

(the troll to be clear)

[tiny kelp]

See #help-35

[obtuse pebble]

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[real shard]

i dont know how to solve this

<@&286206848099549185>

i think we start with nth root, but I got nowhere with that

[obtuse pebble]

@real shard Has your question been resolved?

[obtuse pebble]

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[royal abyss]

,rotate

[real shard]

which one do you have a question on

[royal abyss]

in relation to question c based on the lecture notes i have been provided i do not see a solution to C, can this be solved without using algebra, for context it is in a chapter on percentages and weighted means

[real shard]

since 24/40 i also equal to 6/10

we can assume that he got a 60% of the test and 60% of 30 is 18

wait no im dumb that doesn't work

[royal abyss]

i was trying to get 40% of 70% but my intuition tells me that is wrong

the combination of two different percentages is confusing me

i know now though that i need to figure out what is 22% worth out of 70% but this still doesn't compute in my brain

is (0.7) x (0.22) = 15.4% correct?

[obtuse pebble]

@royal abyss Has your question been resolved?

[royal abyss]

it doesnt seem that is anyway correct...

[obtuse pebble]

@royal abyss Has your question been resolved?

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[rare scroll]

I am not sure if what I have done for b) is correct

And I have the markscheme so I tried working backwards

If anyone could confirm or help me out I would really appreciate it

[obtuse pebble]

@rare scroll Has your question been resolved?

[rare scroll]

<@&286206848099549185>

😭😭

Someboddyyy pleasserr

<@&286206848099549185>

😢

[tame ibex]

the photo is too small

zoom into the question

[rare scroll]

Omg i love u

Ok

Dawn u r my favourite freaky penguin

[tame ibex]

omg i hate this topic

😭

what part you struggling on

part b?

[rare scroll]

Yeah I have the answer too

But I think I’m missing smthn

Like I’ve done it

But idk if I’ve done it right

[tame ibex]

z + z^-1 gives you 2cosx

which when square gives 4cos^2x

then expand the z's

[rare scroll]

Yep

[tame ibex]

and equate

to write cos^2x

[rare scroll]

Wait chillll

[rare scroll]

But there are no Zs anymore!

?

[tame ibex]

thats using de moivres theorem

now ignore that theorem

and just expand it

like z is a variable

[rare scroll]

Oh fr

[tame ibex]

binomial expanision

[rare scroll]

Alr can u give me a minute

[tame ibex]

no worries

gl

the mistakes generally come from the binomial expansions

[rare scroll]

Wait

So

After I expand (z+z^-1)^2

What do I do then

[tame ibex]

yeah

you can start grouping them

the first term is z^2 right

and the last term is 1/z^2

[rare scroll]

Yeah

[tame ibex]

which you can combine

to be a sin or cos term

depending on the symbol\

[rare scroll]

Bomboclat I’m gonna give it a try

Thanks boss

[tame ibex]

with the coefficient of the angle being the power

i think

shady area

[obtuse pebble]

@rare scroll Has your question been resolved?

[obtuse pebble]

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[gray kindle]

if i took the square root of lets say x^2, if x was originally negative, would the output of the square root of x^2 be negative as well? because if you just simplified it first and then plugged in x, it would just be the original answer(sqrt(x^2)) = x, but x^2 always outputs a positive value, so im confused

[unreal musk]

sqrt{x^2} = x is only true when x is nonnegative, in general you have sqrt{x^2} = |x| when x is real

[gray kindle]

the absolute value of x is always positive though

[unreal musk]

In other words, if x was negative, then sqrt{x^2} would be made positive

[gray kindle]

so if i took the limit of something like sqrt(x^2) as x approaches negative infinity the answer would be infinity?

[unreal musk]

Correct

[gray kindle]

wait but my teacher told me its negative

[unreal musk]

(and the square root of any nonnegative number is also nonnegative, hence the absolute value)

[unreal musk]

What exactly did they say? Did they say that sqrt{x^2} = -x when x is negative, or?

(in which case, remember negative * negative is positive)

[gray kindle]

the limit of (3x - 2)/(sqrt(2x^2 + 1)) as x approaches negative infinity is negative 3(sqrt2)/2

my answer was originally positive due to the earlier fact

graphically it isnt positive either

[unreal musk]

Well, if you make use of the fact that $\sqrt{x^2} = -x$ when $x$ is negative (see the above comment), you can rewrite what you have as $\frac{x \qty(3 - \frac2x) }{-x\sqrt{ 2 + \frac1{x^2} } }$

[warm shale]

@unreal musk

[gray kindle]

but you said here

it would be positive

[unreal musk]

The denominator is positive, despite having a negative sign in it

[gray kindle]

im not sure what this means

[unreal musk]

Alright, new question, is -x positive or negative?

[gray kindle]

negative

are you saying negative x as in the context

[unreal musk]

if x = -5, is -x positive or negative?

[gray kindle]

where we're plugging in negative infinity

positive

[unreal musk]

That's what I'm saying here, that...

[unreal musk]

...sqrt{x^2} is always postive, and when x is negative, sqrt{x^2} is -x, in order to "turn the negative into a positive"

[gray kindle]

so basically because we arent actually computing what the output is, we're adjusting it so the sign of the output should be correct

i think i get it now thank you

.close

[obtuse pebble]

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[obtuse pebble]

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[timid silo]

whats the difference of a supremum and a maximum?

[tardy epoch]

https://math.stackexchange.com/questions/1719659/what-is-the-difference-between-a-supremum-and-maximum-and-also-between-the-infi

[timid silo]

yall always got ur links on demand

.close

[obtuse pebble]

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[obtuse pebble]

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[honest creek]

What happens when a disconjuction (or) in an inequality with two opposing inequality signs overlap

[honest creek]

Like if x<-18 or x is greater than or equal to -1

Then they over lap

So what would you do

[deft magnet]

if x<-18 and x>=-1

[light flume]

union

[deft magnet]

^

[honest creek]

it's or though

disconjuction

[deft magnet]

well if they overlap then x can take both ranges

given or

[light flume]

if you treat each case as p and q

the sentence is saying P V Q

or P union Q

[honest creek]

, so you dont simplify

it

??

[light flume]

x<-18 U x>=-1

if you pretend U is union sign

[obtuse pebble]

@honest creek Has your question been resolved?

[obtuse pebble]

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[prisma sail]

this is a derivative, It should be continous since It is differentiable

[prisma sail]

however what exactly do I make out of it 😭 ?

[wary badger]

!xy

[obtuse pebble]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[prisma sail]

I don't understand why does it look like that, I need help figuring out how to check at which point does it increase and when does it decrease

Determine where the function f(x) = 40x^3 - 5x^4 - 4x^5 + 6 is increasing and decreasing.

[wary badger]

this doesn’t ask for a tangent line

if f is increasing then f’ is positive

[prisma sail]

that's the graph of the derivative btw

[wary badger]

if f is decreasing then f’ is negative

[timid silo]

simply just look for where it goes up, then goes down

for the original function, not the derivative

[wary badger]

this isn’t rigorous at all

[prisma sail]

yeah I get that, but I don't understand why does it look like that 😭

[wary badger]

forget the graph

[prisma sail]

oh

[wary badger]

you don’t need it

if $f(x) = 40x^3 - 5x^4 - 4x^5 + 6$ then $f’(x) = 120x^2 - 20x^3 - 20x^4$

[warm shale]

knief

[prisma sail]

still, how do I answer this?

[wary badger]

i’m getting there

[prisma sail]

-20x^2(x+3)(x-2)

[wary badger]

so for f to be increasing, it’s derivative must be positive

and for f to be decreasing its derivative must be negative

[prisma sail]

yeah

[wary badger]

so we need to find the intervals

we need the zeros

of the derivative

and we will work with values in between

this is the first derivative test

i’m assuming you don’t know the second derivative test

[prisma sail]

nope

[wary badger]

and its not really relevant here so don’t worry

[prisma sail]

ok so, I know the derivative is 0 at x=-3, x=2, and x=0

[wary badger]

so we have $-20x^2(x+3)(x-2)$

[warm shale]

knief

[prisma sail]

I can see that from the graph

[wary badger]

who cares about the graph

you won’t always have one available

use algebra

so

let’s look at the intervals in between the zeros

and of course less than the smallest zero and greater than the largest zero

so let’s look at (-inf,-3)

[prisma sail]

kk

how do I even do that

[wary badger]

is f’ positive or negative

just take any value in the interval and plug it in

[prisma sail]

ok

[wary badger]

we only care about the sign

[prisma sail]

hm

[wary badger]

what did you get

[wary badger]

^

[prisma sail]

I don't have my calculator 😭

[wary badger]

consider x=-4

bruh

[prisma sail]

I was doin with algebra

lemme grab it rq

[wary badger]

i don’t even do the arithmetic tbh

i just know the sign

you don’t need a calculator

any value between -inf and -3 will be negative for each term

thus it’s negative

[prisma sail]

oh

[wary badger]

negative * negative * negative is negative

[prisma sail]

It is squared tho..

[wary badger]

there is a -20

i’m considering the following

[prisma sail]

negative * whatever comes out of everything else

ok I get it

mb

[wary badger]

-20x^2, x+3, and x-2

for x less than -3 all three are negative

thus the product is negative

now we look at x between 0 and -3

[prisma sail]

makes sense

hm

negative?

since It is also negative

let x be -2 or sm

[wary badger]

so first is negative, second is positive, third is negative

you get positive

take notes btw for the intervals

it will matter

each one

[prisma sail]

k

[wary badger]

do you get this

[prisma sail]

yes

at least for -2 It is f' is 320

[wary badger]

yea but the sign is what’s important

[prisma sail]

-20

[wary badger]

this

negative * positive * negative = positive

now the next interval

we have 0 to 2

[prisma sail]

between 0 and 2

[wary badger]

mhm

what’s the sign

[prisma sail]

hm

positive

since the last one is negative

[wary badger]

yup

for a short cut you can just see that the zero x=0 is a squared zero so it won’t change sign

it bounces off the x axis

it has multiplicity 2

[prisma sail]

It doesn't really matter since It is 0

[wary badger]

what

[prisma sail]

mutiplicity 2?

what does that mean

[wary badger]

it means x=0 is a repeated root

like technically the function f(x)=x^2 has two roots, 0 and 0

multiplicity two because the root occurs twice

[prisma sail]

yea?

[wary badger]

when that happens the function won’t change sign

it won’t cross the x axis

it will just bounce off it

it’s useful

we couldve skipped the 0 to 2 interval

[prisma sail]

does that work for any derivative with y=0 at x=0?

[wary badger]

nono

it’s not the root being x=0

it’s the fact that that term is squared

even if it was (x-1)^2

then the root x=1

[prisma sail]

oh 😭

ok I get it

[wary badger]

would have the same property

[prisma sail]

It'll just form a parabola at that point

is that it?

[wary badger]

well no

,w graph (x+4)(x-2)^2

[warm shale]

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Requested by c.knief

[wary badger]

it’s not a parabola

[prisma sail]

uh

[wary badger]

it’s just that it doesn’t cross the x axis

at that root

[prisma sail]

ahh I see

[wary badger]

this is an arbitrary graph

[prisma sail]

It will js bounce off

[wary badger]

i was just showing an example

[wary badger]

mhm

[prisma sail]

yeah kinda surprising you managed to pull that off out of nowhere 😭

thx for that tho

I get it now

the derivative from a squared term won't go negative

[wary badger]

that’s important because not crossing the x axis means it won’t change sign

[prisma sail]

towards positive x

[wary badger]

so the sign before the root is the same as the sign after it

so you don’t need to test it twice

like we did -3 to 0 and 0 to 2

but the root x=0 had multiplicity 2 since it was x^2

we know it won’t cross the x axis

so the sign in the interval -3 to 0 is the same as the sign in the interval 0 to 2

[prisma sail]

so the derivative from 2,+inf must be positive

0 and 2 tho 😭

negative

[wary badger]

so negative * positive * positive

[prisma sail]

that's still negative

[wary badger]

yep

[wary badger]

what

[prisma sail]

I forgot for a sec

[wary badger]

this is for x>2

[prisma sail]

I don't quite get it tho

[wary badger]

which part

[prisma sail]

from 0 to 1 it should go up to some finite value

[wary badger]

are we talking about the original

[prisma sail]

then It should start going down to x=2

the derivative

[wary badger]

yea but from the original problem

[wary badger]

this

[prisma sail]

ye

[wary badger]

for x>2 it’s negative

[wary badger]

^

[prisma sail]

this is what I mean

[wary badger]

yea it’s decreasing

because the derivative is negative

[prisma sail]

yes

[wary badger]

are you asking why it’s a maximum?

at x=2 it’s a maximum because the derivative is positive then zero then negative

[prisma sail]

how can you say the interval from 0 to 2 is negative, if we have 0 to some random value being positive

afterwards It is negative

[wary badger]

meaning the function increases to a maximum at x=2 then decreases thereafter

[prisma sail]

OHH

[wary badger]

we said it’s derivative was positive?

[prisma sail]

😭

I forgot about that

I thought the x=2 was at the point where it crossed the x axis

for some reason

yeah It all makes sense

that's on me

yeah the graph makes sense to me now

after x=2 being the maximum

It goes negative

[wary badger]

because it decreases for all x>2

[prisma sail]

-20x^2(x+3)(x-2)

the last term is no longer negative for any x>2

[wary badger]

yup

[prisma sail]

so to answer the question I just write the intervals and the derivative value

e.g for x>2 f' is negative

right?

[wary badger]

but they want increasing/decreasing so what we do is write in interval notation where f’ is positive for increasing and where f’ is negative for decreasing

we had negative for (-inf,-3) and (2,inf)

so that’s where f is decreasing

[prisma sail]

aight

[wary badger]

and we had positive for (-3,0) and (0,2)

[wary badger]

and i wouldn’t write this as (-3,2) btw

because that would imply it’s increasing at x=0

which it’s not

it’s neither increasing nor decreasing

[prisma sail]

oh

-3,2 is increasing tho, although the derivative at x=0 is 0^2

ok wtv

so for -inf,-3 and 2,inf it is decreasing

and from -3,2 It is increasing

yeah I think I get it now

tysm

🙏

.close

[obtuse pebble]

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[warm shale]

w

[timid silo]

what is ur counterargument for the first one?

[lime wolf]

if lim(x->a)f(x) = 0 then the product will tend to 0 as g(x) remains bounded

even if g(x) does not have a limit as long as it does not tend to infinity the product will tend to 0

[timid silo]

wait

the counter argument is if g(x) tends to infinity

[glossy basalt]

looks good to me

[timid silo]

okay

ye

[lime wolf]

that would be it

[wary badger]

what if g(x) was 1/f(x)

[timid silo]

0 times infinity is undefined

ye

true

looks good

[lime wolf]

mb mb

[obtuse pebble]

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[obtuse pebble]

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[kindred quest]

I need some help with getting the formula down for calculating the sample standard deviation, I just can't seem to get it

0.428, 0.388, 0.422, 0.412, 0.389, 0.403, 0,430, 0,434, 0.387, 0.424, 0.436, 0.410, 0.405, 0.460, 0.441, 0.434, 0.419, 0.453, 0.380, 0.435
these are the values

[prisma echo]

It's going to be a pain but you will have to do it by hand

Like each term

[warm shale]

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Upgrade to WolframAlpha Pro!

Requested by xd_senbugha

[prisma echo]

,w standard deviation of 0.428, 0.388, 0.422, 0.412, 0.389, 0.403, 0.430, 0.434, 0.387, 0.424, 0.436, 0.410, 0.405, 0.460, 0.441, 0.434, 0.419, 0.453, 0.380, 0.435

[warm shale]

Results provided by WolframAlpha

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

Requested by xd_senbugha

[kindred quest]

thank you bro

I think I got the next one from here

appreciate the help

[obtuse pebble]

@kindred quest Has your question been resolved?

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[faint marsh]

hi I need help with probs a-c on 1-82

[timid silo]

,rotate

[true grail]

what did u try

[faint marsh]

like

I don’t really understand it -

[timid silo]

what test does a relationship has to pass to be qualified as a function

[true grail]

what about it

what if i asked you what value of x makes

x + 1 = 2 true

[faint marsh]

ye x would be 1

[true grail]

ok now apply the same logic

solve c first

[timid silo]

oh

i was on the wrong quesiton

[faint marsh]

c will be 3

[true grail]

facts but theres also another solution

dyk what the two lines on either side mean

[faint marsh]

not rlly

[true grail]

it just means that whatever value is in those lines will be positive after the equals sign

so |-4| = 4

in that case what other value of x makes that true

[faint marsh]

-3

[true grail]

-3+1 = -2

|-2| = 2

so maybe something else

[faint marsh]

but the lines ?

but

[true grail]

so think about it like this

when you put something in the lines

then it will still act normal

|1-3| = |-2|

but when you take it out of the lines

|-2|=2

the final value becomes a positive no matter what

[faint marsh]

uhm

[true grail]

feel free to ask questions if ur confused

[faint marsh]

like

I don’t understand then -3 + 1 = -2

[true grail]

the original question was

|x+1| = 4

you said that -3 might be an answer

i was like ok lets try it

so then

|-3+1|

now we replaced x with what you said might be right, negative 3

but now how absolute value (the lines) work is that you gotta finish doing everything inside them first, and then you can set it equal to something

so then it becomes |-2|

then u can set it equal to its positive value

|-2| = 2

[faint marsh]

okay

so

:/

I don’t get it

[glossy basalt]

hello

[glossy basalt]

it's alright

[faint marsh]

oh Hi

[glossy basalt]

you know what does those || in |x+1| mean, right?

[faint marsh]

anything that come out is positive right

[glossy basalt]

yea, so what is |-4|

[faint marsh]

4

[glossy basalt]

yea

so what if we wanna get x+1 = -4 so that if we plug that x in, we get |x+1|=|-4|=4

[faint marsh]

yes

[idle bear]

x + 1 = 4 and x + 1 = -4

those r ur two answers

[faint marsh]

yesh

[glossy basalt]

yay!

so what will be your answer for Q (c)

[faint marsh]

3

[glossy basalt]

3 is one of the answers for Q(c)

there is another answer

[glossy basalt]

hint:

[faint marsh]

I keep getting -3

[glossy basalt]

i know

[glossy basalt]

to see, we know that |x+1|=4

so that x+1 can actually be 4 OR -4

because both works, right? @faint marsh

[faint marsh]

oh I see because if we go back we go back by 4

[glossy basalt]

yeaaa

[glossy basalt]

so, what will x be, when x+1=-4

[faint marsh]

-4 or 3