help-38

[vernal fern]

oh dawg

ok just a sec

[earnest briar]

if you put 3 into t, then you get this

you can't divide by 0

[vernal fern]

t ∈(-∞, -13]U (-9,3)

[earnest briar]

not even close

[earnest briar]

show me how you solved for this answer

[vernal fern]

i was using a calculator to help w simplification and didnt

oh

then my reasoning for fucking it up is totally wrong

alright here-

wait

[earnest briar]

do you know how to solve this kind of equation?

[vernal fern]

i was

solving for this

good god

[barren crescent]

@vernal fern i was just doing this one

have u tried asking gemini

thats what helped me

[vernal fern]

Those ai things dont rlly work lol

i might be floundering but u probably want to check your work too

[barren crescent]

oh

[vernal fern]

ok im gonna go watch some khan academy

i think i have more deficits here than i realized

[inland forge]

Just get a common denominator

I guess that's what you did

But it looks more messy in that version

You could have just multiplied the fraction with a denominator of 3 by 3/3, and gotten 21/9

Then just added it

8/9+21/9

29/9

3.2222222.....

[vernal fern]

whaaaat is log?

[inland forge]

I was referring to the 8/9, and 7/3 on

Log is the inverse of exponent

log(100) is 10^x = 100

Assuming the base it 10

[vernal fern]

my head is swimming

[inland forge]

Well yeah

The first three are easy to remember though

[vernal fern]

going back to my original question

i somehow got the right answer

but

[inland forge]

What is the question

[vernal fern]

wrong way of doing it

[inland forge]

The domain is where x has a value

So in terms of a Rational equation it would be where there is not a vertical asymptote

You can turn this into a rational equation

Since you have a radical you can turn it into this

If x is any polynomial, √x = x/√x

So right now you have x to be (1/2) * x +2

So that is ((1/2) * x + 2)/√(1/2) * x + 2

Now to find the Vertical asymptote you set the denominator equal to 0

Also, we have a -2

But since it is being multiplied and not a fraction you can just put in the numerator

So currently the denominator in our fraction is

√((1/2) * x + 2)

So set that equal to 0

√((1/2) * x + 2) = 0

Hint: Square both sides first

So can you tell me what x is in that case

[vernal fern]

alright hold on

baby steps

[inland forge]

Okay

You can re-read it if you need to

Uhhh

I thought his question had a radical

Also, where did you get 2x+4

Ohhhh

Nevermind

That's not his question

I think

What?

But that's not what he asked

[vernal fern]

its like an example question to lead me there

[inland forge]

Ohh

Okay

I just scrolled up

And saw the context

Okay

Did you get the answer

[vernal fern]

honestly i dont think im able just yet

i thought i understood more than i do

[inland forge]

So what you need to do is to get a common denominator

[vernal fern]

that was my initial attempt

[inland forge]

Multiply by t-3 in the fraction with the denominator of t-9

But make sure to do it in the numerator and denominator

So that it is still one

And in the fraction with t-3 multiply by t-9 in the numerator and denominator

So you have

(2(t-3)/(t-9)(t-3)) - 8(t-9)/(t-3)(t-9)

Now both fractions have a denominator of (t-3)(t-9)

Or the other way around

Since multiplication is commutative

Since you have common denominators you can combine the tops of the fractions.

Also, I don't think you need to do this unless it was a question on the assignment

But if you want you can

So now you just need to combine the numerators

Yeah

That's what it should look like

But I think we are getting a bit distracted right now

In order to combine them you would distribute

Umm, you already solved it for him

But I think he should solve the question he pasted, the one with the radical

Really?

In that case, okay

But he said he solved the question incorrectly

But his steps seem correct except for the part where he set it equal to zero, without the radical

But the radical would not make any difference

But he does not understand how

[vernal fern]

i did but the wrong way yknow

[inland forge]

I think

That too

[vernal fern]

i thought it was right my initial qustion was actually about showing it in the proper form for domain

[inland forge]

Wait

So you got it right

[vernal fern]

yeah

but

with the wrong steps

[inland forge]

Well what were the right steps

There is an issue

That is the same as his steps

But he said he did it wrong

I guess

But I think you should teach him the right way

Then actually test if he knows it

[vernal fern]

im a lil dense sorry yall

[inland forge]

Okay

But I have a question

What are we actually trying to solve here

@vernal fern

Can you solve for t

[vernal fern]

attempting

is that a + next to 66

just to show its pos

[inland forge]

It is 6

Yeah

[vernal fern]

66-6t/ x^2 -12x+27

=> 0

yep

guys

[inland forge]

Umm

[vernal fern]

i appreciate it

[inland forge]

He is trying to say not to distribute

Just multiply by the t-3 and t-9 on both sides

Then they cancel

So you just have the numerator

Hmmm

Okay

[vernal fern]

not sure i can be helped until i get a little better at the concept in general..

i will be back once i figure my shit out!

[inland forge]

Okay

To be fair this was all over the place

I don't think we should be bringing up difficult examples that include logarithms until they understand how to get a common denominator

Okay

Bye

[vernal fern]

if theres one thing ive learned its that i have a lot to learn cz dayum

[inland forge]

Yes

I do

[vernal fern]

neop, im gonna go watch a ton of khan academy

[inland forge]

Anyways I think Organic Chemistry tutor is helpful

That's not a gap in logs, that's a gap in inequalities, which I accept was my fault

[vernal fern]

[inland forge]

Okay

Ask if you need any help

At this point you are just trying to be annoying

Okay

Let me see

I will try

What was that

Okay

[scenic delta]

Stop being toxic.

[inland forge]

Exactly

[scenic delta]

Do you really believe that's what happened

If yes, you're really being biased.

Why would you even say this...

Anyways you guys resolve this together I'm not gonna enter this conversation

[barren crescent]

what do pronouns have to do with anything

[scenic delta]

Wdym?

[scenic delta]

Yeah I don't get it

Should I not

It means that I don't get offended if you call me with any pronouns

I'm leaving now.

[barren crescent]

yeesh

[vernal fern]

...

sorry i didnt know when to close the channel

[scenic delta]

It's fine

[vernal fern]

probably shouldve pumped the brakes sooner

[scenic delta]

Did your question get resolved?

[vernal fern]

ive decided to do some supplemental khan academy

[scenic delta]

Oh nvm he reopened it

[vernal fern]

oh

then i cant close it anyways

[scenic delta]

No it's fine, can you not ping me please?

Has your question been resolved?

Yeah it's fine to have personal preferences

[barren crescent]

people have been misgendering dryforest this whole time no

[scenic delta]

Can we stop this aggression?

No?

It's just ok for me if you misuse my pronouns

<@&268886789983436800> should I be mentioning you guys...

[inland forge]

This has gotten out of hand

This is a math discord

[scenic delta]

Or should I just block him because he keeps mentioning me back to this black hole

[inland forge]

Maybe ask this in discussion

[scenic delta]

Yeah

[inland forge]

Yeah

Well I still need to solve this log equation

[scenic delta]

There is a difference between calling someone a "she" and disrespecting them

[inland forge]

Yeah

Okay

That was uncalled for

[scenic delta]

Alright create a new channel and send it there I'll be happy to help

[inland forge]

Okay

[barren crescent]

<@&268886789983436800>

[stoic iron]

.close

[trim jolt]

Channel closed

Closed by @stoic iron

Use .reopen if this was a mistake.

[scenic delta]

For the record I'm heterosexual

[scenic delta]

Thanks...

[north forge]

acting like being called a girl is an insult is awfully misogynistic lmfao

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[karmic sable]

does anybody know how to create a table og values with geogebra or by hand/ calculator?

[hallow crane]

Well you can't really make a table without actually getting the value of the linear function

[scenic delta]

Yeah you have to first find a general formula for it

[karmic sable]

f(x)= 3x + d

or something?

[hallow crane]

But if you want to solve it by hand, there's a more approachable way than this

[scenic delta]

Something like that

[karmic sable]

9=3x+d

[hallow crane]

for an increase of 2 (x value), you notice an increase of 3 (y value)

so for an increase of 1 (x value), you'd expect an increase of 1.5 (y value)

correct?

[karmic sable]

but then there is 2 differece between 4 and 6

[hallow crane]

yes that's what i'm saying

[karmic sable]

and then 3 between 9

6 and 9

yeah

[hallow crane]

4 increasing to 6 is an increase of 2 in the x value right?

[karmic sable]

first it is 2 and then it is 3

[hallow crane]

yeah so an increase of 2 (of the x value) corresponds to an increase of 3 (of the y value)

so can you infer what an increase of 1 (of the x value) would correspond to?

[karmic sable]

whhaaaat

[hallow crane]

let's say this situation works linearly:

if someone works for 2 hours, he makes $3. How much would he make if he works for 1 hour?

[karmic sable]

$4,5

[hallow crane]

How?

He works half the time so his money should be half of what he normally makes as well

2 hours -> $3
1 hours -> $?

[karmic sable]

$ 1,5

[hallow crane]

better

[karmic sable]

sorry i read it too quickly

[hallow crane]

now your question thing is very similar to this

we can still keep it thematic to work and money though if it makes it easier for you to understand:

4 hours -> $9
6 hours -> $12

so what about 1 hour -> $?

[karmic sable]

no wait

[hallow crane]

No you have to use both the things

[karmic sable]

yeah

[hallow crane]

Just look at it like this

[karmic sable]

yes

[hallow crane]

an increase of 2 hours leads to an increase of $3

[karmic sable]

2 hours difference and the money increased by $3

[hallow crane]

so an increase of 1 hour leads to an increase of how many dollars?

[karmic sable]

1 hour is $1,5

[hallow crane]

yes but key word is

an increase of 1 hour leads to an increase of $1.5

the *increase word is key

you can see that

[karmic sable]

aaaa!

[hallow crane]

with let's say:

4 hours -> $9

6 hours -> ?

4 to 6 hours is an increase of 2 hours

which leads to an increase of $3

so you'd have $12 for 6 hours

[karmic sable]

an increase of 2 is an increase of 3

[hallow crane]

anyway

Now can you fill this in?

[hallow crane]

Use this idea

and work on this

[karmic sable]

so if i increase the hours i work by 3 then i get an increase of $4,5

[hallow crane]

yes that's right

[karmic sable]

which is 16,5

[hallow crane]

correct

Now can you fill in the last block too?

well assuming you already filled in:

9 -> 16.5

[karmic sable]

now my money increased to 22,5 so that means my work increased

which can be calculated as

22,5/1.5

?

[hallow crane]

no

use the same idea

[karmic sable]

okay

[hallow crane]

it looks like you strayed from the key focus of the idea

an increase of 1 hour leads to an increase of $1.5

how much did the money increase?

9-> 16.5
? -> 22.5

[karmic sable]

16,5+1,5+1,5+1,5+1,5= 22,5

meaning 4 hours of more work

there has to be a smarter and quicker way

to solve this

i make everything more complicated

[hallow crane]

what i showed you was the "smarter" way lol

but yes that's correct

i mean all you had to do was 22.5 - 16.5 = 6

[karmic sable]

is there an easier way like a pattern that i could mindlessy write down

[hallow crane]

so an increase of $6 which means an increase of 4 hours

[karmic sable]

everytime I see a table?

[hallow crane]

you would make an equation of a line

,, y = mx + c

[solid kiln]

kanna

[hallow crane]

,,m = \frac{y_2 - y_1}{x_2 - x_1}

[solid kiln]

kanna

[hallow crane]

and then you have to algebraically solve for c

[hallow crane]

what i showed you is the optimal way because you only had to fill out some values

[karmic sable]

yeah:>

[hallow crane]

and tbh less work

but the line way is the "standard"/"conventional" way

[karmic sable]

waait this is the

differential

thing

[hallow crane]

Yes, if that's what you want to call it

[hallow crane]

fun fact, we used exactly this in our attempt of the pattern recognition stuff

[karmic sable]

how could i plug in the value for m and y?

[hallow crane]

https://cdn.discordapp.com/attachments/908078008609431674/1217207336095973518/687632422212468806.png?ex=66032fab&is=65f0baab&hm=1d311fd4fbe3cd5eea7aee4df6a900de107c6fda72538b04b1314460d730257e&

First find m

x1,y1 = (4,9)

x2,y2 = (6,12)

[karmic sable]

m is the value that shows by what value the function is increasing

[hallow crane]

yes aka the slope

[karmic sable]

my brain is not braining anymore

[hallow crane]

¯_(ツ)_/¯

[karmic sable]

¯_(ツ)_/¯

[hallow crane]

you're the one that wanted to go the algebraic route

lol

but anyway, your m would look like:

[karmic sable]

okay so the way we solved the problem was less complicated

[hallow crane]

,, m = \frac{12 - 9}{6-4}

[solid kiln]

kanna

[hallow crane]

Owing to this

[karmic sable]

and it would be 1,5?

[hallow crane]

Does that make sense?

[karmic sable]

yes of course

[hallow crane]

Yes indeed

[karmic sable]

omg

[hallow crane]

We kinda leveraged this idea

Red = increase of 3 = increase of price

Blue = increase of 2 = increase of work hours

[karmic sable]

that is so cool

¯_(ツ)_/¯

[hallow crane]

anyway so we have m = 3/2

https://cdn.discordapp.com/attachments/908078008609431674/1217207286712369262/687632422212468806.png?ex=66032f9f&is=65f0ba9f&hm=73a269985cf1321500f8dabaffdc3f3611539bf964489709f270db7c9815b233&

[karmic sable]

which is 1,5

[hallow crane]

we plug m = 3/2

[hallow crane]

no

3/2 is 1.5

,, y = \frac 32 x + c

[karmic sable]

sorry

[solid kiln]

kanna

[karmic sable]

and now

[hallow crane]

finally, you take any (x,y) of your choice

[karmic sable]

i could insert

the value

of 4

and 9

[hallow crane]

yes

,, 9 = \frac 32 (4) + c \implies c = 3

then you solve for c

[solid kiln]

kanna

[karmic sable]

what does it say?

THE DIFFERENCE

THE INCREASE

[hallow crane]

so now you have $y = \frac 32 x + 3$

[solid kiln]

kanna

[hallow crane]

but yeah now you can do the problem with this equation

[karmic sable]

OH

my god

so from now on

i can just put the y value

and find x?

and vice versa

[hallow crane]

To find blue, you'd sub in x = 9 into the line equation we made

and solve for the corresponding y

[karmic sable]

yes

[hallow crane]

To find purple, you'd sub in y = 22.5 and solve for the corresponding x

[karmic sable]

yes

amazing

thank you

[hallow crane]

[hallow crane]

Lol

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[karmic sable]

how can i interpretate this?

[karmic sable]

Between the time intervall 2012 and 1968 the number of wells in a given country increased approximately by 636?

[serene chasm]

yes

rate of change

[pearl tendon]

No

[karmic sable]

and approximately is the key word

[serene chasm]

it means every year about 636 new wells were installed

[pearl tendon]

that is the key phrase that's been missing

each year

[serene chasm]

I was about to type it 😛 sorry

[pearl tendon]

(also, "interpretate" is not a word)

[karmic sable]

lol xD sorry i am so tired at this point

ive been doing math all day

and i have an exam tomorrow

[pearl tendon]

I am so glad you got the right vowel here

heh.. good luck!

[karmic sable]

xD

so wait

Between the time intervall 2012 and 1968 the number of wells in a given country increased approximately by 636 per year?

[pearl tendon]

Think about it.. you know what the word "per" means, right?

Miles per hour... wells per year...

[karmic sable]

as a non native speaker, sometimes i question it

makes sense

[pearl tendon]

"per" indicates you have a ratio

The first item is the numerator, the second item is the denominator

"percent" means the denominator will be 100 (think "centimeter")

You could just as easily break "per" and "cent" apart, and it would fill the same role as the second term in your phrase.. "things" per cent

[karmic sable]

aaaa

[serene chasm]

if you're unsure, you can do the the following calculation:
if you know the number of wells in 1968 was 4000, if you add for 44 years 636 wells you should get to 32000. if you crunch the numbers you get 31984

[karmic sable]

i seee

wait

now i am confused again

[serene chasm]

why's that?

[pearl tendon]

If you divided to get the original 636, you should be able to multiply to get back to the final total of 32,000

[karmic sable]

is it enough if i say this?

[pearl tendon]

The number is slightly off due to rounding

[pearl tendon]

I mean.. I'd accept it

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[thick hull]

Need help with this question. I've done the first part and proven it but i dont know how to go about finding $\nabla r^2$ or $\nabla (\mathbf{a} \cdot \mathbf{r})$ to be able to do the second part

[solid kiln]

I can't believe you've done this

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@thick hull Has your question been resolved?

[thick hull]

<@&286206848099549185>

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@thick hull Has your question been resolved?

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[thick hull]

<@&286206848099549185>

[native aurora]

you can work out $\nabla r^2$ directly

[solid kiln]

Waffloid

[native aurora]

ie r^2 = x^2+y^2+z^2

so $\partial_x r^2 = 2x$, others similar

[solid kiln]

Waffloid

[native aurora]

so $\nabla r^2 = (2x,2y,2z) = 2\mathbf{r}$

[solid kiln]

Waffloid

[native aurora]

and similarly you can calculate $\nabla(a\cdot r)$ by passing into coordinates and computing direclty

[solid kiln]

Waffloid

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[loud bobcat]

any tips on how I can find the domain of this function?

[loud bobcat]

I know that sin(x) can't be -pi/6

not sure how I can write it in interval or <x< notation

[worldly wing]

Well, you want $$\sin x\ge -\frac1{2}$$

[solid kiln]

kheerii

[worldly wing]

Try drawing the graph for y=sinx

And y=-1/2

[loud bobcat]

badly drawn but not sure how to write this as the domain

[worldly wing]

Yeah, so what you have is $$2n\pi-\frac{\pi}{6}\le x\le 2n\pi+\frac{7\pi}{6}$$ where n is any integer

[solid kiln]

kheerii

[worldly wing]

Do you see how that comes?

If you take n=0, you get the middle domain

If you take n=-1, you get the domain on the left

And so on

[loud bobcat]

Oh I see what you mean

[worldly wing]

You can write this in shorthand form as $$x\in\bigcup_{n\in\mathbb{Z}}[2n\pi-\frac{\pi}{6},2n\pi+\frac{7\pi}{6}]$$

[solid kiln]

kheerii

[worldly wing]

So that would be your domain

[loud bobcat]

got it now, thanks

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[wraith hinge]

yo

[wraith hinge]

why was pi invented

like what was the need

[edgy willow]

3.1415926535897932384626433832795028841971693993751058209749445923078164062862089986280348253421170679821480865132823066470938446095505822317253594081284811174502841027019385211055596446229489549303819644288109756659334461284756482337867831652712019091456485669234603486104543266482133936072602491412737245870066063155881748815209209628292540917153643678925903600113305305488204665213841469519415116094330572703657595919530921861173819

[nimble stone]

more accuracy please

[proper kernel]

π was invented to approximate the harder-to-calculate 22/7

[wraith hinge]

no

like

why

wait

what

[proper kernel]

took you a whole 90 seconds to get the joke

we can approximate that with π^4

[nimble stone]

pi is just magically the ratio of circumference and diameter of any circle

[wraith hinge]

ik

but

[proper kernel]

not really magically

[wraith hinge]

why was it needed????????????????????

[nimble stone]

allow me some magic

[proper kernel]

if you needed to build something circular

[wraith hinge]

yeh

[proper kernel]

π is what you would need to multiply by to get from diameter to circumference

[edgy willow]

happy pi day eve btw

[proper kernel]

back then they just used 3 since it was good enough

but people knew it was a bit more than just 3

[wraith hinge]

yeaa

[proper kernel]

thats the first use of π

[wraith hinge]

did u watch mathantics

[proper kernel]

no

[wraith hinge]

oh

anyway

so to

complete the circle's circumference

in a numerical sense

[proper kernel]

sure, to be more precise with knowing how long sides are

heres another Fun Fact

the egyptians have a clean way of getting a right angle using only a rope with 12 evenly marked points

you can bend it into 3 + 4 + 5

and then form a triangle out of that

[wraith hinge]

npo

no

[proper kernel]

lol

[wraith hinge]

thank u doe

byee

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[wraith hinge]

wait

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[wraith hinge]

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[barren crescent]

the choices-

it is not b

the second graph

i know that much

but both a and b look like they start at 21k

[left oriole]

whata should the value be at x=1

[barren crescent]

im inclined to think its a

[left oriole]

(1 year)

[barren crescent]

around 19k

19.5 k maybe

i calculated it i cant fully remember tho

19.7k

[left oriole]

and after say 5 years?

[barren crescent]

mm

did not calculate so far

[left oriole]

that should make it obvious which graph is right

[barren crescent]

one sec

oh

its C

[barren crescent]

...its c right

now im nervous lol

oh nvmd they just went offline

[left oriole]

yea

[barren crescent]

sorry for the ping

[left oriole]

after 5 years i calculate just over $14k

[barren crescent]

yep same

[left oriole]

only one graph even resembles that

[barren crescent]

yeep

thanks bungo!

[left oriole]

sure, gl

[barren crescent]

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[vernal fern]

Explain why it is useful to graph data with a logarithmic scale and why it works. Include two examples of when a logarithmic scale is used.

[vernal fern]

logarithmic scales help visualize data spanning large ranges. by compressing large values, it lets us to see overall trends. This is because log scales focus on percentage changes, which makes them ideal for data like earthquake intensity or sound levels

are my examples too brief?

[nimble stone]

kinda, they likely are for your purposes

[plucky grail]

"log scales focus on percentage changes"?

[vernal fern]

instead of like, raw data

[plucky grail]

you could elaborate on that a bit more

[vernal fern]

truthfully that is the shakiest bit of my ex

i stole a lil snippet from the lesson but they didnt elaborate much further

i might just cut that out

lazy answer i know

but its late and someone needs her 8 hours 😊

thanks all

[plucky grail]

no this is fine, but you could add more detail

imo

[vernal fern]

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[coarse rock]

why does nth term have to – [–3(n – 1)2 + 4(n – 1)]

[sacred sapphire]

sum of the nth term - sum of (n-1)th term

[coarse rock]

never knew that

tysm

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[tulip vector]

hi

[tulip vector]

i got to 2^x = -8

but thats impossible

and when i put it into an online calculator it says the answer is 3

[fathom gull]

first you expanded the bracket right

so what'd you get\

[tulip vector]

actually

i substituted 2^x for u

[wraith hinge]

just write 2^(x-1) as 2^x/2 and simplify you will get it

[fathom gull]

how did u get 3u / 6

its just going to be 3u / 2

[tulip vector]

Oh yeah

thanks

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[twin lion]

hi guys i'm having a bit of a problem on C:
from the previous section the range of g is [0, sqrt(3/2)] so that would make [γ, δ] equal g?
how can i find alpha and beta? is it just getting it to equal [γ, δ] because its bijective? and the h(0) = sqrt(1/2) is throwing me off on where I would use it in this question

[twin lion]

thank you!!

[vagrant prism]

what is the range of g from (b)?

[proven sluice]

Firstly the root should be >=0 so do that and find domain

And since it's bijective codomain=range , so whatever is range of b that's the ans for codomain

[twin lion]

[0, sqrt(3/2)]

[twin lion]

oh so [γ, δ] would just equal [0, sqrt(3/2)]

[proven sluice]

Yes

[twin lion]

and would that mean [α, β] would have to equal [γ, δ]?

[proven sluice]

No

Domain means the values that can be possible

Root cannot be negative right?

It has to be positive

[twin lion]

oh yeah you're right

[proven sluice]

So 1/2+ sinx should be ≥0

By this sinx≥-1/2

So what do u think that x is?

[twin lion]

-pi/6 ?

[proven sluice]

Correct

So x≥-π/6

It can be [-π/6, infinity] or [-π/6,π/2]( if seen for principle values)

[twin lion]

so would that make [α, β] = [-π/6, infinity] ?

[proven sluice]

Yes, generally

[twin lion]

oh sick thank you

one more question if you don't mind

[proven sluice]

Np, sure

[twin lion]

where would h(0) = sqrt(1/2) come in?

is it just there to mess with you?

[proven sluice]

Yeah lol

[twin lion]

omg that had me stumped just wondering where i would use it

thank you!!!

[proven sluice]

Np!

[twin lion]

i think i should be alright for h(x)^-1 since it's just rearranging and swapping the domain and range right?

[proven sluice]

Yep just write x in terms of h(x)

[twin lion]

gotcha have a good day/night !!

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[wraith hinge]

$f(x)=3x^4+4x^3-12x^2$

[solid kiln]

Safiya

[wraith hinge]

I need to find when the function is increasing and decreasing using derivatives

I did this but I'm getting different answers

$f'(x)=12x^3+12x^2-24x$

[solid kiln]

Safiya

[wraith hinge]

$f'(x)=12x(x^2+x-2)$

[solid kiln]

Safiya

[wraith hinge]

$f'(x)=12x(x-1)(x+2)$

[solid kiln]

Safiya

[wraith hinge]

$x=0,1,-2$

[solid kiln]

Safiya

[wraith hinge]

might just honestly be a substitution error but

they have it as + + - +

for -2, 0 , 1

I'm getting - + - +

[loud shell]

Hm, increasing for (-2,0)U(1,inf)

[wraith hinge]

yeah getting the same

increasing then

decreasing (-inf,-2)U(0,1)

[loud shell]

Yea

[wraith hinge]

alright so I'll just assume the answer key is wrong

for this one

[loud shell]

Yes

[wraith hinge]

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[wraith hinge]

someone explain this to me

[wraith hinge]

for a)

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[wraith hinge]

<@&286206848099549185>

[glacial horizon]

Sorry, what is the question?

[wraith hinge]

I'm not getting the same answers

Mines looks more like a line

in the negatives

theirs is weird

even the i) ii) and iii) dont make sense

[glacial horizon]

What you can tell from the graph is that in the first picture is that

• the derivative of f is positive from -4 to 4
• the derivative of f is negative from x > 4 onward
• the derivative of f is 0 at x = 4

[wraith hinge]

is it not negative?

it's going downwards from left to right

[glacial horizon]

This tells you that

• f is increasing from -4 to 4
• f is decreasing from x > 4 onward
• f has a maximum at x = 4

It is going downwards, yes, but the values of f'(x) are still positive

[wraith hinge]

positve $\neq$ increasing

[glacial horizon]

The values are only negative from x > 4

[solid kiln]

Safiya

[glacial horizon]

If you are referring to the derivative of a function, then yes, a positive derivative means that the function is increasing

[wraith hinge]

mm

oh okay this isn't f(x)

[wraith hinge]

how does x have a max at 4?

[glacial horizon]

the derivative is equal to zero at x = 4

[wraith hinge]

sure that's a turning point

but how is it a max

[glacial horizon]

it is positive before, negative after

[wraith hinge]

hmm yeah that makes sense

but that's because it's supposed to be parabolic

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[tidal river]

we want to create a string with the characters 0,1,2,3 with n characters, A - how many different strings is it possible to make?
B - now we want to insert the characters 00 into the string, how many different possibilities are there for a string of length n?
C - how many strings of length 39 can there be such that there will be no two adjacent characters that form the strings 00, 11, 22?

[tidal river]

how do I solve this? in combinatorics, we've been told we need to use a recursive formula but i have no idea how

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.reopen

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[vernal briar]

What have you tried?

(A) shouldn't be too hard. You don't need any recursive formulas for it.

Also (B) isn't very clear. Is it asking about how many strings of length n+2 contain the sequence 00?

[tidal river]

for A i initially thought its just 4^n ,but im not sure if thats right,
for B the string length with the 00 is in total n

[vernal briar]

Yeah that's right

4 options for each character.

I don't understand your answer for B

Like can you clarify what is it asking?

Number of strings of length n, that contain 00?

[tidal river]

i clarified it here in my response, sorry for the misunderstanding

a string of len n that must have the characters 00 in it

i thought it would just be $(n-1)\cdot 4^{n-2}$

[solid kiln]

horizon2.0

[vernal briar]

No, you are overcounting

[tidal river]

but after manually checking it with some values it turend out false, so some recursion formula must be part of solving it

[vernal briar]

Well yes this can be done using recursion.

Ok one sec

So it should be easier to count the number of strings where 00 DOESN'T appear

and do the total number of strings minus that

Can you try doing that?

[tidal river]

let me think it for a min

it doesnt look like a recursive formula but this is just some easy cases to understnd

[vernal briar]

What is f?

What is it supposed to represent?

[tidal river]

f(n) is the next number in the sequence

just a sec

[vernal briar]

Well yes

[tidal river]

idk

i thought i was onto something with this

but i cant find how to get the condition into some algebric expression

[vernal briar]

Try to define your recursive with a sequence that counts the possibilities

a_n - number of strings of length n, that end with a 0 and don't contain 00,

[tidal river]

this is what i was trying to do with the last one

[vernal briar]

b_n number of strings of length n, that end with a 1, 2, 3 and don't contain 00

Find formulas for a_n+1 and b_n+1 in terms of a_n, b_n.

Then you can find a recurrence and solve it.

[tidal river]

i was trying to find the pattern here

[vernal briar]

No

[vernal briar]

Do this

Have you learned how to solve second order recurrence relations?

[tidal river]

maybe? im not sure if we're talking about the same thing (not from an english speaking country)

[vernal briar]

$a_n = \lambda_1a_{n-1} + \lambda_2a_{n-2}$

[solid kiln]

RedstonePlayz09

[vernal briar]

Something like this

Where you know what lambda_1 and lambda_2 are.

[tidal river]

yeah, with the polynomial and the roots, i know this

[vernal briar]

Good

[vernal briar]

.

[vernal briar]

.

Do this

[vernal briar]

.

https://discord.com/channels/268882317391429632/908078124405755914 maybe this guy knows something about "00" lol

[tidal river]

hahaha

[tidal river]

just writing down my reasoning:
if a_n finishes with a 0 than for a_{n+1} = 3*a_n
and for b_n its a similar reasoning such that b_{n+1}=4*b_n

correct me if i'm wrong

[vernal briar]

No that's wrong

You can use a_n to express b_n+1, and use b_n to express a_n+1

Doesn't have to be separate

[tidal river]

is this right then?

since if it ended with 1,2,3 it can have any last digit and if it ended with 0 the last digit can only be 1,2,3

$$b_{n+1}=4b_n+3a_n$$

[solid kiln]

horizon2.0

[tidal river]

i missed the 3 here ignore this one

[tidal river]

and $a_{n+1}=4b_n$

[solid kiln]

horizon2.0

[vernal briar]

Close but not quite

b_n+1, is counting when it ends with 1, 2, 3

So you don't want to add a 0.

And a_n+1 is counting when it ends with 0, so you ONLY want to add a 0

All this while making sure there is no 00

[tidal river]

so its $b_{n+1}=3b_n+3a_n$ ?

[solid kiln]

horizon2.0

[vernal briar]

Yes

And what about a_n+1?

[tidal river]

isnt just b_n then? since it would take some b_n and multiply by 1 to get a 0 at the end

[vernal briar]

Yes lol

$a_{n+1} = b_n$

$b_{n+1} = 3b_n + 3a_n$

[solid kiln]

RedstonePlayz09

[vernal briar]

Now get an expression for b_n+1, using only b_n and b_n-1

And then solve it using what you learned

[tidal river]

i understand, thank you very much!

was it possible in the first place to do it only using a single formula?

[vernal briar]

Hmm lemme think

Like you get just have the b_n probably

Consider b_n+1, you get take a string with length n and add 1, 2, 3 to it.

Or a string with length n-1, and add 01, 02, 03 to it.

The reason you aren't adding 11 to the n-1 string for example, is because you already counted that by adding 1 to the n length string.

But splitting it up into 2 sequences makes it easier sometimes

[tidal river]

making it a whole lot easier, thanks a lot

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[pure isle]

can someone explain what happened here

[vernal warren]

take (x+y) common

[pure isle]

thanks

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[quick hound]

what should i apply for c?

[quick hound]

kinda lost

what substitution to use

just need the general idea

[lament reef]

start by completing the square on the inside

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[red jolt]

i think i had seen a question like this recently but i totally forgot how to solve it

[red jolt]

should i do y = x + 1/y?

<@&286206848099549185>

[scenic delta]

Exactly that

[red jolt]

well i’m having trouble to take derivative of it but let me try it again

[scenic delta]

Maybe try to isolate y first

[red jolt]

come to dms please

still unsolved

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[edgy willow]

the trick is f(x)=x+1/f(x) and you just take the derivative of both sides and solve

[red jolt]

did i do it right?

[edgy willow]

👍 perfect, just plug in 3/2 and you're done

[red jolt]

i got two different answers for that function

= 2 gives the right answer

but did i do it right?

[edgy willow]

uhh yea sometimes these infinite sums give 2 answers from the quadratic but one clearly doesn't make sense usually

like 3/2+positive fraction can't get 1/2 so you throw it out

that's right though

[red jolt]

oh okay

that problem has one long solution

[edgy willow]

then plug in f(3/2)=2 into the derivative thing you got

yea lmao

[red jolt]

i had gone for this solution before i posted here but whole time i was doing a small mistake when taking the derivative so kept getting a wrong answer

thanks for helping 🥂

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[rapid idol]

when it says has solutions x = 1 and x = 2, do i have to plug it in into either log form or exponential form? x = b^y or y = log_b(x)

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[dense gyro]

Isn't exponential form y = b^x

But other than that, you're right I think

[rapid idol]

I think that’s what it would look like if I subbed in 2 for y

I’m super confused lol

[dense gyro]

Nevermind basically what you want to do is solve a system of equations: you have one equation where you plug in x = 1, and another equation where you plug in x = 2

Then you can solve for your two unknowns in your exponential equation

I'm assuming once you do that, you'll get an equation fulfilling the next set of requirements

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[rapid idol]

So do I plug in x=1 and x=2 into the exponential form or log form?

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[vernal fern]

The number of weekly hours spent on a smart device varies inversely with the person's age. If a 20-year-old person spends 52 hours on their smart device each week, how many hours does a 50-year-old person spend on their smart device?

[vernal fern]

does 20.8 hours track

[placid juniper]

yes

[vernal fern]

alrightyyyy

word problems like these r hit or miss for me

thank you very much

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[torn ether]

What should I do from here?

[brazen forge]

its basic system of equations

what do you have trouble with?

[torn ether]

im just not sure wher I should start in the system'

[brazen forge]

first eq is 16=A+C

so A=16-C

use that sub to other 2 equations

ull get 2 eq with 2 variables B and C

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[gleaming flicker]

i just want to confirm if this is correct?

[mystic grotto]

Looks good to me 👌

[gleaming flicker]

alr, thanks

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[torpid moat]

Region A is bounded by the curve y=x^3, the y-axis and the horizontal line y=8. Set up and evaluate an integral expression that gives the volume of the solid generated when A is rotated about the y-axis. Be accurate with setup and answers. Evaluate using the disk method, with dy as variable of integration

[torpid moat]

Would the answer for this be 96pi/5

[olive horizon]

correct

[torpid moat]

thanks, and 2 more questions

Region A is bounded by the curve y=x^3, the y-axis and the horizontal line y=8. Set up and evaluate an integral expression that gives the volume of the solid generated when A is rotated about the line y= 8.

Would the answer for this be 320pi/7?

<@&286206848099549185>

[olive horizon]

what was your integrand?

[torpid moat]

(8 - x^3)^2

[olive horizon]

should be R^2 -r^2 not (R-r)^2

[torpid moat]

That is if the washer method is used, but this is the disk method that I am required to follow

isn't the disk method [R(y)]^2

[olive horizon]

the disk method is a simplification of the washer method if r=0

but you can’t use (R-r)^2 here

[torpid moat]

I can use a disk method when it is rotated about the horizontal axis y = k
V = pi * int (R(x) - k )^2 ds

dx*

can't I do that?

[tribal jetty]

,w integral from 0 to 2 of pi(8-x^3)^2 dx

[solid kiln]

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Requested by svc20060505

[tribal jetty]

you used the same bounds? 0 to 2

[torpid moat]

Yes, but I made a mistake expanding the square

I understand it now, thanks!

Also, I don't understand how to do:
Region P is bounded by the curve y= ln(x-1), the x-axis, and the verticle line x= 10

Set up and evaluate an integral expression that gives the volume of the solid generated when P is rotated about the x axis

[tribal jetty]

use a sketch or a graphing calculator idk

[torpid moat]

I have a sketch

would the radius be r= ln(x-1)?

[tribal jetty]

I think so

[torpid moat]

,w integral from 0 to 10 of ln(x-1)^2 dx

[solid kiln]

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Requested by mimosa_8824

[tribal jetty]

from 0?

where does y=ln(x-1) and y=0 intercept

[torpid moat]

it says log, but ln though

x= 2

[torpid moat]

from 0 to 10; x intercept at x= 2

[tribal jetty]

why from 0 to 10

the area enclosed is from 2 to 10

[torpid moat]

because ohhh 2 to 10

ohhh ok ok

that makes sense

so the answer would be 9*ln(9)^2 - 18ln(9) + 16?

[tribal jetty]

idk

,w integral from 2 to 10 of pi*(ln(x-1))^2 dx

[solid kiln]

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Requested by svc20060505

[tribal jetty]

yeah same answer, you forgot the pi

[torpid moat]

Ok, ig im right then

mhm

right

[tribal jetty]

im blind

you were right after multiplying by pi

[torpid moat]

yeah I just noticed that too

Ok and for the same question, if p was rotated about the line x= 10, the only thing that would change is the radius right? it would become r= ln(9) - e^(x-1) ?

[tribal jetty]

what is p

[torpid moat]

because now the radius is horizontal, and in terms of y so ln(x-1) becomes ye^(x-1)?

[tribal jetty]

oh ok

no

[torpid moat]

the region for which im trying to find the volume

[tribal jetty]

you would be integrating with respect to y

not x

[torpid moat]

mhm

so y= ln(x-1) becomes x= ye^(x-1)?

[torpid moat]

mhm

[torpid moat]

ohh okay that makes much more sense

[torpid moat]

wait but since we're integrating with respect to y, wouldn't the 10 be f(10)?

[tribal jetty]

you integrate now with y bounds

same variable as differential

now you don't use x

[torpid moat]

oh ok

that makes a lot more sense

thank you so much!

[tribal jetty]

it doesn't matter if f is at the right or left of the axis of rotation

[torpid moat]

hmm

[tribal jetty]

(f-axis)^2 = (-(axis-f))^2 = (axis-f)^2

[torpid moat]

right right

[tribal jetty]

you could do it either (r-axis)^2 or (axis-r)^2 same thing, even after expanding: r^2 - 2r(axis) + axis^2 for both

[torpid moat]

right