help-27

[fathom thicket]

yh

dx is an infitessimally small change in x

integration adds up

all these changes

to make something that isnt infitessimally small

[idle tinsel]

oooooooooooooo

[fathom thicket]

its like, imagine if you walked from your house to the supermarket

and i broke down the distance you covered

into infinititly small distances

well when i add them all up

itll just give me the distance from your house to the supermarket

in the case of just

$\int_a^b dx$

[woven radish]

doctor99268

[fathom thicket]

you literally are just going from a to b

so the total change in x

is just b - a

b is your supermarket

a is your house

dx is the small bit of distance that you walked

[idle tinsel]

this is making an incredible amount of sense, i feel like I understand integrals so much better than before lol

[fathom thicket]

now if we want to use integration to find things

lets say

i wanted to use integration to find the area under the graph of f(x)

i know you know it

but imagine you didnt know

you would first just write out what small thing you are looking for

in this case it would just be dA, small bit of area

now

at these small distances, there is no complexity

you can just treat the bit of area as a normal shape

either a square or a rectangle

in this case, you can do a rectangle, since the vertical length is already given

well obv the actual rectangle would be much much much much thinner than this

the height of the rectangle is f(x)

the base of the rectangle is dx

and so dA = f(x) times dx

remember that we want to sum up all these dA

from a certain reigion

x = a to x = b

like how we summed up the distance you walked from your house to your supermarket

so we do $\int_C dA = \int_b^a f(x)dx$

[woven radish]

doctor99268

[fathom thicket]

the C just means

the reigion you want

youll see this alot in maths

its just me saying that this isnt an indefinite integral

there are reigions

i just havent specifyed it yet

@idle tinsel have i lost you?

[fathom thicket]

well you dont need to use C

you can use any letter

just as a single underscore

[idle tinsel]

sorry, i am getting help with two questions at the same time lol

[fathom thicket]

how could you cheat on me

lmao jk

[idle tinsel]

lmao

[idle tinsel]

ok this is making sense

[fathom thicket]

@idle tinsel now

I will walk you into something more complex

Let's say i wanted the length of a curve

Firstly i define what teeny bit i want to sum up

I will call this ds

Then. In the case of a curve

A teeny bit of it

Is just gonna be a straight line

The straight line has length of ds

You can express this

Interms of things we do know

Like dx and dy

We can use Pythagoras

So ds² = dx² + dy²

So ds = √(dx² + dy²)

.

Now to actually

Get it into a usable form

Cuz you can't really integrate that atleast not the way it looks.

You can write dx² + dy² as (1 + (dy/dx)²)dx²

So √(dx² + dy²) becomes √(1 + (dy/dx)²) dx

$\int_a^b \sqrt{1 + \left( \frac{dy}{dx} \right)^2 } dx$

[woven radish]

doctor99268

[fathom thicket]

So if i had the curve of x²

And i wanted to find the length of it

From x = 0 to x = 2

I would do

$\int_0^2 \sqrt{1 + (2x)^2 } dx$

[woven radish]

doctor99268

[idle tinsel]

$\int_0^2 x^2 dx$ would not work?

[fathom thicket]

Now this isn't about you knowing how to find the arc length of a curve

[woven radish]

Price

[fathom thicket]

No

That would give you

Area

Under x²

[idle tinsel]

ooo

[fathom thicket]

You can look at the units

Since dx is being multiplied

This means that your unit is increasing

As in

If y is in distance, and so is dx

Then

ydx will give you units of area

Because distance times distance

differentiation does the opposite

dy/dx would have no units

distance divided by distance

[fathom thicket]

In here

The dy/dx cancels out. So √( 1 + (dy/dx)²) has no units

There is only just dx

So the output will have units of distance

Integration literally just smallifies things you already know

Integration of area comes from the area of a rectangle

Integration of length comes from the Pythagoras theorem

Integration of volume of revolution (where you like rotate your curve around an axis) comes from the volume of a cylinder

You just say, what do i want (length), what is the small version of it gonna look like (straight line), how can i express this interms of what i do know (straight line has length of √(dy² + dx²), then you do the actual integration.

What do i want, area, what is the small version of it gonna look like, a rectangle, what is the area of this rectangle, just f(x)dx.

[idle tinsel]

what if we wanted to integrate whatever is after volume in the 4th dimension?

[fathom thicket]

Well this goes onto multivariable calculus.

[idle tinsel]

ah

[fathom thicket]

Where we deal with integrating 2 or more letters at the same time

Like integral of f(x,y)dxdy

But other than that, nothing changes

It's not like maths knows that we can't see the 4th dimension

[balmy carbon]

you guys are still going xD

[idle tinsel]

lol

might need to call it a night

thank you @fathom thicket

[balmy carbon]

Like how some of those questions turn into a tea party

[idle tinsel]

i think you could be a professor of mathematics irl if you are not already, if that is a field you're interested in lol

[fathom thicket]

This is just the fundamentals of calculus.

When i was at school

Very few people knew it

So when a question came up

It was a trick question

As in it was a really really simple integration question

But worded in a way that

People would be confused

If they didn't know what integration was actually doing

[idle tinsel]

https://www.youtube.com/watch?v=WsQQvHm4lSw
i am going to watch this tomorrow

maybe it will help with fundamentals

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[jaunty mantle]

x needs to be half of what √x is to get to the same y

oh wait

[coral locust]

Can be both
If we see sqrt(2x) as sqrt(2)×sqrt(x), it's vertical stretch by sqrt 2

If we see it as in terms of x, it's horizontal compression

Sorry gtg now

[devout snow]

@restive river Has your question been resolved?

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[sharp holly]

I’m having trouble concluding this issue

[sharp holly]

I need to use simple numerical substitutions to show or not show that the harmonic mean is always the reciprocal of the arithmetic

I used the formulas for both

Using the numbers 3/4/5

For some reason it’s not the reciprocal

[elfin hill]

Yes

It's true

[sharp holly]

I thought so

So when I sub then in I get not the reciprocal

Cause the arithmetic mean is 4

Which is definitely not the reciprocal of the hm

Or am I missing something simple

Or the values that go into are the reciprocal

[stone stump]

so you showed by example that they are not reciprocals of each other. So what's the problem? That's exactly what you were supposed to show

[sharp holly]

Well I just thought the harmonic is the reciprocal of the arithmetic

[stone stump]

well you thought wrong. which the example shows

[sharp holly]

Alrighty just read a couple of places it is

Which is why it made me scratch my head

Ah well

Thanks

[stone stump]

the reciprocal of the hm is the am of the reciprocals of the numbers

[sharp holly]

Yeah must of read it wrong

Thank you 🙏

And the negative of this statement would be the harmonic mean is never the reciprocal

Which is true

[stone stump]

well it could sometimes still be

"by coincidence"

[sharp holly]

Oh right

[stone stump]

the negation of an "always" statement is "there exists at least one exception"

[sharp holly]

Ohhh thank you

Well I don’t know where I can find that exception

[stone stump]

well you gave one

[sharp holly]

Ohhhh I see

[stone stump]

maybe I should have said counterexample instead of exception

[sharp holly]

Yeah you’re all good

Thanks a lot :)

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[lofty monolith]

for part a

[lofty monolith]

do you have to use this similar shapes method

not really familiar w that so would prefer another way lol

[sonic smelt]

I don't think there is a way avoiding similarity of shapes

[lofty monolith]

rip

how would u know to do that lol

is it just because they gave those two lengths at the start

idk on a level papers ive never seen it

[sonic smelt]

Well you need to get an expression for each of the circles and you are supposed to notice that its dimensions have same ratios to the base circle

[lofty monolith]

oh wait their diagram makes sense

ohhhhhhhhhh

yeah lol thanks

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[brazen pollen]

i need help

[sonic smelt]

Let's call the amount of job to be done J (pretty ambiguous, but it's fine)

Then clearly in a day Martin does J/N of the work where as John does J/(N + 2), right?

[brazen pollen]

ok

[sonic smelt]

So, if they worked together, they would do J/N + J/(N + 2) of job

And it's given that this number is also equal to 6J/(N + 6)

So we just solve for N in J/N + J/(N + 2) = 6J/(N + 6)

[brazen pollen]

thank you

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[idle cypress]

hi

[main gull]

• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.

And if you don't have a math question, don't open a help channel like this

.close

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[idle cypress]

.reopen

[devout snow]

✅

[idle cypress]

wait

.cose

.close

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[restive river]

Here in point 2 where $\cup_{i \in I}U_{i} \in \mathcal{T}$ Can i ask what $I$ they're talking about?

[woven radish]

Hin

[restive river]

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[vale merlin]

NOTE do not tell me the answer, just tell me how to solve
These similar triangles are drawn to scale. Find the missing side lengths.

[restive river]

Of course no one will say just the answer, it's not helpful at all.

[devout snow]

@vale merlin Has your question been resolved?

[pastel pasture]

I think you missed an image or something

[restive river]

Okay, how can we find AC?

[torn berry]

sides, angles of similar triangles are in proportion

ie AB/DF = AC/DE

angle ACB = angle DEF

[vale merlin]

how AC

<@&286206848099549185>

[main gull]

It's a right triangle, what theorem can you use to find the missing side of a right triangle?

[idle cypress]

$\matrix5+matrix7$

[woven radish]

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

[idle cypress]

why is this error

[main gull]

Not your channel anymore

Open a new one

[idle cypress]

not?oh sorry just realised once again sorry

[vale merlin]

pythogoream theorm

[main gull]

Can you use that to find the value of AC?

[vale merlin]

yes

i think

[main gull]

What's pythogoream theorm?

[vale merlin]

idk

[main gull]

The formula to be more exact

[vale merlin]

i think its pythogoream theorm

[main gull]

I'm asking you what the formula is

[vale merlin]

idk

[main gull]

Google it if you don't know

Google exists as a resource

[vale merlin]

I got this "The theorem is named for the Greek philosopher Pythagoras, born around 570 BC. The theorem has been proven numerous times by many different methods – possibly the most for any mathematical theorem."

[main gull]

What is the formula for pythogoream theorm?

[vale merlin]

{\displaystyle a^{2}+b^{2}=c^{2}.}

[main gull]

Do you know how to apply that to your triangle?

[vale merlin]

uh

...

[main gull]

Do you know how to use pythogoream theorm?

[vale merlin]

ab + bc = ac

[main gull]

No

[vale merlin]

thx for helping me ima show u something

[main gull]

Can you explain how you got those?

[devout snow]

@vale merlin Has your question been resolved?

[devout snow]

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[restive river]

How do u go about computing [
\int_{-\pi}^{\pi} \abs{\pi - \abs x} \dd x
]

[violet wind]

graph it probably

[sonic smelt]

Break the interval up, no?

Actually wait you don't really have to

Just use the symmetry

[restive river]

Oh yeah it's gonna be a triangle

Isn't it

[dark dawn]

graphing seems liek the best

[dark dawn]

ye

[violet wind]

yeah

[restive river]

Symmetrical over the y axis

[sonic smelt]

Why am I the only one not wanting to graph that

[violet wind]

idk why they included the outermost set of abs bars, they're useless

maybe to stop you from splitting the integral up

[sonic smelt]

No I even realized you don't need to split it up

[restive river]

Probably to scare you

[arctic field]

bars

[restive river]

Oh yeah @arctic field is dead scared of vertical lines

[arctic field]

[violet wind]

||||||||||||||||||||||||

[sonic smelt]

Just do $\int_{-\pi}^{\pi}\abs{\pi-\abs{x}}\dd{x}=2\int_0^{\pi}\abs{\pi-x}\dd{x}=2\int_0^{\pi}(\pi - x)\dd{x}$

[restive river]

Yeah sorry sorry sensitive topic ikik

[woven radish]

A Lonely Bean

[sonic smelt]

And here I would do the graph

Hm

[dark dawn]

or just do pi * pi

[sonic smelt]

Actually wait

King's rule

[violet wind]

answer = 10

[arctic field]

g

[violet wind]

that's what i said

[dark dawn]

[restive river]

but why isnt it the queen's rule

[dark dawn]

https://tenor.com/view/engineer-fixed-tape-gif-14723651

[restive river]

guys petition to make a queen's rule?

[sonic smelt]

$\int_{-\pi}^{\pi}\abs{\pi-\abs{x}}\dd{x}=2\int_0^{\pi}\abs{\pi-x}\dd{x}=2\int_0^{\pi}(\pi - x)\dd{x} = \int_0^{\pi}2x\dd{x} = \pi^2$

[violet wind]

no

[restive river]

[woven radish]

A Lonely Bean

[arctic field]

the queen died

its the king now

[restive river]

[sonic smelt]

If you come up with something new, sure

[arctic field]

just integrate (pi - x)

[sonic smelt]

Why if you can use King's rule

[arctic field]

its an easy integral

[violet wind]

king's rule?

[restive river]

wait wot

[arctic field]

rule for kings

[violet wind]

,w king's rule

[restive river]

just

[sonic smelt]

The u = a + b - x sub

[woven radish]

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[restive river]

why cant u like

[arctic field]

the shitty name for "you can reflect"

[sonic smelt]

$\int_a^bf(x)\dd{x}=\int_a^bf(a+b-x)\dd{x}$

[violet wind]

ah

[restive river]

[
\int \pi -x = \int \pi - \int x
]

[woven radish]

A Lonely Bean

[sonic smelt]

I can

[restive river]

[sonic smelt]

But that's a slightly bit more of work than just x

Soo

King's rule

[restive river]

dont want to do the power rule anymore

does the king's rule instead

[arctic field]

friendship ended with power

[restive river]

you are so deep and wise mister snow

[arctic field]

,w integral from -pi to pi of |pi - |x||

[woven radish]

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[restive river]

approximately g

[polar chasm]

aproximately 1

[arctic field]

,w g

[sonic smelt]

What? Why is wolfram approximating an integer

[woven radish]

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[sonic smelt]

[violet wind]

,w 10

[woven radish]

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[restive river]

[
g = \pi^2 = 10
]

[restive river]

[polar chasm]

sin(x)=x, sin(pi)=0=pi, pi^2=0^2, pi^2=0

[sonic smelt]

10 = 1 * 0

Shorter proof

[restive river]

i never understood the sin(x) = x meme

any big brain person care to explain?

[sonic smelt]

x is the linear approximation of sin(x) around 0

[arctic field]

,w plot sin x and x

[woven radish]

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[violet wind]

,w sin(0.01)

[restive river]

oh

[woven radish]

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[polar chasm]

I use it to solve most limits, sin(x)=x, cos(x)=1, ln(x)=x-1, x is tangent to sin(x) at 0

[arctic field]

thats just taylor with more steps

[dark dawn]

simple pendulum moment

[sonic smelt]

Yeah I was gonna mention this approximation is used in the pendulum's diff eq

Because its solution is non-elementary

[zenith jacinth]

bro ?

[violet wind]

it doesn't depend on angle trust me

[sonic smelt]

Don't take this channel seriously

[arctic field]

take this channel very seriously

[violet wind]

don't set theta to 179 degrees just trust me

[sonic smelt]

Trust me the solution is a sinusoid

[violet wind]

pendulum = simple harmonic oscillator or smth

[restive river]

sorry i am an engineer, forgot to introduce myself

[wooden veldt]

Hello an engineer

[sonic smelt]

Hopefully that's sarcasm

[restive river]

no i am a very very serious person.

[zenith jacinth]

people were memeing about that g = pi^2 = 10, but I've never seen a single person actually do it :x

[restive river]

added the period there for extra effect too

[restive river]

it is maaaybe a good idea if u r like

[violet wind]

yea everyone knows pi isn't sqrt(10)

[restive river]

in a no calc exam

[violet wind]

it's 4/sqrt(phi)

[polar chasm]

If you are engineer, why dont you do one term riemann sum instead of actually integrating it

[restive river]

[arctic field]

,w 4.0/sqrt((1 + sqrt(5))/2) - cbrt(31.0)

oh bruh

[restive river]

nothing better than a wolfie wall

[violet wind]

it interprets phi as a variable lol

[sonic smelt]

,w 4/sqrt((1 + sqrt(5))/2)

[woven radish]

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[violet wind]

,w phi^2 at phi = phi+1

[zenith jacinth]

close enough

[woven radish]

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[zenith jacinth]

maths in engineering is boring anyway

i thought i would learn manifolds

[sonic smelt]

I wanted to say something but forgor

[arctic field]

[sonic smelt]

Oh, ye

If you are an engineer name all engines

[restive river]

[violet wind]

stockfish

[restive river]

saving that

[arctic field]

its in hwh

[sonic smelt]

alphazero

Or whatever it's called

I forgor

[restive river]

the frick is hwh

[violet wind]

deep blue

[restive river]

oh homework help

was i meant to know that

[arctic field]

yes

[restive river]

snow are u getting ur homework done in 'hwh'

[sonic smelt]

https://www.youtube.com/watch?v=CzTCqW89Pxs

[zenith jacinth]

[arctic field]

[wooden veldt]

!help

[devout snow]

Please read #❓how-to-get-help

[restive river]

damn sigma...

mean

anyways enough meming

is it memeing or meming

anyways

.close

[devout snow]

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[zenith jacinth]

both is gud

[sonic smelt]

I thin memeing

[wooden veldt]

Mememeing

[sonic smelt]

Actually now that I am exposed to such thought, meming sounds more natural than I expected

[devout snow]

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[grizzled python]

I understand how to solve this question, but the addition of the white checkers are confusing me. How would you use pascal's triangle when the white checkers are obstructing the paths?

[devout snow]

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[grizzled python]

.reopen

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✅

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@grizzled python Has your question been resolved?

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[sinful copper]

HIIII

[sinful copper]

this is very simple

but i forgot how to do it

anndddddd

i dont understand it when i search it up

sorry :))))

ive done

4(x^2 - 8) + 5

and then

4(x - 4)^2 -16 +57

but

i dont think its right

https://tenor.com/view/cat-the-cat-he-dance-he-dance-gif-24077288

[restive river]

u went wrong here

factor out 4 from 4x^2-32x

what do u get?

[sinful copper]

x^2 - 8x?

[restive river]

yea

4(x^2-8x)+57

then complete the square for x^2-8x

and expand back out the constant

[sinful copper]

okay thank you :)

il keep it open if i need more help but then i will clsoe it]

have an awesome day

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[restive river]

If sin theta = 4/5, find the exact value for tan theta

[restive river]

Need help

[cosmic breach]

Just find theta using sin-1 and use it to find tan

[restive river]

could i use the pytagorean theorm instead?

because i figured sin is opp/hyp

so 4^2 + b^2 = 5^2

b=3

i think i solved it actually

would it be 4/3?

[cosmic breach]

Good job

[restive river]

thx

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[lament flint]

Is this a power series?

[cosmic gust]

First simplify the integral

$\int_1^\infty ln(x) e^{ln(x) (n - 1)}dx$

$u = ln(x)$

$e^u = x$

$\frac{du}{dx}x = 1$

$du x = dx$

$\int_0^\infty u e^{u(n - 1)} e^u du$

$\int_0^\infty u e^{un} du$

$v = -un$

$-\frac{v}{n} = u$

$\frac{dv}{du} = -n$

$-\frac{dv}{n} = du$

$\int_0^\infty \frac{v}{-n} e^{-v} \frac{dv}{-n}$

$\frac{1}{n^2} \int_0^\infty v e^{-v} dv$

$\Gamma(2) = 1! = 1$

So the question is now

$\sum_{n=1}^\infty \frac{1}{n^2} = \frac{\pi^2}{6}$

[devout snow]

@lament flint Has your question been resolved?

[cosmic gust]

I'm not sure if I have solved it in the intended way, but here you go

[woven radish]

Oğuzhan

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[fierce pelican]

Is this statement true ?

[fierce pelican]

f can have any number of stationary points without having a single real root

given that degree of polynomial is n

so does that mean n-1 complex roots of f are responsible for those stationary points

[pseudo basin]

youll need at least one stationary point if you wanna go without real roots

[fierce pelican]

i need or it at least exist ?

@pseudo basin

[pseudo basin]

if f has no real roots, then it has at least one stationary point

[fierce pelican]

what about at most condition ?

@pseudo basin

[pseudo basin]

well the number of turning points of f is the number of roots of f', so at most n-1

[fierce pelican]

thanks

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[wooden axle]

Option C is correct right?

[wooden axle]

But the answer sheet says 4 option

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[lofty monolith]

,rotate

[lofty monolith]

Is this wrong

[small raptor]

is that $e^x^2$

[woven radish]

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

[final laurel]

,w integrate e^(x^2)

[lofty monolith]

Yeah e to the power of x squared

[woven radish]

Results provided by WolframAlpha

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Requested by keto11#8349

[small raptor]

yea

thats like

horrible

to integrate

[lofty monolith]

Oh

[small raptor]

you get error function

[lofty monolith]

Why is there pi

I tried checking online aswell

😭

[small raptor]

because you cannot integrate e^x^2

[lofty monolith]

HUHH💀

[small raptor]

Its not possible

[lofty monolith]

Why?

[final laurel]

try doing a u sub for e^x^2 you can't cancel out the 2x

[lofty monolith]

It seemed like a by parts

Oh

I thought u can

Because of the x squared aswell

[frosty cradle]

,w integrate (x^2 * e^(x^2))

[woven radish]

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[lofty monolith]

someone sent this to me and it’s from a past paper apparently but I’m gonna verify now💀

I got first part at least lmaoo

Thanks

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[restive river]

Hello

[restive river]

I got this math thing

1. During the three stages of production, the following amount of water is used to produce one ton of products:
   in the first phase 6,9 hectoliters, in the second phase 645,7 liters and in the third phase 31,2 dm^3.
   a) What is the total water consumption per ton of products?
   b) How much water will be needed for 264 kg of products? Express the result in Hectoliters.

I dont know how to start

[pseudo basin]

the total water consumption is the sum of water consumptions per phase

naturally, you'll need to convert them to the same unit before adding them.

@restive river

[devout snow]

@restive river Has your question been resolved?

[restive river]

Alr then it is 1 366,9 Litres

A= 1 366,9

[pseudo basin]

1366.9 liters per ton.

[restive river]

Now i suppose the B is meant to be

x/1 366,9 = 264/1

[pseudo basin]

264/1000, surely?

[restive river]

What?

[pseudo basin]

the target production is not 264 tons, it is 264 kilograms

[restive river]

Yes

[pseudo basin]

a ton is 1000 kilograms

[restive river]

Yes

[pseudo basin]

(x liters)/(1366.9 liters) = (264 kg)/(1000 kg)

if you want to do it that way

(instead of just multiplying the consumption per ton by the 0.264 tons of product you want to produce)

[restive river]

Ahhh

Yes

We do it by three thing

ahhh and that is

360,8616

[pseudo basin]

"we do it by ___"

does this mean "Doing this with any other method is prohibited under threat of failure on the entire assignment"?

surely not.

[restive river]

I mean no

But i know this method only

we never learned any other method

[pseudo basin]

if you had the question

Apples cost 35 Kč per kilogram. You buy 5 kg of apples. How much do you pay?

would you also go through the trouble of making a proportion like this?

[restive river]

5*35

[pseudo basin]

exactly

[restive river]

175

[pseudo basin]

i mean you COULD do it like x/35 = 5/1

[restive river]

oh

[pseudo basin]

but it's wasted effort imo

and just like yours

[restive river]

Thats also possible

[pseudo basin]

you found the total water consumption as 1366.9 liters/ton

and you're producing 0.264 tons of product

the numbers are no longer integers but that makes no difference

[restive river]

How did you get Kč :D?

[pseudo basin]

i happen to know the names and symbols for some currencies around the world.

for the czech koruna in particular it's because my dad went to czechia once when i was a kid and brought back some coins from there

[restive river]

Even Koruna Česká?

[restive river]

You all go to Prague 😞

[pseudo basin]

i was never there, only my dad. on a business trip

i don't remember where he went

[restive river]

Ohhh yeah

Prague

Cuz thats the business

[pseudo basin]

probably

[restive river]

No one ever goes to Silesia

[pseudo basin]

anyway, yeah.

the proportion thing is helpful if you're not yet comfortable with thinking of fractions as numbers in their own right.

[restive river]

The machine works continuously and produces one product in 14 min 7 s. How many products does it produce:
a) in 24 hours? Only list the number of complete finished products.
b) in 264 days?¨

I also got this one

I think i understand this one now

So

24 hours: 86 400 seconds

14 mins and 7 secs is 847 seconds

that means

102 per day

now 264 * 102

26 928

[restive river]

a

[restive river]

b

Right?

@pseudo basin

[pseudo basin]

,calc 264*86400/847

[woven radish]

Result:

26929.87012987

[pseudo basin]

,calc 86400 - 102*847

[woven radish]

Result:

6

[pseudo basin]

it looks like there is just enough extra time accumulated over these 264 days to produce a single extra product

[devout snow]

@restive river Has your question been resolved?

[devout snow]

@restive river Has your question been resolved?

[devout snow]

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[unreal pollen]

I need help with this question, I tried reading the step by step instructions but I don't see how they got the answer

[restive river]

whats 8 as a power of 2

[sonic smelt]

And 32 as well

[unreal pollen]

2^3?

[restive river]

yea

what about 32?

[unreal pollen]

I know it's 5/3 but how? How do I use that power to get 32? What's the steps?

[restive river]

whats 32 as a power of 2

once u make them the same base its just algebra

[unreal pollen]

2^5

[restive river]

u can log both sides

alr so

$2^{3^x} = 2^5$

[woven radish]

hibyehibye

[restive river]

so

$2^{3x} = 2^5$

[woven radish]

hibyehibye

[topaz axle]

@unreal pollen
(2^3)^x = 2^5
2^(3x) = 2^5

[restive river]

do u agree ?

[unreal pollen]

yes i do

ohh

[restive river]

so from here u can log both sides

3x=5

[topaz axle]

like yeah, you have to just know both those powers

[unreal pollen]

okay I get it now, just powers then logs afterwards

okay thank you guys

.solved

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[ornate crypt]

Question: Determine if the sequence is monotonic and if it is bounded.

[ornate crypt]

I am still a confused on how to determine if it is monotonic and want more clarity

[cosmic gust]

The bottom increases faster than the top after n = 12

[wooden salmon]

Try to compute $a_{n+1}/a_n$

[restive river]

a_n+1 = a_n * 12/(n+1)

[woven radish]

Micni9

[ornate crypt]

To find that after n=12 it increases faster than the numerator, do we randomly plug in or how can I know this?

[restive river]

write it as a recurrence relation

[restive river]

you get this and notice its 12/(n+1)

[cosmic gust]

First use ratio test to see if it goes to 0

[cosmic gust]

Then check which part increases faster

[ornate crypt]

Oh I see

[cosmic gust]

$\frac{12^n}{n!}$

[woven radish]

Oğuzhan

[cosmic gust]

Imagine it like

$\frac{12 \cdot 12 \cdot 12 ... \cdot 12 \cdot 12 (n times)}{1 \cdot 2 \cdot 3 \cdot ... \cdot n}$

[woven radish]

Oğuzhan

[cosmic gust]

As you can see in here

The number in the denominator are smaller than 12 when n < 12

So the bottom will be bigger

[ornate crypt]

So therefore it is monotonic

[cosmic gust]

After n = 12 the denominator will get much bigger than the top because it will start to multiply by 13 14 15 16 instead of 12 12 12 12

[cosmic gust]

After 12

[ornate crypt]

And before 12 it is not

[cosmic gust]

You can see it here in the graph

It's not exactly 12 or the graphing calculator is wrong

But as you can see around 12 it starts to decrease

[ornate crypt]

Sorry I'm slow in the head but can we go back on solving for 12x/x!

[cosmic gust]

Which part

[ornate crypt]

finding the ratio

[cosmic gust]

Ratio Test?

[ornate crypt]

yes

a_n+1 / a_n

[cosmic gust]

$\lim_{n\to\infty} |\frac{a_{n+1}}{a_n}|$

[woven radish]

Oğuzhan

[cosmic gust]

Do you know how it works?

[ornate crypt]

not really

[cosmic gust]

Basically

If this limit's result is smaller than 1

it converges

If it's 1, then it's inconclusive and we can't decide

If it's larger than 1 it diverges

You can think of this as if next term divided by the last term is bigger than 1

Then the next term is bigger because the ratio is bigger than 1

[ornate crypt]

ah ok

[cosmic gust]

So our function doesn't go to 0 near infinity

If we apply it

[ornate crypt]

can you show me example using the problem above

[cosmic gust]

$\lim_{n\to\infty} |\frac{12^{x+1}}{x!(x+1)} \frac{x!}{12^x}|$

[woven radish]

Oğuzhan

[cosmic gust]

the formula?

[ornate crypt]

the algebra

[cosmic gust]

$\lim_{n\to\infty} |\frac{12}{(x+1)}|$

[woven radish]

Oğuzhan

[cosmic gust]

Oh wait I used x instead of n

My apologies

[cosmic gust]

What do you mean?

[ornate crypt]

like how did we get here

is it just algebra?

[cosmic gust]

From here

Yes it's just algebra

[cosmic gust]

Since we know a_n

[cosmic gust]

And as you can see this gives us 0

So at near to infinity it's 0

So it converges

Which can be seen in the graph

[ornate crypt]

i see

i get the conclusion but i suck at algebra

lol

[ornate crypt]

i can't do the algebra to get here.

give me some time

[cosmic gust]

You just substitute the value of a_n+1 and a_n to there

[ornate crypt]

so a_n+1 is (3^n+1 4^n+1)/(n+1)!?

and a_n is given

then simplify?

[cosmic gust]

Yep

You can bring 3 and 4 together

so it becomes 12^(n + 1)

[ornate crypt]

oh i see

then how about the bottom x!(x+1)

[cosmic gust]

(x + 1)! can be written as x! (x + 1)

Think it as

for example for x = 4

1 * 2 * 3 * 4 * 5 = 1 * 2 * 3 * 4 * (4 + 1)

[ornate crypt]

oh ok

they are equivalent

[cosmic gust]

Yes

[ornate crypt]

so 12n!/(n+1)!

[cosmic gust]

Yes

[cosmic gust]

wait no

[ornate crypt]

fuc

[cosmic gust]

12^(n+1)

[ornate crypt]

this is embarrassing i can't do basic algebra lol

[cosmic gust]

Algebra is just about practice

[ornate crypt]

ok so can you guide me step by step? let's start with (a_n+1)/(a_n) which is ( (3^n+1*4^n+1)/(n+1)! ) / ( (3^n * 4^n) / n!)

right?

then I multiply the numerator by reciprocal of denom
((3^n+1 * 4^n+1) / (n+1)!) * ( n! /3^n * 4^n )

combine (3^n+1 * 4^n+1 * n!) / (n+1)! (3^n * 4^n)

cancel out common factors (3* 4n!)/( (n+1)! )

i still get 12n! / (n+1)! ?

what am i doing wrong

[wooden salmon]

$(n+1)!=n!*(n+1)$

[woven radish]

Micni9

[ornate crypt]

so I can also write it as 12n! / n! (n+1)

[wooden salmon]

the ratio should be 12/(n+1)

[wooden salmon]

yup

[ornate crypt]

oh oops i see it now

didn't know you can cancel factorials out like that

ok thanks now it makes sense :D

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[frail lake]

so im trying to prove this by induction
Im not going to write the base case to save time
(but i wrote it down on my paper)
so for hte assumption i have assume for some k > 1, that the product of these k consecutive integers is divisible by k!
this means that k! | (n)(n+1)...(n+k-1)
then i said let Lk! = (n)(n+1)...(n+k-1) for some integer L.
then consider k + 1. We have (n)(n+1)...(n+k-1)(n+k)
I can rewrite this according the assumption to the following:
Lk!(n+k)
Then since n >= 1, I can say that (n+k) = [(k+1) + (n - 1)]. Thus Lk![(k+1) + (n - 1)]
now im confused on where to go and i can't really make any progress

[restive river]

product from 1 to k = qk! for some integer q, then product from 1 to k + 1 = product from 1 to k * k+1

inductive hypothesis

u get q(k+1)!

[red heath]

i saw something about it

wait

[restive river]

urrr

maybe im dumb given the reactions lol

[wooden veldt]

its not product from 1 to k

its product of k consecutive positive integers

[restive river]

ohhh

[red heath]

i dont know where did i get it from but the exact problem

[frail lake]

wait so did i do anything wrong thus far?

[wooden veldt]

it looks fine but you do seem to be at a road block

[frail lake]

yes. So was the path i took wrong?

[wooden veldt]

it just might be that induction isn't very easy here

[frail lake]

oh

[wooden veldt]

you esentially need to show that Lk!(n-1) is divisible by (k+1)! which isnt immediately obvious to me

[frail lake]

is there another route I can take besides induction?

[red heath]

this?

[frail lake]

oh shit, didnt even see that lol

my b

[red heath]

np

[frail lake]

how do you know that it equals (s+k-1)!/(s-1)!?

that doesn't look obvious to me

oh wait

nvm

[red heath]

it is starting from number s and multiplying every number to k afterwards

[restive river]

ohhh this is smart

[frail lake]

it makes sense

[red heath]

there were (s-1)(s-2).... besides so

we just divide by (s-1)!

[frail lake]

where is this proof from?

just curious

[red heath]

i cant remember where did i get it from

it just seemed cool and i took it

not mine

[topaz beacon]

its used in some olympiad proofs as a sleek way of proving that

[frail lake]

lol

that's interesting. Ive never done math olympiad

i prob should though, i just feel like it would make me feel bad about myself lol

[topaz beacon]

it makes me feel bad, dont worry

[red heath]

where did you come across with this problem

some olympiad books or?

[topaz beacon]

this feels like a class problem

[frail lake]

yes this is from my discrete math class

[red heath]

oh okay

[frail lake]

wait what's binomial (s-1, k)?

[red heath]

C

[frail lake]

like s-1 choose k?

[red heath]

yeah

(s+k-1)!/(s-1)!*k!

thats the formula

s-1 choose k

[frail lake]

okay got it

thank you so much. This is a really genius proof lol

[red heath]

yep its cool

[frail lake]

.close

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[restive river]

?

[fair minnow]

!status

[devout snow]

What step are you on?
1. I don't know where to begin
2. I have begun but got stuck midway
3. I got an answer but I'm told it's wrong
4. I got an answer and would like my work checked
5. I have a question about someone else's worked solution
6. None of the above

[fair minnow]

can you let me know how far into the question you are?

[restive river]

i think im confused by the bounds

can i say f(-inf) = 10 + inf - 0?

[fair minnow]

i think you've made a mistake at the f'(x) stage

let me double-check

oh no you did get

x=+/-1

right

i'm not really sure what you're doing after that, though

[restive river]

did the 1st derivative test to check for local min/max

[fair minnow]

would you not just substitute the maximum value's x value you found into f(x) = 10 - x - 1/x

i dont think your working on the right is necessary

but otherwise you're correct

it's 8

[restive river]

.close

[devout snow]

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[rigid rain]

Can someone explain the answer to 13C?

[devout snow]

@rigid rain Has your question been resolved?

[devout snow]

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[devout snow]

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[visual solstice]

.close

[devout snow]

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[devout snow]

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[topaz quiver]

is this correct?

[graceful elm]

no?

It's x - ipi

[topaz quiver]

how do I manipulate the -e^-x such that only x remains

[graceful elm]

(don't quote me)

[topaz quiver]

like im simplifying it or something

[graceful elm]

ln -(-e)^-x

[topaz quiver]

oh ok alirght

thanks

.close

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[graceful elm]

If you put a negative number in ln you get a complex number (most of the time)

[devout snow]

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[wooden axle]

How to find the value of this function?

[devout snow]

@wooden axle Has your question been resolved?

[toxic flower]

you have to break the integral at integer points

[pastel pasture]

Yeah just do it geometrically

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[wooden axle]

I tried it but failed

[toxic flower]

like anything given in question is a and b are integers or so?

[wooden axle]

No no I don't know how to face floor x

[toxic flower]

you have to break it at integer points

[wooden axle]

Suppose i take 2.1 = x

[toxic flower]

hm

[wooden axle]

Floor 2.1 + (- floor 2.1) = 2+(-3) =2-3 = -1 ??

[toxic flower]

yes

[wooden axle]

Is this correct?

But now I'm having difficulties with x why?

[toxic flower]

but remember limits are opposite in both of the integrals

[sonic smelt]

floor(2.1) + (-floor(2.1)) = 0
floor(2.1) + floor(-2.1) = -1

[toxic flower]

hm ye lol

[wooden axle]

Yeah i understand it

Thanks for giving me that deep knowledge

Can I change limit of floor -x a to b?

Adding negative sign?

[toxic flower]

ys

[toxic flower]

you forgot to change limit*

[wooden axle]

Opps sorry

Last one

[toxic flower]

not correct still

why you put negative sign to your limit

[wooden axle]

It's by mistake

Ahh i didn't add it

Ahh wait

I am doing it so quickly

[toxic flower]

you didnot even noted corect question

[wooden axle]

Didn't even noted?

I'm learning English too these day

[toxic flower]

bruh

[wooden axle]

I hope this is correct. Wrote it with patience

[sonic smelt]

Should be from -b to -a in the latter integral

[toxic flower]

why the hack are you putting equals sign in between

[wooden axle]

So that I can replace the negative value with positive value and add

[toxic flower]

;-;

you indian?

[wooden axle]

Ha bhai

[toxic flower]

ha toh hindi me samjhata

[wooden axle]

We can talk over VC if you prefer

[toxic flower]

dekh agar koi integral de rkha hai $\int_a^bf(x)dx$ then you can replace it by switching sign of function and replacing limits like $\int_b^a-f(x)dx$

[woven radish]

Hack With Techno Boy

[wooden axle]

@sonic smelt

[sonic smelt]

Why does x in the first integral magically turn into floor(x)

[wooden axle]

Opps i left floor sign@sonic smelt

[sonic smelt]

The rest is fine

[wooden axle]

@sonic smelt

[sonic smelt]

x goes from a to b, it is unsafe to assume that there exists some k such that floor(x) = k for all x on the interval

Unless a, b are both on an interval [k, k+1)

And, btw, integral of 2k+1 from a to b is just (2k + 1)(b - a)

No idea how you got (2b + 1) - (2a + 1)

So $\int_a^b\floor{x}\dd{x}+\int_b^a\floor{-x}\dd{x} = (2\floor{a} + 1)(b - a)$ if $\floor{a}=\floor{b}$

[woven radish]

A Lonely Bean

[toxic flower]

how ?

[sonic smelt]

Went on the same path as Arjunn but with the assumption that I mentioned

[restive river]

what will be 0 raised to the power zero and its graph?

[sonic smelt]

1. Post your question in an available help channel
2. 0 raised to the power of zero is undefined
3. Numbers don't have graphs

[restive river]

Sorry i didnt knew it was occupied im new

[wooden axle]

Check this one

@sonic smelt

[sonic smelt]

It's fine, the available help channels are below the label "MATH HELP (AVAILABLE)"

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