help-23

[worldly fiber]

you are a real babe

thank you

[novel pasture]

And height is fi/x + fi

[worldly fiber]

(babe is what i call blessed people)

Yes!

i had that

phew

[novel pasture]

Btw we solved this problem in another server

And its way way longer than this

[worldly fiber]

so this isn't the right answer?

:c

[novel pasture]

Nah

[worldly fiber]

😭

[worldly fiber]

would you please explain it?

I promise i'll understand

if you got time

I have really pondered

[novel pasture]

I can inv to the other server if u want and you can have a look

[worldly fiber]

YES

WOW

THANK YOU

[safe radish]

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[cunning thistle]

hi

[cunning thistle]

How would i start this problem. I know that theres 26 letters and 10 digits to chose from

[fickle pendant]

can you reduce it to 5 individual choices

and knowing that e.g. if you have 10 choices for A and 5 for B, then you have 10*5=50 total choices

[loud crag]

what have you establis so far? a? should be straightforward permu question

[safe radish]

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[cunning thistle]

Permutation?

[safe radish]

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[cunning thistle]

.reopen

[safe radish]

✅

[cunning thistle]

How would I set up the permutation problem though

[thorny condor]

i failed my math

[loud crag]

so you know it hahs to be form 0ABCD or 0ABCDE, right? then you know
$ digit 1 = {0, 1}$
$ digit 2 = {A, B,.....}$
$ ... $

so how many unique str?

[safe radish]

@cunning thistle Has your question been resolved?

[cunning thistle]

im a bit confused with the dollar signs and the set notation

[safe radish]

@cunning thistle Has your question been resolved?

[cunning thistle]

im working on another problem as well thats somewhat similar, should i put it here or start a new channel

[hazy roost]

So consider for a

[hazy roost]

How many potential digits exist for the first spacing

Please reply to me when you see this so that I can be notified that you responded

[cunning thistle]

10 for the first spacing if wanted. Theres the option for zero as well. Theres 26 for the 4 or 5 different letters

[hazy roost]

Yeah. And you can have two types of plates. Digit and 4 letters or digit and 5 letters right

So we have two outcomes where we have to add the possibilities, right?

So let's start with the first scenario

A digit and 4 letters

[cunning thistle]

10 * 26^4

since theres 4 spaces

[hazy roost]

So that's all of the first plate scenarios

Then for the second plate scenarios, it should be?

[cunning thistle]

just 24^5? since theres no numbers?

[hazy roost]

That's tricky because of the phraising

I believe it is a digit followed by 5 letters

Phraising is atrocious

[cunning thistle]

the biggest problem im having is trouble understanding the maximun amount of characters it needs

this is optional homework but i have a test tomorrow that im feeling uncomfortable about

[hazy roost]

I believe a plate could be 0aaaa or 1bbbbb as examples

With this presumption it should be 10×26^5

Which we add with 10×26^4,

[cunning thistle]

i got 123383520

which is expected for adding two big exeponets together lol

[hazy roost]

Yeah that sounds cool

[cunning thistle]

its incorrect 😦

[hazy roost]

So perhaps the error is in our understanding of the question

I dare try 10×26^4+26^5

Which is 16,451,136

[cunning thistle]

nope. 3/10 tries used. so I have a lot of options

[hazy roost]

Looking at fluffy flareons concept, they said the only digits are 0 or 1. Let's try 2×26^4+2×26^5

Then perhaps 2×26^4+26^5 if that doesn't work

Former is 24,676,704. Latter is 12,795,328

[cunning thistle]

nope 😦

[hazy roost]

Okay perhaps here is another understanding

A plate doesn't have to have a number. So possible plates are aaaa, bbbbb, 1cccc, 2ddddd. If that makes sense

So add those differing possibilities

I get 135,721,872 from 26^5+26^4+10×26^5+10×26^4

[cunning thistle]

IT WORKED!

tysm

[hazy roost]

Np. Issue was with how they phrased the question

Should have be a plate can have a digit followed by 4 letters or 5 letters, or a plate can have [no digit followed by] 4 letters or 5 letters

[cunning thistle]

for the second one, I think i just need to decrement the 26 by one for every exponent it exists since it cant repeat

[hazy roost]

Yeah that sounds right. So 26×25×24×23 for the 4 letter plate

The way the number is often represented is through 26!/(26-n)!, where in is the ammount of letters you are using

[cunning thistle]

tysm. im going to take a break for now

.close

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[hazy roost]

Np.

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[barren laurel]

halp

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[sturdy hornet]

.

.open

k

[sturdy hornet]

I need help with finding the area under a curve, in this case its the below image

[icy lance]

progress?

[sturdy hornet]

im completely lost tbh

.close

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[pliant nacelle]

Need help with a partial differentiation question

[safe radish]

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[proven matrix]

My wants me to do my cousin who mainly speaks Arabic’s math homework. Problem is I don’t know how to do long division. Pls help!!

[royal kiln]

That's a tall order for a Discord text channel

I suggest you watch a video about it, try it, and teach your cousin instead of doing it for them

[proven matrix]

I did

Thanks for the suggestion

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[neon burrow]

I need help with 3 variable linear systems. I've worked this problem out over and over and I cant seem to get the right answer.

[safe bluff]

what's the problem

[neon burrow]

1. 3x + 3y + z = 26
2. x - 3y +2z = -12
3. 8x - 2y +3z = 10

These are the three systems and that's my work above.

I know that X = 2 but I can't figure out how to get there.

[safe bluff]

why'd it go from 30x+11z=82 to 30x+9z=82

[neon burrow]

Hmm

I went back and caught a mistake earlier. let me look

Ah yes

9z is the correct number

Even though it shows 30x+11z=82 I used 30x+9z=82 when solving a 2 variable

[safe bluff]

are you sure?

i think your work to get 30x+11z=82 was correct

[neon burrow]

1b. 3(8x-2y+3z=10)

That equals 24x-6y+9z=30. I made a mistake earlier and was trying to focus on another variable to eliminate instead of Y.

oh....

oh shit

that's it

[safe bluff]

right, you were eliminating y

but you still got the equation with 11z in it

[neon burrow]

you're a legend

Thank you

[safe bluff]

yeah looks right

gj

[neon burrow]

any tips for mitigating future mistakes?

[safe bluff]

i mean i think this was just not checking the work carefully enough

if youre confident your method was correct the only thing to do is just go line by line and see if anything got messed up

[neon burrow]

does it matter which ones i pick?

Between systems 1, 2 and 3

[safe bluff]

not for this kind of question

your choice might make it a little faster

but it should get the same answer

[neon burrow]

Gotcha ok

Thanks again mate

Appreciate you

[safe radish]

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[lean otter]

can you set up an equation foe the circle

for

[obtuse spruce]

i already know that it is proabllay pi/3 on the unit circle

[obtuse spruce]

but i thought that answer C and D are the same position on the unit circle ?

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[haughty finch]

Can someone give me some intuition about modules? I have studied them for 2 semester, but I couldnt understand more than "its something like a vector space over rings" and "a ring acting on an abelian group"

[haughty finch]

.close

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[subtle saffron]

Help pls

[subtle saffron]

I hope this doesn't go into hidden section again 💀

[obtuse plover]

That’s a test 💯

[empty gyro]

<@&268886789983436800> this is meme level ban

[hard crest]

looks like it's a review of a completed test

[empty gyro]

What specifically are you asking help on?

[subtle saffron]

Exactly

[subtle saffron]

The question?

[empty gyro]

you mean this?

[subtle saffron]

Yes

[subtle saffron]

I'll make sure to do the crop part next time

[empty gyro]

!show

[safe radish]

Show your work, and if possible, explain where you are stuck.

[empty gyro]

!status

[safe radish]

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 was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[subtle saffron]

I know nothing

[empty gyro]

How did you get 3x+8 then?

[subtle saffron]

I just solved it

[empty gyro]

solved what

[subtle saffron]

To get value of y

[empty gyro]

how?

[subtle saffron]

From the equation given above

Let me send u the pic.

[empty gyro]

4x+7y=14?

[subtle saffron]

Yes

That equation

Wait..

[subtle saffron]

I couldn't find the page where I had to solve it🥲

It was yesterday's test

I got this thi

Tho*

I had tried solving it again in the morning

@empty gyro

Oh I got the original one too

Looks like I had kept the value of the given coordinates (-3, 8) in the formula y:mx+c (which seems very wrong now cause my is slope and idk what I did yesterday)

[safe radish]

@subtle saffron Has your question been resolved?

[subtle saffron]

.close

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[candid oar]

Where is this function continuous?

[candid oar]

I know its continous for every (x,y), where y Isnt equal to 0

But I cant figure out for which points (x,0) its continous

[safe radish]

@candid oar Has your question been resolved?

[neon pewter]

for a fixed value of x, what happens when you make y a small positive value? what happens when you make y a small negative number?

[candid oar]

-x^2/y goes to negative infinity for y>0

And it goes to positive infinity for y<0

[neon pewter]

and what does the behavior of this exponent tell you about how the value of your whole expression will change

i.e., what is "x*e^{-infinty}" and what is "x*e^{infinity}"

[candid oar]

xe^{infinity} "is" infinity, xe^{-infinity} "is" 0

[neon pewter]

exactly, so this is to say that on one side of (x,0) you will get points close to zero, and the other side you will get points close to infinity

so what does this say about continuity at (x,0)

[candid oar]

So its not contunuous for any (x,0)

[neon pewter]

mostly! we need more caution at (0,0)

[odd vapor]

Can you sent bazik of math

[neon pewter]

since this part only really holds when x is non zero

[candid oar]

What do we do for (0,0) then?

[odd vapor]

All the basics of mathematics

Please

[neon pewter]

at (0,0), we would know the function is discontinuous if we could find arbitrarily small values for x and y where the function would take on a large value

so is there any path we could approach (0,0) from that would not converge to 0?

[odd vapor]

I am very need

[candid oar]

So for a small x we use a negative y thats close to 0

as a negative y approaches 0, e^(-x^2/y) goes to infinity

And since x is constant, xe^(-x^2/y) goes to infinity

[neon pewter]

this worked when x was fixed, but will this still work when x goes to 0?

one way that you can approach this is to parameterize a path, like we could take the points (x,-x) and send x to 0, this would be a path that goes to zero

and allong this path we take values xe^(x), but this function approaches 0 as x goes to 0

so if we want to show discontinuity we would need a different path

and one way to think about this is what value could we put in the exponent of e^() so that it goes to infinity as we go to 0

[candid oar]

Maybe x = y^2?

[neon pewter]

if we had x=y^2 then our exponent would be e^(-y^3) which goes to 1 as y goes to 0

[candid oar]

What do you suggest?

[neon pewter]

so you want the exponent to get large when x goes to zero

so something like 1/x

is there some choice of y we could make to get the exponent to be 1/x?

[candid oar]

y = x^3

[neon pewter]

yep! (-x^3) might be slightly cleaner to look at as it would cancell the negative sign though

so if you approach (0,0), along the path (x,-x^3), what happens to f(x,-x^3)?

[candid oar]

f(x,-x^3) = xe^(1/x) for y thats not 0

[neon pewter]

Exactly

And as the limit path you are only never has y=0, it will always take that value

And what happens to that when you make x small?

[candid oar]

It goes towards infinity

[neon pewter]

And how is this information useful in the context of our trying to sho continuity?

[candid oar]

Its discontinuous in (0,0)

[neon pewter]

Yep!

[candid oar]

Thank you

[neon pewter]

[candid oar]

.close

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[lean otter]

Find the Total number of ways in which 20 balls can be put into 5 boxes so that the first box just contains 1 ball.

[lean otter]

The answer key says it’s 20 * 4^19

I’m getting 5 * 4^19

By principle of counting, there are 5 ways to put the first ball into 5 boxes but 4 ways to put each of the other 19 balls to be put, so 4^19.
So 5 * 4^19

I understand that one way to account for the answer in the key is the fact that we can choose any one of the 20 balls to be put into the first box. But I feel like I can’t see where that would fit in with what I’ve done

I understand now lol

.close

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[frigid osprey]

Hey guys, i've just had a test and they asked me if these matrixes were stepped, reduced or not any of the before. I got a 0,5/1 and I don't know why and I don't know if i should go to the revision about this

[obsidian oracle]

Think about matrix A again

[frigid osprey]

Yeah is the one i was thinking about

i've said none bc there is no 1 in the 3rd row

Isn't there a need to be all the principal ones in the matrix to be stepped/reduced?

[safe radish]

@frigid osprey Has your question been resolved?

[safe radish]

@frigid osprey Has your question been resolved?

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[plucky swan]

anyone mnd guiding me

[plucky swan]

everything ive tried has turned out wrong

i know this much ig

@raven vessel

[potent seal]

That's right

It's a right angled triangle with other two angles as 45⁰

Can you think of any property?

[plucky swan]

I dunno anything, OW and WV are obviously equal then since it is isoceles

[potent seal]

Yes, we are gettin somewhere

[plucky swan]

I've tried various methods but I'm lost

[potent seal]

And also it's a right angled triangle

[plucky swan]

SOH CAH TOA?

[potent seal]

Yeah, that or Pythagorean

Anything works

[plucky swan]

how would i get the point w from that tho

[potent seal]

Also you know the modulus of 4+i

[plucky swan]

root 17

[potent seal]

We will come to that
let's take it as x,y for now

[potent seal]

Yes

So what's the modulus of OW?

[plucky swan]

not sure

OV is not nessecarily equal to OW

[potent seal]

It's not equal

But you can use soh cah toa

Or Pythagorean

[plucky swan]

okay lemme try

[potent seal]

Sure

[plucky swan]

Yeahh alr I dunno where to go after getting the modulus

I have like near 0 experience with vectors but gotta so

[potent seal]

We will get to it

It's alr

But did you get the modulus?

[plucky swan]

yep root 17 + 1/root2

for both VW and OW

[potent seal]

Ig it's sqrt(17/2)

cos 45⁰ = 1/sqrt(2)

[plucky swan]

yep

[potent seal]

Yes so

[plucky swan]

oh mb i typed +

[potent seal]

The length of hypotenuse is also given

Ah np

Ok we assumed W as (x,y)

And we know it's modulus

[plucky swan]

i and j components....?

[potent seal]

How can we form an equation now

[potent seal]

Ye

[potent seal]

Relating x and y to modulus?

It's the same way you found modulus for 4+i

[plucky swan]

modulus = root of x^2 + y^2

[potent seal]

Yess

[plucky swan]

work backwards alr

[potent seal]

Ye

[potent seal]

You could square both sides to get rid of the sqrt

So it would be
x^2 + y^2 = 17/2

Works?

[plucky swan]

yep

mb for long waits

[potent seal]

It's alr

Ok so we have 1 equation

[plucky swan]

and we need a second probably for a sim equation

[potent seal]

To find another... We can do the similar thing for VW

[plucky swan]

yep so 17/2 also equal x^2 + y^2, no??

[potent seal]

Yeah that's the first equation

But now between points V and W, the distance is the same as OW

[plucky swan]

yep root17/root2

[potent seal]

Yeah

So how do you find the distance between 2 points

[plucky swan]

|Vw|

w-v, no?

[potent seal]

No I mean

Based on their coordinates

[plucky swan]

x1 - x2, y1-y2 maybe

[potent seal]

Have you seen the distance formula?

[plucky swan]

Errr nope

Oh you mean

root x1-x2 squared one??

[potent seal]

Yesss

• y2-y1 square

Sqrt of that

[plucky swan]

I'm still kinda confused as to how we can use this to get w but i know nothing atm so

[potent seal]

Just get the relation

If it's difficult to type then just send the pic of the equation when you get it

[plucky swan]

I gotta dip g I havent been focusing here

if you're able and you've done it could yhou send line a small line just explaining your method in dms

i gotta dip

i apprecite it alot tho still

.close

[safe radish]

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[potent seal]

Ok

[safe radish]

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[noble void]

As for A, by drawing the probability tree I got 1/5 is that right?

[safe radish]

@noble void Has your question been resolved?

[thin bridge]

show your tree

and which part(s) of the tree you're looking at

[noble void]

I'm looking at 1/5

[thin bridge]

taking just the 1/5,
that'd be the probabilty that the second ball is white given that the first ball is white
which isn't what the question is asking for

also note that there are two branches where the second ball is white

[noble void]

The problem is, it's not stating from which side it's coming from

I mean it's only saying the second ball is white

[thin bridge]

yes

so you'd need to consider all cases where that occurs

[noble void]

In that case what should I do?

[thin bridge]

and not just take one, or a random one

there are two ways to get second ball being white
WW
RW

find the probability for each event

and add them

[noble void]

So 1/5+2/5 is what you mean?

[thin bridge]

no

probability of getting WW isn't 1/5

[noble void]

Oh wait so 2/6 * 1/5 + 4/6 * 2/5

[thin bridge]

yes

[noble void]

And the third question I did the same but P(w1|w2) using the formula

for b I got 8/15 by doing 2/6 * 4/5 + 4/6 * 2/5

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[unique valley]

Bit confused with the final proposition, is it saying r is determined by the lambda blocks or that r is determined by the minimal polynomial itself?

[safe radish]

@unique valley Has your question been resolved?

[merry sleet]

it just makes the link between the two

[unique valley]

I’m more so asking say for example we have lambda blocks of J_1(lambda)…J_5(lambda) but the minimal poly is (x-lambda)^6.

Max(1,2,3,4,5)=5 but clearly the highest power of (x-lambda) to divide the minimal poly is 6

[merry sleet]

thats not possible

thats what the theorem says

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@unique valley Has your question been resolved?

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[indigo tundra]

I did this idrk what to do

[fleet condor]

,tex .log rules

[indigo tundra]

Idk

Oh wait

I can use quotient rule right

[fleet condor]

you cant use quotient rule at all

[indigo tundra]

oh

😭

[fleet condor]

sec

use change of base rule on the bottom, to make it base 5

[indigo tundra]

wat

[fleet condor]

do you see the base change rule on my tex?

change the bottom log_15 to be log_5

[indigo tundra]

how

[fleet condor]

do you see the base change rule on the tex?

it changes a log_b to two log_c

[indigo tundra]

yes

[fleet condor]

you should end up with a fraction on the denominator

with the top and the bottom both being log_5(soemthing)

[indigo tundra]

😭

[fleet condor]

the bottom two logs should be log_5 not just log, and you should show that the bottom division happens first with parenthesis

but yes, thats right

now do you know how to simplify fraction/fraction?

[indigo tundra]

how log base 5

oh yeah we can make it any

[indigo tundra]

no

oh is the top part 1

Idk

[fleet condor]

no?

$\frac{\frac ab}{\frac cd} = \frac{ad}{bc}$

[indigo tundra]

log_5(m)/log_5(m) at the top tho

[fleet condor]

i was going to ❌ this but i guess its ambigious and could be right

can you show work

[indigo tundra]

uhh I haven't rlly written anything else

[fleet condor]

okay, when you do let me know

[indigo tundra]

I get the same thing fr

@fleet condor

[fleet condor]

the log_5(15) should have moved all the way to the top, why did it get stuck down there

its d in my above example of simplifying

[indigo tundra]

oh

I thought it was c

and 1 was d

is the answer D idk

@fleet condor

[fleet condor]

your fraction is (log_5(m) / 1) ////// ( log_5(m) / log_5(15) )

lots of division for the middle one

[indigo tundra]

OK

I got D

@fleet condor

[fleet condor]

ok

[indigo tundra]

is it right

[fleet condor]

is all of your work correct? if so then it is

||its correct, but you should understand why and not have to ask to confirm||

[indigo tundra]

yes

Ty

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[sweet sand]

Hey, can anyone help me figure out matlab?

[sweet sand]

I want to plot (0.5258,-0.0117) with my differential eq graph

if I plot the point after i first plot my diff eq graph, the differential equation graph disappears

but I cant figure out how to plot them together either, because the differential equation isnt defined by one single variable so i cant exactly follow tutorials online

[raven heart]

that's weird I pretty much copied your code and I can have the graph and the point

I'm using octave but that shouldn't change much

@sweet sand

[sweet sand]

:')

Maybe ill refresh my computer its a slow one

thank you though

OH

I RAN THE WRONG FILE

holy shit

[raven heart]

lol

[sweet sand]

ok

nevermind im good

sry bout that

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[edgy lodge]

Hi I’m struggling to find inverse of this

I thought I should find the range of each line and then use that as the domain conditions

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[neon pewter]

what have you tried so far?

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@lament flare Has your question been resolved?

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[lean otter]

Please help. This is all I got so far

What did I do wrong?

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[oak cedar]

Some context. Parts a and b respectively ask to show that the sample covariance between a scaling instrument and predicted residuals is 0, and to show that two different forms of an instrumental variable are algebraically equivalent. One using the predicted value y and the other using observed y. My problem arises in part c where it asks to test the validity of one scaling instrument (z1) given another (z2). I got to the final line, and now am unsure whether I take an expectation here or if I am even correct in not cancelling the cov(x,z) term as I did in part a. Would greatly appreciate some guidance here.

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@oak cedar Has your question been resolved?

[oak cedar]

<@&286206848099549185>

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@oak cedar Has your question been resolved?

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[fickle trail]

is multiplicity the same thing as eigenvalue?

if so, i have 2 eigenvalues?
λ_1 = 1 λ_2 = 2?

here is the full question:

[fleet condor]

not really?

[fleet condor]

its how many times its the root of the characteristic polynomial

[fickle trail]

oh, then how do I find the eigenvalues?

[fleet condor]

so your characteristic polynomial looks something like $(x+4)x^2$, right?

[fickle trail]

this matrix looks like this to me
1 1 -2 0 0 0 0 0 0

[fickle trail]

you subtracted λI from A?

0-λ 0 0 4 4-λ -8 4 4 -8-λ

am i allowed to do RREF on A first? or will that change the answer

1-λ 1 -2 0 0-λ 0 0 0 0-λ

[fleet condor]

you cant

but yes, do det(A-lambda*I)

@fickle trail

[fickle trail]

OK

i will come back to this one

tyvm

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[earnest glacier]

are general equations always supposed to be written using the smallest positive angle?

[earnest glacier]

for example, I'm doing a question to solve for theta in cot(theta) = -2.2 radians, where theta is a real number. The first angle I got was about -24.5 degrees, and I made the general equation -24.5 + 180n, where n is an element of an integer, but the correct general equation was made using the positive angle 155.5.

is this just preference, or is it a rule to form general statements with the smallest positive angle?

[trim swan]

it's just preference, doesn't really matter

[earnest glacier]

oh ok, thanks 👍

[trim swan]

no problem 👍

[earnest glacier]

also, I am having trouble visualizing angle restrictions like this:

-360 <_ theta < 360

those should be greater than/equal to and vice versa

for these restrictions, when I try to understand them I just think that it's a single line because if you were to go -360 it would bring you back to the same point as going 360 degrees around a circle

[trim swan]

well that is two full rotations around the circle

[earnest glacier]

oh

so it's like if you go -360 that line would be 0

then you go around another -360 and it would be 360?

sorry if that's confusing

[trim swan]

well, I think you might be misunderstanding the point of the restriction

yeah, -360, 0, and 360 are coterminal angles

all three of them are the same position on the circle

and yeah your picture makes sense

360 is a full turn counterclockwise

-360 is a full turn clockwise

and 0 is doing nothing

so all three end up in the same place

but you're probably being asked for all the solutions to some equation between -360 and 360 or something

[earnest glacier]

ohh

I'm not sure why I didn't get that before, it was just a bit difficult to understand visually for me 😅

thanks again

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[trim swan]

no problem 👍 hopefully that made sense lol

[earnest glacier]

yeah now I get that theta has to be an angle between -360 and 360, just strange because they're coterminal angles

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[dense rain]

Hey! I need help with this question. find the lagrange error remainder for f(x)=e^(3x) at x=2 centered at 0 (using the 3rd degree)

[dense rain]

stuck on finding the max value of the n+1st derivative
and how to use that to get an answer

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[dense rain]

@raven vessel

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@dense rain Has your question been resolved?

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[dense rain]

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[glacial tinsel]

Hello, I've done this problem a few times now and can't seem to get it right

[glacial tinsel]

For the third side c, I get 9.25 each time which I'm pretty sure is right, but the angles can't be right since they don't add to 180

this was my first attempt, my second attempt I got 84.4 for A

[thin bridge]

can you show all the work you're doing

[glacial tinsel]

yeah sure it's a little unorganized

I used the law of cosines

oh I should mention the picture is my second attempt, different from my answers in the screesnshot

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[peak crane]

How would u tackle this problem

[peak crane]

T-14 x (1/t) = 5

[fair nacelle]

Solve for t?

[peak crane]

Yea

[fair nacelle]

So basically, (t-14)/t = 5

Is that correct

[peak crane]

No

1/t

Mb

[fair nacelle]

So 14/t is the second term?

[peak crane]

Mhm

[fair nacelle]

Correct?

[peak crane]

Yea

Yea

[fair nacelle]

Try multiplying the whole equation by t

Do you know how to solve quadratic equations?

[peak crane]

Yea

[fair nacelle]

Did u multiply by t?

You shoukd get the following

[peak crane]

Yea that’s what I got

Thanks

[fair nacelle]

Type .close

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[full oar]

is there more context?

[torpid ivy]

1. this is how you could do n1

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[dreamy finch]

what's the quickest way to go about this problem?

like i was thinking letting u = <a, b, c>, v = <d, e, f>, and w = <g, h, i>
and then computing the two results

but is there an intuitive understanding to skip these calculations

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[dreamy finch]

<@&286206848099549185>

[untold ermine]

try to use 3 that are not linearly independent in R^3 im not pretty sure if this is true for this case

[vital steppe]

I think you can use the indicial notation

[dreamy finch]

what's that?

[vital steppe]

For example the product u x v can be expressed as $\varepsilon_{ijk} u_j v_k$

[flat frigate]

milo_schwarz

[vital steppe]

But if you haven't seen this stuff before nevermind

[dreamy finch]

okay thank you

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[spring horizon]

where do I start?

[pulsar spade]

It's essentially a system of equations. Let's call the best cruise speed x and the economy cruise speed y.
You know that:
Trip 1: 2x + 3y = 850
Trip 2: 3x + 2y = 900

[spring horizon]

ohh ok

and then i use substitue method?

[pulsar spade]

That is one way to go about it, yes

[spring horizon]

ohh ok tyty

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[fickle abyss]

why is it not $\frac{0.42 \cdot 0.512}{0.512}$

[flat frigate]

yomiko

[fickle abyss]

i get that the probability would just be 0.42(1), but having just the probability of 0.42 on the numerator seems counter intuitive

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[real iris]

Find all positive integer solutions of $x$ and $y$ in the equation: $y=\frac{x^{2}+7}{x+3}$

[flat frigate]

Sb

[real iris]

I want to do this without trial and error

Domain is that $x\ne-3$

[flat frigate]

Sb

[real iris]

Yes but then how do we know for certain that a very large positive integer is not also a solution?

[flat frigate]

Sb

[real iris]

The answers are: (x,y)
1,2
5,4
13,11

<@&286206848099549185>

[pseudo elm]

try doing polynomial division. where will that get you?

[flat frigate]

Sb

[real iris]

I get this

[pseudo elm]

by the way i think you made some error while checking your answers

[real iris]

Wdym?

[pseudo elm]

y=5 and x= 4 is not a solution for example

[real iris]

mb I swapped the x and y values

[pseudo elm]

hmm where did that +16 come from?

[real iris]

Remainder

[pseudo elm]

why does it say +7?

[pseudo elm]

here you've written +1

in the numerator

[real iris]

My bad

it is supposed to be 7

[pseudo elm]

alright

[flat frigate]

Sb

[real iris]

Updated

[pseudo elm]

so the result of the polynomial division is x-3 with a remainder of 16

[real iris]

Yes

How do I do this?

The only way I think I can get the solution is to use trial and error but there has to be better way

[pseudo elm]

so with the polynomial division you've arrived at this step

(x+3)(x-3)+16=x^2+7

there is something you can do here to make y appear on one side

what can you do?

remember y=(x^2+7)/(x+3)

[real iris]

y= (x+3)(x-3)+16?

[pseudo elm]

no because that would mean that y = x^2+7

[pseudo elm]

you can divide by a certain expression here to make sure (x^2+7)/(x+3) appears

[flat frigate]

Sb

[pseudo elm]

exactly!

now you can simplify

perfect

do you know how to finish from here?

[flat frigate]

Sb

[real iris]

No

[pseudo elm]

alright

so since x is supposed to be a positive integer

x + 3 is also a positive integer

integer + integer = integer

so now all you really gotta do is check when 16/(x-3) is an integer

it is still a tiny bit of trial and error but now you only have to check a small amount of cases

[real iris]

Right

What cases should I try?

[pseudo elm]

so when would 16 be divisible by a number?

or rather

what are the divisors of 16?

[real iris]

16,8,4,2,1,

[pseudo elm]

exactly

don't forget negative divisors as well

so, if you want 16/(x-3) to be a whole number as well, then x-3 has to divide 16

makes sense right?

if it doesn't then it's some sorta decimal

[real iris]

Yes it makes sense

[pseudo elm]

and then y would not be an integer

so - when would x-3 divide 16? and that's how you get all your cases

[real iris]

I also have another question if you have time

[pseudo elm]

it's a bit late for me sorry

i'm sure you can find someone else though

maybe with some time

[real iris]

Ok thanks for your help

<@&286206848099549185>

[safe radish]

@real iris Has your question been resolved?

[real iris]

<@&286206848099549185>

Someone pls help

[drowsy karma]

$P\left( a \right)=pa+q\text{ , and }\text{ }P\left( -a \right)=-pa+q\text{ hence}\\P\left( a \right)+P\left( -a \right)=2q\Leftrightarrow q=\frac{1}{2}\left( P\left( a \right)+P\left( -a \right) \right)$

[flat frigate]

Joanna Angel

[drowsy karma]

and

$P\left( a \right)=pa+q\text{ , and }\text{ }P\left( -a \right)=-pa+q\text{ hence}\\P\left( a \right)-P\left( -a \right)=2pa\Leftrightarrow p=\frac{1}{2a}\left( P\left( a \right)-P\left( -a \right) \right)$

[flat frigate]

Joanna Angel

[drowsy karma]

etc

[real iris]

Thanks you didn't have to do the whole thing for me but anyways

Also how do we do part b?

[real iris]

.close

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[subtle robin]

My clock chimes two times $15$ minutes after the hour, four times $30$ minutes after the hour and six times $45$ minutes after the hour. The clock also chimes eight times on each hour in addition to chiming the number of times equal to the hour. (So at $2:00$ p.m., the clock chimes $8 + 2 = 10$ times.) Starting at $12:05$ a.m., how many times does the clock chime in a $24$-hour period?

[flat frigate]

龖

[subtle robin]

when they say after the hour do they mean after XX:00 or after 12:05

[soft verge]

Anyone can help me with a quadratic question

[last heath]

!occupied

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@subtle robin Has your question been resolved?

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[jagged creek]

How would I go about solving this?

[jagged creek]

Any help is appreciated

[plucky elk]

Do you know how to add vectors

[jagged creek]

Not at the moment

I am watching yt vids on it tho

Would this be how to solve vectors?

[jagged creek]

Or a way to solve them?

Any help is appreciated d

[plucky elk]

Yes that's how you add a+b

[jagged creek]

Got it okay

Imma solve for the dot product now

[jagged creek]

And these?

[safe radish]

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[jagged creek]

How would I go about solving this?

[cosmic grove]

calculate a-b ?

i guess b = -2i + 2j

[jagged creek]

But what does comp mean?

[cosmic grove]

I have no idea

give the whole context

[jagged creek]

Okay, thanks tho

Oh

That is literally the whole context

[cosmic grove]

comp isnt a word tho

[jagged creek]

If you want the entirety of it

Here:

It's question 2c

[cosmic grove]

maybe it is projection

projection of (a-b) on c

$\operatorname{comp}_{\bold{c}} (\bold{a} - \bold{b}) = \left( (\bold{a}-\bold{b}) \cdot \bold{c} \right) \bold{c}$

[flat frigate]

Herels

[jagged creek]

Hmm

Thanks

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[flat frigate]

Ava

[timber moss]

you're just negating the second term

distribute the negative to the (y-x) and you'll see

now compare that to the left hand side

[merry sleet]

similar?

-(y-x) = this

right?

yeah ok

you can add in any order

a+b = b+a

.

wdym lol

[timber moss]

you're just factoring out a -1 from the term

[merry sleet]

you are surprised ok

but do you understand?

ok nice

you can distribute the -1

to what you want

the equality will stay true

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[safe parrot]

I can’t seem to do the F(x) correctly

Would love some advice 🥺

I’m thinking of using this

But I’m not getting the same as wolf

[potent seal]

$\int_0^\pi \frac{sinx}{2 - sin^2 x} dx$

[flat frigate]

Lorentz

[potent seal]

This the q?

[safe parrot]

Yes

[potent seal]

The second step seems correct

Have you tried to substitute cosx = t?

[hazy elbow]

wait

[hazy elbow]

you can simplify much better

wait

[safe parrot]

I just want step 3 to be correct

[hazy elbow]

nvm it won't work

[potent seal]

$-\int_1^{-1}\frac{dt}{t^2 + 1}$

[flat frigate]

Lorentz

[safe parrot]

Why minus at before integral?

[potent seal]

If cosx = t
Then -sinx dx = dt

That's why

[safe parrot]

why -?

[potent seal]

...

The numerator is sinx dx

[safe parrot]

sinx = 1

[potent seal]

And that's -dt

[safe parrot]

no? It’s cos^2x

Sorry

Meant sinx

Yes

[potent seal]

Yes

[safe parrot]

How does it become -dt?

[potent seal]

And the limits from 1 to -1, I hope you got why it's that

[potent seal]

cos x = t
-sinx dx = dt
sin x dx = -dt

[safe parrot]

Where does -sinx dx come from?

[potent seal]

Differentiate both sides

Are you familiar with substitution?

[safe parrot]

No

And yes

Just sub and then revert

[potent seal]

Yeah

So I substituted cos x = t

[potent seal]

Coz if you have dt/dx, it's -sin x

dt/dx = -sin x
So dt= -sin x dx

Which is the negative of numerator

[safe parrot]

Is this what you mean?

x = cos^2x

1 = sinx

[potent seal]

Why did you put sinx = 1

[safe parrot]

Because the formula requires it

But w/e

Keep it as sinx

It still becomes that

[potent seal]

Which formula? I'm confused now

[potent seal]

This?

[safe parrot]

This

[potent seal]

Yes so

[safe parrot]

I just replace 1, a and x respectively

And get that answer

[potent seal]

This will just be tan^-1 t from -1 to 1

Right

[safe parrot]

Why -1 to 1?

[potent seal]

No wait after you substitute and change the lower and upper limits you don't need to revert back

[safe parrot]

I got 0 and pi

[potent seal]

You can directly put in the new limits in the new variable you made for substitution

[safe parrot]

I don’t want that

I want to know how I get

[potent seal]

You'll get the same ans since it's definite integral

[safe parrot]

-sinxtan^-1(cosx)

[potent seal]

Well ok let's say you revert it back

[potent seal]

Where did the -sinx come from

[safe parrot]

?????????

It’s the primitive function

[potent seal]

This is just tan^-1(cos x) as t= cos x

[safe parrot]

I want to more

Know

How

I

Get it

Thank god

[glacial cairn]

The primitive of sin(x)/(2-sin(x)^2) is not -sin(x)arctan(cos(x))

[safe parrot]

THANK GOD IS NEL

I don’t understand this part

[safe parrot]

if I want to integrate this

[glacial cairn]

Didn't you do that last week or so

[potent seal]

Wait yeah

The q looked familiar

[safe parrot]

I’m about to submit it but I don’t understand it

I checked my answer on paper and did it wrong somewhere

I just don’t understand how I get F(x) from the function

[potent seal]

#help-18 message

[safe parrot]

Yes, I understand trig 1 formula

We substitute cos^x with u

And sinx with 1

[glacial cairn]

No, that's not how it works

[safe parrot]

[safe parrot]

Isn’t this the formula used to find F(x)

[glacial cairn]

Yeah but you can't just substitute sin(x) with 1

$\int \frac{\sin(x)}{2-\sin(x)^2} ,dx = \int \frac{\sin(x)}{2-(1-u^2)} \cdot \frac{1}{-\sin(x)} ,du$

[flat frigate]

Nel

[safe parrot]

I have no idea what that means. I had cos^2x in the denominator

$\int \frac{sinx}{2-(cos^x +1)}$

[flat frigate]

Merineth

[safe parrot]

$\int \frac{sinx}{2-(cos^2x -1)} = \int \frac{sinx}{1+cos^2x}$

[glacial cairn]

That's ok

[safe parrot]

Ok

[glacial cairn]

This is wrong, should be 1-cos^2(x) in the parentheses

[safe parrot]

Yeah mb

[glacial cairn]

$\int \frac{\sin(x)}{1+\cos(x)^2} ,dx = \int \frac{\sin(x)}{1+u^2} \cdot \frac{1}{-\sin(x)} ,du$ with $u = \cos(x)$

[safe parrot]

Fml this boy

[flat frigate]

Merineth

[safe parrot]

Like so?

[flat frigate]

Nel

[glacial cairn]

No, it's 1 - cos^2(x), not cos^2(x) - 1 ...

[safe parrot]

But u = cos^2x not u =cosx

[glacial cairn]

If you want to use this formula then you need u^2

[safe parrot]

What it’s that multiplication you are doing?

[glacial cairn]

It's what dx becomes

dx = du / -sin(x) because du/dx = d/dx cos(x) = -sin(x)

[safe parrot]

I’ve never ever seen something like this in my life, neither my so also

dx du etc

[glacial cairn]

Yeah that's why the last time I kind of side-stepped the use of substitution

[safe parrot]

Oh I see but why can’t I just use the formula directly?

That’s what I’ve do so far in the course

[glacial cairn]

Because you have sin(x) in the numerator, not 1

[safe parrot]

And previously in my life

Oh okay

So I want to get it to be 1 before doing anything?

[glacial cairn]

Well, also because you have cos(x)^2 instead of x^2

You can't just use a formula for integrating something with x^2 to integrate something with f(x)^2, whatever f is

[safe parrot]

OH so that is why you multiply with 1/-sinx

To get 1 in the numerator

[glacial cairn]

I mean I don't multiply by 1/-sin(x) from nothing

This comes from the substitution