help-10

[low cipher]

So to formulate a proper question,
if i have played 157 cards, how can i calculate how many decks are still left in play

[flat rune]

you just want to know how many cards are left, you're not concerned with which cards have been played or if there even is any full deck (all distinct 52 cards) left?

after youve played 8 cards , there is a possibility you have no more full decks

otherwise you just take 416 - 157 , divide it by 52 and round down

[low cipher]

thats not what i mean

it doesnt matter what cards have been played

just how many

[flat rune]

okay then you take (416 - X)/52 and round it down to the nearest whole number

where X is the number of cards you've played

[low cipher]

okay

ill try that

thank you

[obtuse pebble]

@low cipher Has your question been resolved?

[obtuse pebble]

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[final sonnet]

Not sure how to approach this

[high lily]

consider the period of cot

add/subtract integer multiples of that from the argument until you get something more recognisable

[final sonnet]

yeah i get that so thats what i tried

The first thought i had was to simpliyfy it to 5pi/3 + 5pi/3 but that obviously doesnt work

[high lily]

The first thought i had was to simpliyfy it to 5pi/3 + 5pi/3 but that obviously doesnt work
that's not representative of what i described

nor is 25pi/6 equal to what you wrote

[final sonnet]

Yeah got that but what would be an alternative

[high lily]

what i wrote...

[final sonnet]

this?

[high lily]

that combined with the line above it

[final sonnet]

Could you go in a little more in depth ?

[high lily]

do you know the period of the cot function

[final sonnet]

What do you mean by period?

[flat rune]

@final sonnet maybe read this https://www.cuemath.com/algebra/periodic-function/

[final sonnet]

so its period is pi?

if so how does that help me ?

[flat rune]

yes the period is pi, that means that for any x, cot(x) = cot(x + pi) = cot(x + 2*pi) = cot(x - pi)

you can just add or subtract pi as many times as you want

[final sonnet]

ngl my mind just exploded

[final sonnet]

Ok

But i dont see how this helps me tho ?

Like it gives me context but how do i apply it ?

[flat rune]

what ramonov was saying was that you can change 25pi/6 to something else which might be more recognisable

but i guess it depends if youve ever dealt with the cot function

[final sonnet]

yeah

[final sonnet]

something like what tho?

I got that part

trying to simplify it to a fraction more recognizable

But i dont even know what that would be

[flat rune]

are there any values of x where you know what cot(x) is?

[final sonnet]

i have this resource to help

So thats like

What im trying to turn 25pi/6 to ?

to a fraction recognizable from there?

[flat rune]

yes, exactly

[final sonnet]

the problem is im not sure how i can modify 25pi/6

[flat rune]

is it the fraction sign that confuses you?

like do you generally have trouble with fractions?

[final sonnet]

no its the fact i cant seem to simplify the fraction

say i have cos(7pi/12)

I could turn that into cos(3pi/12 + 4pi/12) ?

[dusk quail]

Isnt thats a radian value?

[final sonnet]

and then pi/4 and pi/3

yes im working with radian

[dusk quail]

Yea sth like that

[final sonnet]

yeah but i dont know how to repeat the process with 25/6?

[flat rune]

no this doesnt help you at all

anyway 7pi/12 is a bad example i think

[dusk quail]

Bro like thats a very shit number

[flat rune]

because it doesnt appear in that chart of yours. nothing there is divided by 12

[dusk quail]

25×30>360

U can find some values but i dont think it makes sense :/(by breaking it ofc)

[final sonnet]

but it turns into this

so then i would have my chart?

i think

[flat rune]

how does 7pi/12 turn into pi/4 and pi/3? do you mean pi/4+pi/3?

[dusk quail]

💀

[final sonnet]

yes

[flat rune]

oh right

[dusk quail]

It is

Bro is sleeepy

[flat rune]

but this isnt the same as ramonov was talking about at all

youre not using the period of the function

youre just rewriting a fraction

[dusk quail]

Btw dont u guys think that doesnt make sense?

[dusk quail]

.

[final sonnet]

well then im completely lost

[flat rune]

now im lost as well

[flat rune]

did you understand this?

[final sonnet]

not well enough

[flat rune]

for any x, that is true

so put in the x from your problem

what do you get

write it out!

[final sonnet]

theres no x?

[final sonnet]

you mean

i make it = to x?

[flat rune]

theres an x in what i wrote

now you put in something instead of x

for example

the thing from your problem

[final sonnet]

25pi/6 + pi?

[flat rune]

since what i wrote is true FOR ALL x, it will also be true for the number in your problem

[flat rune]

what about cot? write the full thing cmon

[final sonnet]

im not sure what you mean ?

[flat rune]

it's very simple

take what i wrote. the equation i wrote

replace x with something

lets try 24pi/6? or maybe 23pi/6?

no lets try 25pi/6

that seems reasonable

[final sonnet]

one sec

[flat rune]

what i mean is.. i wrote cot(x) = cot(x + pi) = cot(x + 2*pi) = cot(x - pi)

you write that now, here, while i watch

but instead of 'x' you write '25pi/6'

[final sonnet]

cot(25pi/6) = cot(25pi/6 + pi) = cot(25pi/6 + 2*pi) = cot(25pi/6 - pi)?

[flat rune]

right right

the point is that these are all the same

as well as... cot(25pi/6 - 2*pi)

cot(25pi/6 - 3*pi)

cot(25pi/6 - 4*pi)

cot(25pi/6 - 5*pi)

etc etc

[final sonnet]

right

which means?

[flat rune]

can you rewrite one of those?

like simplify it

[final sonnet]

what like make it -25pi?

[flat rune]

for example what is 25pi/6 + pi

[final sonnet]

26pi/6?

[flat rune]

write it as one fraction

nope

[final sonnet]

25pi/6 + pi/1 so 31pi/6?

[flat rune]

yeah!

31pi/6

but thats not any easier to work with

its even worse

what if we do minus instead of plus

25pi/6 - pi?

[final sonnet]

19pi/6?

[flat rune]

yes

do it again

19pi/6 - pi

[final sonnet]

13pi/6

[flat rune]

keep going

[final sonnet]

7pi/6 then 1pi/6

[flat rune]

yeah

pi/6

[flat rune]

which appears here

[final sonnet]

ok and then cot means we are looking for what in here

[flat rune]

so cot(25pi/6) = cot(pi/6)

[final sonnet]

yeah

[flat rune]

since we can just subtracted pi (or add pi) as many times as we want in cot

and then maybe you can take it from there?

[final sonnet]

this tho

like

what am i even looking for as an answer?

like

what it wants the angle ?

[flat rune]

do you know the definition of cot?

[final sonnet]

no

[flat rune]

why are you trying to solve this problem again?

[final sonnet]

part of the class im in

[flat rune]

have they taught you about cot before?

[final sonnet]

its online so yeah without further explanation but the explanation given didnt suffice so i tried to learn it onm y own

but never actually understood it

its the same with all the other ones

[flat rune]

what about

sin? or cos?

[final sonnet]

arcsec csc sec sin cos tan arctan arcsin

arccos

like

why so many freaking words

[flat rune]

so youre basically no longer asking how to solve cot(25pi/6)

youre asking "tell me about trigonometry"

which is kinda beyond what i can do here :v

[final sonnet]

wish my teacher gave us more than a 7 minute video but yeah thats fine

Ill read more about it tommorow

see what i can come up with

[flat rune]

i'll just tell you that cot(x) = cos(x)/sin(x)

[final sonnet]

ok

so for instance here

cos is x sin is y ?

so then

it would be

divided by 0.5

so sqr3 divided by 1 ?

[flat rune]

you like to divide things by 1

but yeah

just say sqrt3 :p

[final sonnet]

yeah it helps me visualize it

[flat rune]

why dont you test it

[final sonnet]

?

[flat rune]

thats a good theory

[final sonnet]

test it ?

[flat rune]

like simply go to google and type cot(pi/6)

,cot(pi/6)

[final sonnet]

well i need to understand it tho

[flat rune]

,,cot(pi/6)

[final sonnet]

then answer wont make me pass the exam

[flat rune]

oh i thought you could use it here

yeah yeah you have understood a bunch of things now (hopefully)

but this is just to test your answer

[final sonnet]

yeah

Well i understand that not the rest but ill check that out on my own obviously

thank you very much

[flat rune]

https://www.wolframalpha.com/input?i=cot(pi%2F6)

see it is sqrt(3)!

[flat rune]

this is correct too

[final sonnet]

yeah i got that part from the 7 min video he gave

[flat rune]

at 0 degrees, or radians, you get cos is 1 and sin is 0

[final sonnet]

basically the only thing i fully understood

[final sonnet]

yeah

[flat rune]

i also always forget what cot/csc etc etc means

so i have to look it up

theyre all based on sin and cos though

[final sonnet]

1/sin

is csc

arc means the opposite from what i understood?

So arcsin is sin^-1

same for the others

sec idk what it is yet

[flat rune]

yeah so arcsin(1/2) will get you pi/6

since sin(pi/6) = 1/2

[final sonnet]

arcsin 1/2 is -1/2?

Like

if i go with my chart

the opposit of sin1/2

is basically what its asking

[flat rune]

huh wait

where do you look at for sin(1/2)

[final sonnet]

if you ask me sin(1/2) = ?

ill answer

pi/6

because this

[flat rune]

no you read it the wrong way

the pi/6 is what goes inside sin

and 1/2 comes out

sin(pi/6) = 1/2

[final sonnet]

oh right okay

[flat rune]

however with arcsin, its the other way

arcsin(1/2) = pi/6

thats all

[final sonnet]

right

[flat rune]

so actually answering sin(1/2) is hard. its not on the chart

[final sonnet]

then calculator wise its just sin-1

well id just do calculator then i think

[flat rune]

yeah its sometimes written sin^(-1)

$sin^{-1}$

[warm shale]

reking

[final sonnet]

one more question

if sec is = to 1/cos

Then arcsec = cos/1?

[flat rune]

this confuses me

ohh

[flat rune]

youre writing is equal to

[final sonnet]

yeah that

[flat rune]

okay lol

yes?

[final sonnet]

so then arcsec would be the flipped fraction

since its the opposite

[flat rune]

you mean arcsec x = cos x?

thats not true

[fierce lagoon]

This is the same as $\cot{\left(\frac{\pi}{6}\right)}$

[warm shale]

Umbraleviathan

[final sonnet]

is what i mean

[cedar lichen]

arcsecx ≠ 1/secx

[fierce lagoon]

No that's not what it works

arcsec(x) is the inverse of sec(x)

[final sonnet]

yeah

and sec is the thing at the top so 1/cosx

so the inverse is the -cos?

[fierce lagoon]

Inverses are not just flopping the fraction and then multiplying by -1

[flat rune]

inverse doesnt mean that you flip the fraction

or put a minus sign in front

exactly :.p

[final sonnet]

What does it mean then ?

-cosx?

[fierce lagoon]

It means you flip the function across y=x

That's what inverses are

[flat rune]

look at your chart

[cedar lichen]

If secx = y then x = arcsec y

[fierce lagoon]

You take a function, flip it cross y=x

[cedar lichen]

If you're asking about a relation relating arcsec and arccos, there's arcsec(x) = arccos(1/x)

[final sonnet]

i just want like formula type thing

i understand more like like

like sec is fine right

iget that 100%

1/cosx is simple

But then how would i wirte the function of arcsec

[flat rune]

here

[final sonnet]

1/cosy?

[cedar lichen]

You can't write arcsec in terms of cos as far as I'm aware

Just arccos

Idk maybe you can, but it wouldn't be nice

[final sonnet]

so ?

[flat rune]

not exactly

[fierce lagoon]

Idk what you're trying to do

Arcsec is arcsec

[final sonnet]

but what does it mean ?

It means

the opposite of sec

and sec is 1/cosx?

so what is the oppositie of 1/cosx?

well

yeah

[fierce lagoon]

If you wanna find the inverse using 1/cos

It's arccos(1/x))

[final sonnet]

if i dont want to do that it would be ?

[fierce lagoon]

$\arcsec{(x)} = \arccos{\left(\frac{1}{x}\right)}$

[warm shale]

Umbraleviathan

[flat rune]

if you dont want to do that you have to do something else i guess :v

but thats what arcsec is

[final sonnet]

ok

yeah ill try to look more into it tommorow

thank for you help ill close room now

.close

[obtuse pebble]

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[delicate lion]

https://www.youtube.com/watch?v=n4KKgKpp--0&list=PLUl4u3cNGP60UlabZBeeqOuoLuj_KNphQ&index=64
Optimal marriage problem (graph theory)
3:40 - 6:00
N and T is rogue, contradicts the fact N is optimal for K? Thank you

[obtuse pebble]

@delicate lion Has your question been resolved?

[obtuse pebble]

@delicate lion Has your question been resolved?

[obtuse pebble]

@delicate lion Has your question been resolved?

[obtuse pebble]

@delicate lion Has your question been resolved?

[delicate lion]

Hello guys, anyone knows?

[void pelican]

maybe ask in #discrete-math @delicate lion

[delicate lion]

I did. no response

[void pelican]

F

[delicate lion]

Hello guys, anyone knows?

[obtuse pebble]

@delicate lion Has your question been resolved?

[obtuse pebble]

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[lost fulcrum]

I am working on an excel sheet to calculate damage done but certain weapons in a video game. I want to know the real rounds used per hour for a turret. One of these does 300 rounds per minute with a magazine size of 10 rounds and has a reload time of 2.5 seconds. I worked this out the long ways to figure out what the answer should be for this particular weapon but I need a formula I can put in an excel cell so I can calculate the value of many more weapons in this game.

[trail musk]

Does the 300 rounds per minute account for the reload time already

[lost fulcrum]

no

[trail musk]

And you reload after every 10 rounds fired?

[lost fulcrum]

thats what I am trying to figure out. that is just continious firing

yes

[grizzled shore]

So how long does it take for fire 1 bullet

If it can do 300 continuously in a minute

[lost fulcrum]

I figured at 300 rpm I could fire 10 rounds in 2 secs

[trail musk]

,calc 300/60

[warm shale]

Result:

5

[grizzled shore]

So 5 rounds a second

So every 2 seconds it needs to reload

[trail musk]

5 rounds a sec, which means 5 rounds every 4.5 sec with reload time

[grizzled shore]

So each set of 10 rounds + reload takes 2s + 2.5s

No no there’s 10 in a magazine

[trail musk]

Yeah you're right

[grizzled shore]

So it takes 4.5s to fire 10 rounds

How many 4.5s are there in an hour?

[lost fulcrum]

14 in 60 seconds

[grizzled shore]

,calc 60/4.5

[warm shale]

Result:

13.333333333333

[grizzled shore]

Sus

[lost fulcrum]

So in my excel sheet RPM is cell U2 Magazine size is X2 reload time is Y2 and the formula I am putting in AA2 which is Rounds Consumed per Hour

those cells all have variables for the different weapons

[grizzled shore]

.

Well how did we find the 4.5s

We took rounds per minute

[grizzled shore]

Divided it by the minute to see how much time per round

In this case it is 0.2

0.2s per round

[lost fulcrum]

thats how I came up with the answer the long way

[grizzled shore]

Then we multiplied it by the number of rounds in a magazine

So 0.2*10 = 2s

Then we added the reload time

Then we divide the number of seconds in an hour to the time we just got (2s + 2.5s)

To find the number of rounds shot over 1 hour

[lost fulcrum]

yeah thats not working out to match my other results

I should be at around 8400 rounds per hour

what I am getting is 300rpm/60=5 Then mag size of 10/5=2 then adding the two together is 2+2.5=4.5 then the total seconds in an hour is 3600/4.5=800

In this screenshot I am starting with 60 seconds then I minus 2 seconds to fire 5 10 rounds which takes me to 58 seconds then I reload with takes 2.5 seconds which takes me to 55.5 seconds and so on

every 2 in the screen shot represents 10 rounds and there are 14 2's

14 * 10 bullets is 140 bullets per minute * 60 minutes = 8400

ahh wait....

[lost fulcrum]

that 800 needs to be multiplied by 10

so 8000

I guess thats close enough

.close

[obtuse pebble]

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[rough dirge]

how would i set this up, im not a big word problem guy

[obsidian isle]

Have you tried drawing a picture?

[proven zephyr]

"how would i set this up"

i think he dont know how to

[rough dirge]

ya im confused on how to set it up

ill probably be able to figure it out with it setup just confused on that step

[high lily]

start by drawing the ground

and then a pole

label that pole with a length of 19ft etc

try and represent all the given info on the diagram

[rough dirge]

alright

.close

[obtuse pebble]

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[obtuse pebble]

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[haughty shore]

For calc

[haughty shore]

how do you know what kind of region it is

between type 1 and 2

[tardy epoch]

Can you be more specific with an example of each or precise definitions

@haughty shore

[haughty shore]

1<=x^2+y^2<=9

i think thats both

would i be right

[tardy epoch]

What is the difference between type 1 and 2

[haughty shore]

uh type 1 is dydx type 2 is dxdy

right

[tardy epoch]

Every book is different

Find the definition in your book

[haughty shore]

yeah it says pretty much this

[haughty shore]

this would be both right?

[obtuse pebble]

@haughty shore Has your question been resolved?

[haughty shore]

<@&286206848099549185>

.clos

.close

[obtuse pebble]

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[brittle warren]

hey can anyone teach me integration

[tardy epoch]

Google Khan academy videos on integrals @brittle warren

[obsidian isle]

Add tiny rectangles

That's all there is to it

[tardy epoch]

Only tiny in width.

[obsidian isle]

You can also use trapezoids if you feel like it

[obtuse pebble]

@brittle warren Has your question been resolved?

[obtuse pebble]

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[obtuse pebble]

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[quiet ruin]

could someone please teach me how to do (c)

[spiral ether]

you know the lengths of AO = OB = R using part b)

now just apply the formula arc length = $r \theta$ (which isn't the same as $R$ or $\theta$ in the diagram)

[warm shale]

findingsouth

[spiral ether]

and then you can solve using your GDC, most likely

[obtuse pebble]

@quiet ruin Has your question been resolved?

[quiet ruin]

using which numbers though? sorry im confused

[spiral ether]

for that arc length AB, your radius will be R and your angle will be 2 theta

[quiet ruin]

let me try attempt it again, brb

[obtuse pebble]

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[obtuse pebble]

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[mellow frost]

Can anyone help me in question no. 2?

[mellow frost]

Mention me if someone come

[midnight pumice]

i try

[mellow frost]

Ok

[obtuse pebble]

@mellow frost Has your question been resolved?

[kindred oasis]

Have you already done something? Do you need help with 2a or 2b?

[mellow frost]

I just don’t know how to solve these

I solved basic questions

Like this

Can you explain me from basic?

[kindred oasis]

Do you need help with 2a, 2b or 2c?

[mellow frost]

I searched on YouTube but didn’t find any good tutorial

[kindred oasis]

What do I have to explain?

[mellow frost]

2b

I know how to solve 2a

Help me with 2b(i,ii,iii)

[kindred oasis]

From Khan Academy

https://images.app.goo.gl/QsekXWR8qpbSWa8r7

[kindred oasis]

Here you can see it better

[native ginkgo]

Well

There is another way to tackle this

[mellow frost]

How

[kindred oasis]

You should be able to solve 2c without limits @mellow frost

Then for the others you can follow the graph I sent you before

[mellow frost]

Can you tell me how to solve this?

[kindred oasis]

That's how you solve it

[kindred oasis]

This is better

[native ginkgo]

Yeah I agree just there are formulas online about mathematics intergovernmental it could help you a lot in the journey. I use this for my codes to solve local proximity distance so you could use powers of the 10s then dividing making a graphical chart and start adding up the math then search up the basic formula for the question then BOOM.

[kindred oasis]

What

[mellow frost]

Everything is going above my head lol

[woeful crane]

I dont that works on simple limits

[kindred oasis]

Lol

[native ginkgo]

I’m in advanced mathematics

So my professor thought us there way

[kindred oasis]

Just have a look at the picture I sent you, it takes 30 seconds, is really easy to understand and you will solve your question in a minute

[native ginkgo]

Basically like kung fu

[kindred oasis]

Ahah

[solar dune]

Well you can always manipulate by the asymptote if you want to you can even formulate the limiting factor using the conjugates in distributing the identifier in trig

[native ginkgo]

Could work

[solar dune]

Sometimes you just have to extrapolate your ping in such a way that it makes the mathematical equation appear earlier and it allows you to parry easier

[native ginkgo]

That is pretty true

How about warry ?

@cedar dirge

[kindred oasis]

Your method is better though

[native ginkgo]

Yeah ik

[solar dune]

Warry is harder than parry but you can always industrialize the Ford Focus

[kindred oasis]

@mellow frost have you tried to solve the problem with the chart I sent you? What did you get?

[mellow frost]

Im trying

[native ginkgo]

You mean a graphical chart is way better

[solar dune]

In calculus, the Leibniz integral rule for differentiation under the integral sign, named after Gottfried Leibniz, states that for an integral of the form

where

−
∞
<
a
(
x
)
,
b
(
x
)
<
∞

[kindred oasis]

Let me know when you're done 👍

[obtuse pebble]

@mellow frost Has your question been resolved?

[obtuse pebble]

@mellow frost Has your question been resolved?

[obtuse pebble]

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[obtuse pebble]

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[feral sentinel]

hi can someone tell me how this works
im trying to use the maclaurin series to solve this
but i dont get the working here

[timid silo]

You don't understand how mclaurin series work?

[feral sentinel]

i know that we need (1 + ...)^n for us to use the formula

but im not sure how did we get (2 + x)^-2

like the working xD

[timid silo]

Did you do that working

[feral sentinel]

oh no not me

found it online

[royal basin]

then whoever did that work is lying to you

raising something to the power of 1/2 is not the same as raising it to the power of -2

[feral sentinel]

ohhh

[timid silo]

We could derive the mclaurin series ourselves

[feral sentinel]

hmm so would it be like

3 + (1 + x)^1/2

then we expand (1+x)^1/2

[timid silo]

???

[feral sentinel]

omg wait

nvm we cant just take out 3

[timid silo]

$$\sqrt{4+x} \neq 3 + \sqrt{1+x}$$

[warm shale]

Legolas

[feral sentinel]

ahh yes sry

[timid silo]

well we know the mclaurin series is

$$\sum_{n=0}^{\infty} \frac{f^{(n)}(0)}{n!}x^n$$

[warm shale]

Legolas

[feral sentinel]

yes

do i have to memorise that formula

or can i always refer to it

[timid silo]

well it comes from the formula of taylor series

$$\sum_{n=0}^{\infty} \frac{f^{(n)}(a)}{n!}(x-a)^n$$

[warm shale]

Legolas

[timid silo]

You should try to have it in the back of your mind yk

[feral sentinel]

ohh okay i will try

[timid silo]

so we only want to expand until the x^3 term

[feral sentinel]

is this equivalent to what u wrote

[timid silo]

$$\sum_{n=0}^{3} \frac{f^{(n)}(0)}{n!}x^n$$

[warm shale]

Legolas

[feral sentinel]

oh wait nvm this is strictly for (1 + x)^n

[feral sentinel]

yes

[timid silo]

so we have the function sqrt(x+4) right

[feral sentinel]

yes

[timid silo]

we want to keep evaluating it's derivatives at zero

[feral sentinel]

we differentiate it 3 times

wait wat

[timid silo]

yes

[feral sentinel]

oh

yes

then we find

f(x) and f'(x) and f''(x)

and f'''(x)

[feral sentinel]

then we sub into this

[timid silo]

$$\frac{f(0)}{0!}x^0+\frac{f^{1}(0)}{1}x^1+\frac{f^{2}(0)}{2!}x^2+\frac{f^{3}(0)}{3!}x^3$$

[warm shale]

Legolas

[timid silo]

Yes

pretend that those represent derivatives plz

[feral sentinel]

ohhhh

yessir

i understand

thank u

wait actually can i ask one more

[timid silo]

this is just f, first derivative, second derivative, third derivative

[timid silo]

sure

[feral sentinel]

for this small angle approximation thingy

why can we do this

is it because since the mac series expands infinitely the x becomes insignificant or somethin

yes

[timid silo]

for small values of x, only the x term matters

because the -x^3/3!+x^5/5! .... terms will all be really small

if x is small, like less than 1, then x^3 will be even smaller

[feral sentinel]

oh rightt

[timid silo]

and then we divide it by 6, so it will be even smaller

so x^3/6 will be neglebible

[feral sentinel]

ohhh i see

[timid silo]

and so will all the other terms

[feral sentinel]

so a) we can just convert sin2x to sinx

yessir so it will become so small that we can jut ignore it

[timid silo]

sin 2x would be 2x

[feral sentinel]

wait wut

OH

[timid silo]

Look at the graph

sin 2x is really close to y=2x for small values of x

[feral sentinel]

oh so if we remove sin

there wont be much difference

[timid silo]

only for small values of x of course

[feral sentinel]

then when we add all the other expressions the difference will get even smaller

[timid silo]

yes

[feral sentinel]

ohhhh

can i see an example for tan pls

[timid silo]

tan(x) is sin(x)/cos(x) right

what is the first term of the mclaurin of sin and cos

we know the first term of sin(x) is x

what about cos(x)

[feral sentinel]

1

i think

[timid silo]

so tan(x) is sin(x)/cos(x)

[feral sentinel]

OH

okay me get

wait but why is cos(x) approximation so much different

from sin and tan

[timid silo]

well try deriving the mclaurin series for cos(x) and sin(x) yourself

the graph of tan x is close to the graph of y=x for small vals of x

[feral sentinel]

hmm

wait so cos(x)'s approximation

is the first two terms

of the mac series

[timid silo]

well for really small values of x

they are almost one

,wolf cos(0.001)

[warm shale]

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[timid silo]

,wolf cos(-0.001)

[warm shale]

Results provided by WolframAlpha

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Upgrade to WolframAlpha Pro!

Requested by Legolas#2686

[timid silo]

so yeah technically 1

,wolf cos(-0.000000000000000000001)

[warm shale]

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[feral sentinel]

ohh wait so the "-x^2/2" is just to show that the value is really really close to 1, but not 1

[timid silo]

so we could just say cos(x) approximately 1 for** really **small values of x

[feral sentinel]

oh wooot

[timid silo]

nah it's just to add precision

[feral sentinel]

ohh i see

[timid silo]

every term we add to the mclaurin series adds extra precision to our answer

[feral sentinel]

ah okei, but the reason why we didnt add 1 more term to the tan and sin one

is because the value is literally x?

[timid silo]

well we could've, but we are dealing with small values

[timid silo]

it says if x is small enough to neglect x^3

so that's why

[feral sentinel]

hmm okay i understand

wait then what about part c

would it be

would it be x sec x?

oh wait no

[timid silo]

Sin(x/2) becomes x/2

[feral sentinel]

oh yea

[timid silo]

Sec x is 1/cosx which is approximately 1 for small values of x

So it’s just x/2

[feral sentinel]

ahhhh

okay

then for part d

its uh

pi/4 cant be ignored right

its a biig number

so itll just be x + pi/4?

[timid silo]

Yes

[feral sentinel]

OH

okay tysm sir

[timid silo]

Well we said tan(x)~x

So tan(x+π/4) approximately equal to x + π/4

Same thing

[feral sentinel]

ohh i see

oh wait LASTTTT

question

all these expansions

are just the maclaurin series right

like for sin(x) = x - x^3/3! + ...

its as if we differentiated it and subbed the values into the maclaurin series

its just an easier way to go about it

[timid silo]

I don’t get what you mean

[feral sentinel]

oh um like all these terms are derived from the maclaurin series formula

[timid silo]

Yes

[feral sentinel]

ohhh

okay

i get it now

thank u xD

.close

[obtuse pebble]

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[onyx sentinel]

need help in proving that the given number is irrational

[onyx sentinel]

(2 - root 2) / (5 - root 5)

Need to prove here that the given number is irrational

already proved that root 2 is irrational

can someone tell what to do next?

[devout sable]

Do you know rationalisation

[onyx sentinel]

yes

[devout sable]

$\frac{ (2-\sqrt 2) \cdot ( 5 + \sqrt 5)} {(5 - \sqrt 5) \cdot (5 + \sqrt 5)}$

[onyx sentinel]

yes

then?

[kind hawk]

brackets please

[devout sable]

Sry for that i’m on phone

[onyx sentinel]

i do know that we need to compare them with rational numbers and then arrive at a contradiction

[warm shale]

Deep.

[devout sable]

Yes

[devout sable]

Can you simplify this?

[onyx sentinel]

yea

gimme a min

yea

so

i am getting

( 10 + 2 root 5 - 5 root 2 - root 10) / 20

[devout sable]

Correct, so now if we show root 5 and root 10 irrational the can we say that this is irrational?

[onyx sentinel]

yea

so

can i tell you what i did before this

so

what i was planning to do is

that i prove root 2 is irrational

and then simplify and transpose terms in such a way that i get root 2 on one side and rational numbers on the other

and then arrive at a contradication

[onyx sentinel]

yea

[timid silo]

(irrational numbers arent closed under addition though)

[onyx sentinel]

hmm

ok

then what do i do

[obtuse pebble]

@onyx sentinel Has your question been resolved?

[onyx sentinel]

.close

[obtuse pebble]

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[velvet ibex]

Given a point E(k^2, k, 2) lies on the plane

[velvet ibex]

find the value of k such that k < 0

I have equated i, j and k so

k^2 = 7 + 8s - 3t
k = 2 + 5s - t
2 = 3 + s + 4t

I've been trying to solve these three simultaneous equations for the past hour or so

[royal basin]

sounds a little overkill

but also may i suggest an alternative

[velvet ibex]

sure, go ahead

[royal basin]

1. find the normal of the plane
2. find the equation of the plane in point-normal form
3. plug in your point (k^2, k, 2) into it

[velvet ibex]

normal of the plane through the vector product?

or cross product

whichever you call it

and by point-normal form, Cartesian form? like ax + by + cz = d?

[royal basin]

close enough

either way you get an implicit equation

[forest yacht]

if you want some visualization

[velvet ibex]

not sure whether or not I am even doing it correctly @royal basin @forest yacht

I think I found the cross product

[royal basin]

,w (8,5,1) × (-3,-1,4)

[warm shale]

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Requested by Ann#0413

[royal basin]

looks like you made a sign error somewhere

[velvet ibex]

the last bit

ah I see

[royal basin]

you can shorten this by a factor of 7

and take (3, -5, 1) as your normal

3x - 5y + z = d

find d from knowing that the point (7,2,3) lies in your plane

[velvet ibex]

I haven't fully learnt the cross product yet

is the cross product the normal vector I get

like what's seen on @forest yacht's diagram

[royal basin]

yes

[forest yacht]

yes it is

[velvet ibex]

d is the dot product between the position vector 7, 2, 3 and the normal

right?

[royal basin]

nyeh if you wanna put it that way sure

[forest yacht]

a vector produced by cross product of 2 vectors is perpendicular to them

[velvet ibex]

is there another way to put it?

I've been learning this myself so I haven't really had anyone to give me a proper explanation

my syllabus says r dot n is equal to p

[royal basin]

your syllabus probably says something else rather than Just That as if it were gospel

[forest yacht]

point normal form ax+by+cz=d, you can just put components into x y z or use dot product. but they are actually equivalent

[velvet ibex]

I've arrived here

[forest yacht]

seems legit

[velvet ibex]

then I just substitute in what the question originally told me to find

to find k

[velvet ibex]

I was attempting simultaneous equations before because my book covers the vector product immediately after this question so

and my original vector was where the coefficients in s were all negative

because that's how I found that direction vector

if the coefficients of s were all negative would have it made a difference?

[forest yacht]

a plane is generated by the all scalars of the 2 direction vectors (linear combination)

[velvet ibex]

ah

with scalars there's no direction so it wouldn't have mattered right

and there's the answer

is what my worked solutions said to do

[velvet ibex]

@royal basin

[forest yacht]

looks so cumbersome lol

[velvet ibex]

that's what I have been doing for the past 10 or so questions

last question I just summed everything up

[royal basin]

cbf rn

sorry

[velvet ibex]

and the working looks like

@forest yacht would the cross product have been more efficient?

I think that took several attempts for me to get correct

question was to find p given a point B that lies on the plane

[forest yacht]

normal vector is how you define a plane. it's natural to do so

[velvet ibex]

yeah

the next exercise is all about the cross product

in future, should I stick to the vector product for planes

and the dot product for lines

in general

or is it not as clear cut as use this or this

[forest yacht]

stick to the definition, that's math

and a plane is defined by normal vector

[velvet ibex]

I vaguely remember this

if a vector is (a1,a2,a3) then a1x + a2y + a3z = d

I'll keep this in mind

[forest yacht]

wdym btw

[velvet ibex]

I don't know

nevermind the question

the dot product is useful for both

but next time I won't think of doing simultaneous equations

for this type of problem again

[forest yacht]

you want some derivation of vector form and normal form of the plane?

might help memorizing

[velvet ibex]

my one looks something like this so far

that's how much I know about the vector product

I know how the scalar one is derived

[velvet ibex]

if you have some already then that's fine

if you have to make them then just give me some things I should go find resources for

I made this one quickly last night I think so

[forest yacht]

given a plane $\vec{r}=\vec{r_0}+s\vec{u}+t\vec{v}$, the normal vector $\hat n$ is perpendicular to directional vectors $\vec{u},\vec{v}$. so the dot product $\vec{r}\cdot\hat n=(\vec{r_0}+s\vec{u}+t\vec{v})\cdot\hat n=\vec{r_0}\cdot\hat n$

[warm shale]

JellyShark

[forest yacht]

let $\hat n=(a,b,c), \vec{r}=(x,y,z)$. compared to the normal form $ax+by+cz=d$, $d=\vec{r}\cdot\hat n=\vec{r_0}\cdot\hat n$.

[warm shale]

JellyShark

[forest yacht]

if you to get familiar with it, you can try to derive the formula of cross product yourself

or ask somebody else or internet how to do it. knowing how a formula comes from help comprehension a lot.

[velvet ibex]

alright

I'll set some time myself to derive the cross product

I think I have a video or two that derives it

I just haven't had time to watch it yet

thanks for the help today @forest yacht and @royal basin

.close

[obtuse pebble]

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[rocky kite]

how do you do this im so stuck

[timid silo]

Look

Notice that DB is the altitude of the equilateral triangle

And it is the diameter of the circle

So by trig you can find AC in terms of the radius

[rocky kite]

ok

[timid silo]

Like focus on ADB and try to find AD

If you know that DB is 2r where r is the radius

[rocky kite]

i found AC in terms of the radius but how would i use the radius to find FE

[timid silo]

OFB is isosceles

You could draw an altitude on FB from O

Then it would be a right triangle with 30 60 90 as well

[rocky kite]

oh i got it

i forgot OF was the radius as well lol

[obtuse pebble]

@rocky kite Has your question been resolved?

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[tribal beacon]

The answer should be 2 no ? the computer says I'm wrong

[timid silo]

It should be 2

[tribal beacon]

No but ?

[reef grotto]

no

dude

just subtract number 1 from number 2

lol

[timid silo]

Subtract second equation from first

[reef grotto]

no need for sub

[timid silo]

Yes

[tribal beacon]

Ty

.close

[obtuse pebble]

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[eternal gulch]

Determine the value for the number “a” so that lim x—>2 [ (3x^2 + ax + a + 3) / (x^2 + x - 6) ] will exist.

[eternal gulch]

That’s how it looks

I’m not@sure how to solve this

[novel knoll]

What does denominator approach?

[harsh remnant]

Factor the denominator first

[eternal gulch]

Oops right sorry forgot to show that

I got (x-2)(x+3)

But I’m still not sure what to do

Sorry calculus is like super complex for me I’m rlly dumb at it

[eternal gulch]

What do you mean by that?

[novel knoll]

I mean what I said?

What does it approach

[eternal gulch]

Ohhh oops

It approaches 2 sorry

[novel knoll]

Really?

2^2+2-6=2?

[eternal gulch]

Ohhh my bad I thought you meant the limit

Sorry

It’s 0

That’s what I don’t get thooo

I thought maybe factoring might help

But it didn’t I still get a denominator of 0

[novel knoll]

And it changes sign if approaching from 2 from above or below

So only way limit can exist is if numerator also approaches 0 to cancel that out

[eternal gulch]

So if it’s a 0/0, that means the limit can exist ?

Can I make the numerator = 0, and then get the value for a theough that?

[novel knoll]

Yes

[eternal gulch]

Sorry if this is dumb

But wouldn’t that be undefined ?

[novel knoll]

Indeterminate

[eternal gulch]

I thought anything divided by 0 is undefined

[novel knoll]

Not undefined

[eternal gulch]

Ohhhh

Okok

Thanks sorry

[novel knoll]

lim x->0 of x/x

Is not undefined

For example

[eternal gulch]

Ohhhhh

Wait yeah ur right that makes sense

Thank u

[uneven flicker]

but the denominator has x-2, so x -> 2, itll become 0

[novel knoll]

What?

[eternal gulch]

Where did u get x - 2

[uneven flicker]

x^2 + x -6

what are the factors for this

[eternal gulch]

Yeah

X-2

[novel knoll]

Yes what?

We already went over this

[uneven flicker]

oh

[novel knoll]

What?

[eternal gulch]

And x+3

[uneven flicker]

(x+3) (x-2)

[eternal gulch]

Yeah

No he explained it

[uneven flicker]

no but like

[eternal gulch]

A 0/0 is still a limit

[uneven flicker]

if x -> 2

[uneven flicker]

true

[eternal gulch]

I thought it was undefined

[uneven flicker]

but does the upper part reach 0

oh you got a

you can make it reach it

[eternal gulch]

Yeahhh

[novel knoll]

????

[uneven flicker]

my bad

[eternal gulch]

Yeah nw

Thanks guys

[uneven flicker]

thank him

[eternal gulch]

I get it now imma try solving

[uneven flicker]

i didnt do shit

[eternal gulch]

Yeah thanks sm scrapeprof

I actually get the concept too now tysm

[uneven flicker]

wont they need to know lhospital rule

to like say 0/0 can be a limit

?

[novel knoll]

Why

[novel knoll]

.

Hopefully one can compute this limit without L’H

[obtuse pebble]

@eternal gulch Has your question been resolved?

[eternal gulch]

Wait quick question

How would I find the limit

[eternal gulch]

Is there any other way than that

@novel knoll @uneven flicker

Nvm

Mb for the @

.close

[obtuse pebble]

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[distant ravine]

Linear Algebra
The video shows how to find the image of a transformation. So he takes all the vectors in B, does the transformation on them and says that the span of those vectors is the Image of T. But first, he checks that they're all linearly independant, by putting them in a matrix and ...(idk what the term for what he does is).
Anyways, what I don't understand is: Why does he put the vectors in the matrix horizontally and not vertically?

[distant ravine]

In which situations do you think "I need to put the vectors vertically / horizontally into a matrix"?

[warm canopy]

When it comes to images, it's always columns

The column span of a matrix is its image

You "put vectors in horizontally" when you're solving a system of equations by representing it as a matrix

[distant ravine]

Okay, but this is about images, and it's rows, not columns.

[warm canopy]

In the picture you sent, he's put them in as columns no?

[distant ravine]

Nope. Rows.

[warm canopy]

Ah, you can do either

So what he's done there is put them in as rows so that he can put that matrix into REF to see if they're linearly independent

You can do the same thing by putting them in as columns and doing column operations instead i believe

[distant ravine]

Column operations?

Like, same as usual but like, adding them horizontally and stuff?

[warm canopy]

Yeah

[distant ravine]

I just don't understand why he'd decide to do that instead of vertical

What the motivation for doing either is

Wait a minute, he does row operations

He says so in the video

[warm canopy]

Okay so in that case

[distant ravine]

oh wait I'm getting confused

nvm lol I'm tired

[warm canopy]

What's also true is that because of the nature of those vectors, they're both the columns and rows lol

Look at the matrix

Both columns and rows happen to be those 3 vectors

So it can look like he's put them in as columns or rows

He's put them in as rows and doing row ops

You could pit them in as cols and do column ops

Works the same way

But if you actuslly want the mateix representation of T, you need to put them as columns

[distant ravine]

Can I always put vectors into a matrix either like this or like that?

[warm canopy]

If you jusy want tl see if they're LI, then yes

[warm canopy]

Otherwise, see this

[distant ravine]

So it has to be columns, not rows

[warm canopy]

If you just want to see if vectors are LI, you can make them rows or columns and do row or column ops respectively