help-0

[crimson field]

Much less 200

Can’t prove it tho

If you figure out a nice proof ping me

[wintry sluice]

💀

bruh

tell me how you came to this conclusion

[crimson field]

Ancient wisdom

Known as trial and error

[wintry sluice]

pour some into my brain

[wintry sluice]

oh

alr ig

also one more help

can you tell me how to apply component and dividendo

[crimson field]

Just use the formula

I’m gonna sleep gl

[wintry sluice]

alr

thanks

[wintry sluice]

bruh

.close

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[dusky igloo]

when I do the integral from -999 to 999, I get 0. Why does the integral not converge when it goes from -infinity to +infinity? How can I show that it does not converge

[hushed locust]

when you have an integral with two improper bounds like that, you have to split it somewhere in the middle

for example, if we choose to split it in the middle, we can write:

,,\int_{-\infty}^\infty x^3 \odif{x} = \int_{-\infty}^0 x^3 \odif{x} + \int_0^\infty x^3 \odif{x}

[ocean seal]

pnoןɔ

[hushed locust]

then we need both improper integrals to converge individually in order for the overall improper integral to converge

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[dusky igloo]

Thanks

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[heady trail]

Any idea in regards to 22b?
I was thinking two senarios, assuming the two are A and B

1. A is in one of the team, so 5/9
2. A is umpire, so 8/9

But I just can't think further, it is given 22a=310

[arctic lintel]

i think it is easier to find the probability they are on the same team and then subtract that from 1

[heady trail]

Yeah I was thinking about that also

So in this senario that would be 310(4/9) right?

Or 9C4?

Here is when I became a bit loss on the answer of 210

[arctic lintel]

wait how did they end up getting 310 for 22a

[heady trail]

(9C1)(8C4)/2

9C1 is for the umpire

Remaining is 8C4, divided to avoid repetition

[arctic lintel]

that equates to 315

and how does dividing avoid repitition here

see i went 9C4*5C4, as we have 9 we can choose 4 of them to be on a team, then choose 4 more out of the remaining 5 ot be on a team and the last guy will be the ref

but that gets 630

[heady trail]

My fault, I misremembered the answer. But it should be 315 according to the textbook

That was what I and the person I asked got to and made the most sense

[arctic lintel]

aight i see the repitition now

yeah 315 seems right

[arctic lintel]

is the final answer 210?

oh didnt mean to reply to myself

[heady trail]

Yes

[arctic lintel]

aight

so we want to find all the ways that they are on the same team and subtract it from the total possible ways of arranging the team

[heady trail]

Yup

[arctic lintel]

if we make those 2 people on 1 team already that means we have 7 people left out of those 7 people 4 people are going to be on the other team so 7C4, out of remaining 3 players 2 of those players are going to be on the same team as the other 2 so 3C2, last guy is the umpire therefore there are 7C4*3C2 ways they are on the same team

therefore if we subtract 7C4*3C2 from 315 we should get all the ways they aren't on the same team

7C4*3C2 = 105, hence 315-105 = 210 ways they arent on the same team

[heady trail]

That made so much more sense

Thank you!

[arctic lintel]

aLL G

[heady trail]

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[queen quiver]

I claim help 0!

Can someone explain this simplification step here? I don't really get how the difference of two cubes is applied here

Nevermind I understand it

[lone heart]

@queen quiver Has your question been resolved?

[queen quiver]

I am in Calculus 1

Would I need to know how to demonstrate why lim x->0 for sin(2x)/x = 1 via Squeeze Theorem as well

I guess it'd be enough to know how via L'Hopital's

Why is it c=2

[queen quiver]

Nvm

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[north heart]

Can someone help me understand why if x belongs to rational cos(int pi) beomces +-1.Why cant it be zero?

Cant x times n! be a fraction?

[next spade]

x*n! cant be a fraction when n->infinity

[north heart]

Thats where i dont get it

[next spade]

if x is rational then x = p/q, then q!*x would be an integer, you agree?

[north heart]

yes

[next spade]

then any n greater than q, n!*x would be an integer

[north heart]

ooooh

tysm

[next spade]

np

[north heart]

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[mental solar]

Does anyone have a video / guide

[mental solar]

On how to find the domain and range of a function

[barren portal]

post your question

also try khan academy

[wanton nova]

Just search on youtube

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[slim chasm]

how do i work out sd again???

[ocean seal]

Δyssrupt

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[flat roost]

If a function has no turning points, can i say that if f(5) is smaller than f(10) the function is increasing?

[flat roost]

just an example

[lavish cave]

Yes, well depends on what you mean by 'increasing'

[flat roost]

"increasing function"

[lavish cave]

But if a function is strictly increasing then if a < b, f(a) < f(b) for all a, b

Increasing can be like $a \le b, f(a) \le f(b)$

[ocean seal]

south

[lavish cave]

So both work if a is not equal to b

[worn fox]

You also would likely need continuity

[lavish cave]

Yeah ofc

[flat roost]

oops

ignore what i wrote

but this is the question

[lavish cave]

Ah this is a rational function so f'(x) = 0 is impossible

[flat roost]

Ye

[lavish cave]

The idea is that if f'(x) > 0 for all x then it's increasing

If f'(x) < 0 for all x it's decreasing

[flat roost]

How do i prove that?

[lavish cave]

And otherwise it's neither

[flat roost]

This could happen if there are turning points right?

[lavish cave]

Yes

[flat roost]

But i proved there are neither

[lavish cave]

You must have $\lim_{x \to a} \frac{f(x) - f(a)}{x - a} > 0$

[ocean seal]

south

[lavish cave]

So $\lim_{x \to a} f(x) - f(a) > 0$

[lavish cave]

Cool so that's it

[ocean seal]

south

[flat roost]

Mind running through what should i write here?

Thanks

Find f'(x), set it = 0, prove no turning points

[flat roost]

And then prove increasing/decreasing, but what is "a" here

[lavish cave]

Don't worry about that

a is just an arbitrary x value

We need it for the definition of the derivative

[lavish cave]

Yes so if there are turning points that are not saddle points, it cannot be an increasing or decreasing function

You need to find the second derivative or sketch a sign diagram

[flat roost]

question didnt explicitly say find 2nd derivative though

[lavish cave]

I think they're taking increasing to mean that the function can be flat

Yeah that's the normal definition

Like with this definition y = x^3 is increasing

But it has a critical point at x = 0

The critical point is just a saddle point and not a turning point

[flat roost]

Point of inflection?

[flat roost]

You can prove this by seeing that the slope of x^3 at, say, x = -5 and x = 5 are both positive, hinting no change of slope right?

[lavish cave]

Yeah

[flat roost]

thanks

but increasing vs decreasing is my proof okay or not

like showing that for a>b f(a) > f(b)

[lavish cave]

As long as you get the idea that f(x) having turning points almost always means that it is neither increasing not decreasing, you're good

[flat roost]

Ah i see

Thanks

I understand it now

.close

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[lavish cave]

No worries

[lone heart]

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[tiny bay]

hello i have a plane

x = 1 + s - t
y = 2 + s + t
z = 3 - s + 2t

and i want to rewrite it to this form

ax + by + cz + d = 0, how do it (also what is this form called, math english is not my main)

[lavish cave]

So the plane has (x, y, z) = (1, 2, 3) on it if you set s = t = 0

Now the direction vectors come from (s, s, -s) and (-t, t, 2t)

And we want to take the cross product of those two vectors

But it would be easier to take a scalar multiple out from them

So we can just do (1, 1, -1) cross (-1, 1, 2)

,w (1, 1, -1) cross (-1, 1, 2)

[ocean seal]

Results provided by WolframAlpha

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Upgrade to WolframAlpha Pro!

Requested by sooouth.

[lavish cave]

This is your normal vector

So a = 3, b = -1, c = 2

Substitute in the point (x, y, z) to find d

[tiny bay]

hmm thanks i am going to try this is new and confusing for me

[lavish cave]

No worries

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[tiny bay]

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[chilly hedge]

Could someone explain me this?

[chilly hedge]

116=244+200-441,81 cos

how did the -382 come out? i got -328

and what after that?

[lone heart]

@chilly hedge Has your question been resolved?

[chilly hedge]

<@&286206848099549185>

[alpine sable]

-328 is correct yeah

[chilly hedge]

so did the teacher miswrite it?

[alpine sable]

only case yeah

[chilly hedge]

and how do i get 42,06°?

[alpine sable]

its cos right?

[chilly hedge]

yea

[alpine sable]

$$\cos\left(\alpha\right)=\frac{b}{c}$$
$$\alpha=\cos^{-1}\left(\frac{b}{c}\right)$$

[ocean seal]

Project_Nova

[alpine sable]

if i remember correctly it should be this 😭

[chilly hedge]

so how tf do i put this into my calculator? 😭

[alpine sable]

let me pull out mine holdon

actually nah wait

[chilly hedge]

yea i got that, how do i put that into? what do i write?

[alpine sable]

one sec

[gray isle]

shift cos

[alpine sable]

please save them i cant

😭

[chilly hedge]

yea?

[alpine sable]

that is how you get it on the calculator

[chilly hedge]

no i swear im starting to tweak out

[alpine sable]

press shift on the calculator

then cos

[chilly hedge]

so im here at -328= -441,81 cos

[alpine sable]

yeah

oh wait

bruh.

[chilly hedge]

What do i do from here?

[alpine sable]

has it been 441.81 this entire time

im writing it yeah one moment

im crying 😭

[chilly hedge]

a2=b2+c2 -2bc cos alpha

116=244+200-441,81cos alpha

i got -441,81 from an other thing

but thats 100% correct

[alpine sable]

$$-328=-441.81\cos\left(\alpha\right)$$
$$\cos\left(\alpha\right)=\frac{328}{441.81}$$
$$\alpha=\cos^{-1}\left(\frac{328}{441.81}\right)$$

[ocean seal]

Project_Nova

[alpine sable]

although uh

the erm

final result doesnt make sense to me

[chilly hedge]

wait

i got it?

[alpine sable]

i think you have to press another button

[chilly hedge]

this last thing worked

i dont even know the name im failing math

[alpine sable]

yep

[alpine sable]

i think its inverse trig

[chilly hedge]

but then how tf did the teacher get 42,06

is he acoustic?

[alpine sable]

no

because he wrote -382 instead of -328

lil mistake i believe :P

[chilly hedge]

but when using shift cos is it okay to leave the - behind?

[gray isle]

leave what - behind

[chilly hedge]

that 30,17

[chilly hedge]

on the example above its 328, and mine is -328

i can leave that - behind?

[alpine sable]

because both side are negative

we can multiply it with a minus one

both side

and - disapears :D

or when you uh divide

if top and bottom are negative it cancels eachother out

[chilly hedge]

ahh

makes sense now

thanks

[alpine sable]

[chilly hedge]

.close

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[steep rover]

im supposed to use a definition and not the wronskian, which states a pair of functions are linearly independent on an interval if neither of them are a constant multiple of each other

on that interval

how do i show that this one is linearly independent/dependent?

on the interval (0,1)

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[orchid oasis]

hi! i just need some clarification on why $sin^2t$ needs to be derived with the chain rule and not the power rule? I know how to do both methods I'm just not sure why the chain rule is the one applied to this problem

[ocean seal]

kne

[orchid oasis]

oh lol i didn't think it would do that to the whole paragraph

[naive valley]

if you did just the power rule, you would get 2sin(t), right?

[orchid oasis]

yes

[naive valley]

but the problem is that you're taking the derivative with respect to t, not with respect to sin(t)

[orchid oasis]

ohhhhh i see

thank you!

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[swift saddle]

Does x + x^2dx = 1 + 2x or x + 2x?

[swift saddle]

Does the dx apply to the entire expression or just the x^2 because no parenthesis

[lone heart]

@swift saddle Has your question been resolved?

[swift saddle]

This is why I don’t like help channels

[fathom dragon]

Hi

Could you explain the context to your problem?

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[cyan pecan]

HHIII

[cyan pecan]

Helloooooooo im hereee im backkkk

exam tmr im running so late with time

i was hoping to get some help on my review

[jovial dawn]

?

[cyan pecan]

its sending

lol

[jovial dawn]

ahh

[cyan pecan]

ta da!!

[jovial dawn]

what help do you need?

[cyan pecan]

well

1

[jovial dawn]

do you know how to add matrices?

[cyan pecan]

im not sure what i need to do

im not familiar no

[jovial dawn]

do you know what matrices are?

[cyan pecan]

sortaaa

[jovial dawn]

then, what are they?

[cyan pecan]

theyre what number one is

i dont rly need to learn this i need to just know how to solve it for my exam tmr

its a bullshit class

just need to get a good grade on exam so i can finish this semester

[jovial dawn]

well, look at the first question

[cyan pecan]

looking

[jovial dawn]

and at the answers, what do you think fits the best

[cyan pecan]

do i do -7 plus 6

[jovial dawn]

if the sign between the matrices is addition

[cyan pecan]

and 1 + 2

[jovial dawn]

so the position a1,1 is -1

position a1,2 is 3

[cyan pecan]

u lost me

lmfoaoaoa

[jovial dawn]

a[row][column]

is the index

so when i refer to a11 its the top left elemnt

which is -7

and in the second matrix its 4

[cyan pecan]

okay

i with tya

let me google something

[jovial dawn]

so when you do addition on matrices, they need to be of equal rowsXcolumns

if they are not, you cannot do addition

so in this case the matrices are both 2x2, meaning you can do addition

you understand this far?

[cyan pecan]

ITS

C

right

jumped ahead

BUT ITS C!

[jovial dawn]

indeed

[cyan pecan]

so what about # 2

[lone heart]

@cyan pecan Has your question been resolved?

[cyan pecan]

YESSS

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[spring dawn]

I still dont get fundamental principle of counting 😭

[shy dove]

For the first show, how many choices does she have?

[spring dawn]

this one I still get it

[spring dawn]

3, but I dont really get the questions I suck at word problems 😭

[limpid turret]

Thrice means three times

[spring dawn]

ik but I dont get why it's based on first stop second stop and third stop

[oak parrot]

you only have 2 options left after the first stop

[shy dove]

Ah, yeah that‘s quite common. Trick is to read them thoroughly two or three times, and write down what you know about the problem, and what the problem wants you to find

[spring dawn]

is it bec it says there "ways to arrange her tour schedule"?

[oak parrot]

yes

[shy dove]

Yes, and usually you don‘t repeat a city twice in a tour

[spring dawn]

ohh then I get it how about this one, wait give me a sec

A model selecting her outfits purchased 6 blouses, 4 skirts, and 3 blazers. How many different new outfits consisting of a blouse, a skirt, and a blazer can she create from her new collection?

[spring dawn]

I got confused on the different new outfits part

[shy dove]

Well one outfit is blouse and skirt and blazer

And little trick, in maths, when you hear "and", think multiplication, and when you hear "or", think addition

[spring dawn]

ohhh

[spring dawn]

thanks so how do I solve this?

[shy dove]

Well, we know that one outfit is blouse and skirt and blazer, how many choices does she have for each?

[spring dawn]

6 for blouse 4 for skirts and 3 for blazers

is it.. 3x3x3?

Haha since it needs to consist all 3 of them and there r only 3 blazers...

[shy dove]

Well, theoretically she can have the same blazer but with different blouses, so it wouldn‘t be 3x3x3

[spring dawn]

then how 😭

is this question easy?

[shy dove]

If you‘d like you can use smaller numbers and like draw the thing. Say you have 1 triangle, 2 circles and 2 squares, how many groups of 3 different shapes can you make. You might notice a pattern

[shy dove]

"Easy" is relative. It‘s easy for some and harder for others

[spring dawn]

yeah ur right, I was absent when they taught this

[shy dove]

Wait my example might not even work the same way, ignore this

[spring dawn]

oh ok lol

but what was the answer to ur example?

[shy dove]

Things like these are not really fresh in my memory. My mind tells me 4, but I couldn‘t explain it intuitively. I asked in the helper channel so someone else might come here and explain better :)

[spring dawn]

Thank you though 😄

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[fathom harbor]

how does htis prove anything???

[fathom harbor]

I thoguht this was a reciprocal function

oh wait is ti because it's a reciprocal function the deonminator must always be positive or 0?

approaching 0**

[merry iris]

since the slope of the graph is always positive (strictly/monotonically increasing), tangents to the graph cant be negative because that would imply the graph has a negative slope at some point

[fathom harbor]

ok that makes enough sense

do you think I could dm you like 5 questions later?

[merry iris]

no

[fathom harbor]

A lil confused on some stuff but I'll try them again

[merry iris]

just ask here

[fathom harbor]

ok

tysm

.close

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[fathom harbor]

.reopen

[lone heart]

✅

[fathom harbor]

how is F slowing down?

I thought it was speeding up

or is it slowing down comparatively to E-F

[merry iris]

the speed is decreasing

speed is absolute value of velocity/slope

in other words it's getting flatter

[fathom harbor]

but this is displacement over time graph I thought

[fathom harbor]

S is denoted as displacement in my curriuclm but doing different questions is S supposed to denote speed?

[merry iris]

it is a bit hard to see though lol

[fathom harbor]

oh

Wait now I see what you mean was a little bit confused with that one tyty

[merry iris]

idk it depends

my institution uses x(t)

[fathom harbor]

ah fair enough

.close

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[fathom harbor]

.reopen

[lone heart]

✅

[fathom harbor]

nvm

.close

[lone heart]

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[fathom harbor]

.reopen

[lone heart]

✅

[fathom harbor]

I don't get

the last part on how they sovled for the derivitive

I understand chain rule is used here yes

but I don't get (-3/(2sqrtx)

that part

[silent stratus]

kk so what did you get after applying Chen Lu

[fathom harbor]

I wasn't sure how to do the (-3/2sqrtx

part

Just that part

I understand how to solve the rest of the equation

[silent stratus]

ohh so differentiating -3/2 sqrt(x)

[fathom harbor]

yeah

[silent stratus]

sqrt(x) is x^1/2

differentiating that you get 1/2sqrt(x)

[fathom harbor]

but hten why does the 3 go to the top in this case?

.close

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[bleak harbor]

Okay this is a random question I thought of. I don't know if such a problem can be answered or not as I do not have a high enough math education to answer such a problem but I thought I might as well ask in case anybody wanted to take a crack at it.

The problem is as follows:

You are playing a game with your friend. You setup bins with balls in them and your friend picks balls from those bins.

A bin can hold up to 5 balls.
You can drop up to 5 distinct types of balls in the bin and/or you can drop multiple of the same type of ball in the bin without restriction. The bin just cannot hold more than 5 balls.

In the game, if your friend picks a certain type of ball, they will either automatically lose, automatically win, or have to pick from another bin. This other can be a bin picked from previously or a bin that has not yet been picked from. When your friend picks from a previously accessed bin, they pick with replacement so it will still have the same makeup of balls from when they first picked from it.

There can be multiple types of balls that result in an automatic loss or an automatic win.

Given that you can place any number of bins before the game starts and can determine from what bin your friend starts picking from, is it possible to create a setup of bins for any given rational probability of your friend's victory?

If you need clarification for the problem, let me know.

[bleak harbor]

As a side note lemme just say that this would be a REALLY boring game to play.

[untold crane]

Do you allow infinite number of bin ?

Ah no my bad you said rational so it’s useless

[bleak harbor]

You can have as many bins as you want

The probability of winning just has to be a rational number

E.g: If I wanted the probability of my friend winning to be 1/7 I could make the setup this:

-Bin 1: 50% chance of going to Bin 2, 50% chance of Losing
-Bin 2: 1/4 chance of Winning, 1/2 chance of Losing, 1/4 chance of going back to Bin 1

[untold crane]

I think the fact that your friend can go back to any bin he want destroys your game

Because then it’s not determinist

[bleak harbor]

Your friend does not choose to go back to any bin he wants

You decide when your friend goes back to a previous bin

[untold crane]

Before the game starts ?

[bleak harbor]

Yes

[untold crane]

So why not juste placing the same bin after

[bleak harbor]

You decide everything before the game starts. The game is 100% deterministic. Your friend isn't making any actual choices.

The problem isn't to just make it as hard as possible for your friend to win because you could just keep placing more and more bins. The problem is asking is it possible to make the probability of winning any rational number you wanted.

Like if I wanted to make my friend have a 832/1731 probability of winning could I make a setup such that that occurs? Can I do that for every rational probability?

[untold crane]

This is impossible to make 1/2 chance to win

Because there is 5 ball

[bleak harbor]

You don't have to place all 5 balls into one bin.

You can place up to 5 balls in a bin.

So you could choose to place 4, 3, 2, or 1

[untold crane]

Okok

What kind of game is that 😂

[bleak harbor]

Yeah no it was just a random problem I thought of lol

I was just wondering if anybody had any idea how to approach something like this

[untold crane]

I don’t think you could do 1/7

Ah yes you just did

[bleak harbor]

-Bin 1: 50% chance of going to Bin 2, 50% chance of Losing
-Bin 2: 1/4 chance of Winning, 1/2 chance of Losing, 1/4 chance of going back to Bin 1

^This results in a 1/7 probability

[untold crane]

In this case the friend begin in wich bin ?

[bleak harbor]

Bin 1

[untold crane]

Since there is an infinite loop

You calculated the infinite sum ?

[bleak harbor]

Yes that probability converges to 1/7

[untold crane]

Damn you’re committed

[bleak harbor]

When your brain makes up a random problem that you're not able to solve as a means of procrastinating:

-Bin 1: 50% chance of going to Bin 2, 50% chance of Losing
-Bin 2: 1/4 chance of Winning, 1/2 chance of Losing, 1/4 chance of going back to Bin 1

1/8 chance of winning directly (that is to say getting the 50% chance of going to bin 2 and then the 1/4 chance of winning)
Given that you're sent back to Bin 1 (which is a 1/2 * 1/4 = 1/8 chance), there is another 1/8 chance of winning.

As you can see, this results in the probability of winning being an infinite geometric series of 1/8 + 1/64 + 1/512 +... = (1/8) / (1- 1/8) = 1/8 * 8/7 = 1/7

But this is just for a 1/7 probability. The question I have is: can this be done for any rational number probability?

[untold crane]

Yep I calculated it too

So you generated 1/(x-1) when x=8

With 5 balls I don’t think we could genrate other interesting number with this pattern

I think I would go for the binary decomposition intuitionally 😂

I don’t know but I hardly see a way to make an algorithm that make the good configuration for any given rational number

[bleak harbor]

Yeah I was trying to think and I just couldn't see a way to generalize any of this

[untold crane]

Even just for all 1/x

(Wich would be incredible if it’s worked for that)

But that’s a fun game actually 😂

I mean

The game is to generate a given rational number

[bleak harbor]

The game itself is incredibly boring lol

the concept of the game is interesting

Imagine playing this game just going through a bunch of bins your friend laid out and picking out a ball from each of them lol

[untold crane]

That’s actually something I would love doing while procrastinating

[untold crane]

I think I would laugh and then after 3h just 🔫🔫 in his head

[bleak harbor]

lol

I sent the question to one of my friends and he came up with a way to create a 1/13 probability (discrepancy in printed result is due to trial size)

Bin 1: 50% chance of losing, 50% chance to go to Bin 2
Bin 2: 50% chance of losing, 25% chance of going back to Bin 1 or going to Bin 3
Bin 3: 50% chance of winning, 50% chance of going back to Bin 1

Because initially when I thought of this problem I thought "there's no way you could create a probability X/Y where Y is some prime number that wasn't 2, 3, or 5" simply because of the limitations of what probabilities of each of the 3 options you could make for each bin but 1/7 and 1/13 at the very least have solutions (or seem to anyways).

But the ability to loop back to previous bins yields some...interesting results.

[lone heart]

@bleak harbor Has your question been resolved?

[charred flint]

this would fall under markov chains, but it kind of reminds me of problems where you try to make probilities like 1/3 with coinflips

if you're able to feed things back into each other you probably get everything, it's sort of how continued fractions work

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[wet solstice]

Can someone remind me the rule on how to factor things like this

[jagged cobalt]

difference of two squares

a^2-b^2=(a+b)(a-b)

[wet solstice]

ok thx

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[hearty compass]

<@&286206848099549185>

[crimson field]

Do you know what supplementary angles are

[hearty compass]

No I asked my teacher when we get to that and she said in a few weeks

[crimson field]

It means they add up to 180 degrees

[zinc haven]

!15min

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Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.

[zinc haven]

is this a test?

[hearty compass]

No

Ixl

<@&286206848099549185>

[alpine sable]

stop being a snitch bruh

[zinc haven]

??

[alpine sable]

still need help w this?

[zinc haven]

who are you?

[hearty compass]

Yes

[alpine sable]

ok

first, supplementary agles are those that theyre added up to 180 degrees

like if you have an angle thats 70 degrees, the supplementary angle would be 110

cus adding them together would give 180

[hearty compass]

Ok

[alpine sable]

okey. vertical angles are the pairs of opposite angles formed when two lines intersect

okie i think

KLI AND NPM

[hearty compass]

That not an option there

[alpine sable]

then i think its MNP AND JKI

[hearty compass]

It was wring

[sweet ether]

JKI and LKN

U can see all (hopefully lol) the vertically opposite angles here

[lone heart]

@hearty compass Has your question been resolved?

[hearty compass]

No

[lone heart]

@hearty compass Has your question been resolved?

[lone heart]

@hearty compass Has your question been resolved?

[hearty compass]

<@&286206848099549185>

So scalene

?

Ok thank you

[dense ore]

hello

help

[flat robin]

help

im here to help

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@hearty compass Has your question been resolved?

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[noble notch]

how do i do Q8

i am completely stuck

[nimble fern]

did you try to draw it out?
first consider when will |a+b|=1

[lavish cave]

Yeah, notice that if you have vector a be (1, 0)

Then you will have an equilateral triangle if a + b lies on the unit circle

In fact, two equilateral triangles: one on top and one on the bottom

[noble notch]

wdym one of the bottom

if they are negative?

[lavish cave]

Yes

So theta is actually not the anticlockwise angle
It can be both anticlockwise and clockwise

So only $0 \le \theta \le \pi$ is possible

[ocean seal]

south

[noble notch]

yeah ok

how do u reduce that inequality

to find the answer

[lavish cave]

So what's the angle between the two vectors when |a + b| = 1

If both of them start from the origin

[noble notch]

90?

or 0

[lavish cave]

Nope

Did you even draw it

[noble notch]

yes

idk how correctly tho

[lavish cave]

Ok can you show me then

[lavish cave]

Nope, cause sqrt(1^2 + 1^2) = sqrt(2)

|a + b| = sqrt(2) and not 1

[noble notch]

ik

what did u mean by unit circle then

[lavish cave]

So that doesn't satisfy |a + b| = 1

[lavish cave]

A circle with radius 1 centered at the origin

[noble notch]

oh right

i’m actually dumb

[lavish cave]

So it makes an equilateral triangle if you draw it out

[noble notch]

so the magnitude of a and b is equal but not 1?

[lavish cave]

Nah, the question says that a and b are unit vectors

So they both have magnitude 1

[noble notch]

then how does that work

then

if |a+b|=1

[lavish cave]

One sec

This is the equilateral triangle I was talking about

[noble notch]

oh ok

so the angle is 60?

[lavish cave]

Yeah so then if a and b start at the origin

[lavish cave]

That's not the correct angle though

[noble notch]

120

[lavish cave]

Yes

So it's option D

[noble notch]

oh

i get it now

ty

[lavish cave]

No worries

(Imagine rotating b closer towards a: the black resultant vector will still be inside the circle)

[noble notch]

yup ok

thx

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[last rampart]

can someone explain to me how to do this

apparently the answer is 400.1

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[crisp fulcrum]

hii

[crisp fulcrum]

i am doing an integral

but i get a weird answer

$\int_{-\pi/2}^{\pi/2}{e^{-\tan^2 x}\dd x} = e\pi \mathrm{erfc}(1)$

[ocean seal]

what is an 🍓

[crisp fulcrum]

this is the correct answer

[hollow warren]

looks correct

[crisp fulcrum]

but i get this but without the erfc(1) factor

[hollow warren]

with which method

[crisp fulcrum]

i did a substitution where u = tanx

[hollow warren]

then taylkor series?

ive done this exact integral and had the same issue before

[crisp fulcrum]

then dx = du/(1+u^2)

[crisp fulcrum]

i used contour integration

[crisp fulcrum]

from there on i used contour integration

[hollow warren]

can u send a pic or smth?

[crisp fulcrum]

uh the way i wrote it down is kinda shitty but sure

[hollow warren]

ok

using semicircle contour i assume?

[crisp fulcrum]

yes

[hollow warren]

the integral over the arc doesnt go to 0 in this case

[crisp fulcrum]

it does not?

[hollow warren]

theres some criteria the function needs to meet for it to go to 0

lemme check it again

[crisp fulcrum]

im gonna do it all again but less shitty

[crisp fulcrum]

jordans lemma

[hollow warren]

our complex function is $f(z) =\frac{e^{-z^2}}{1+z^2}$

[ocean seal]

Obotron

[hollow warren]

doesnt satisfy the form required for jordan's lemme

[crisp fulcrum]

is this correct

e^(-R^2 e^(2iθ)) decays rapidly as R goes to infinity

1/(1+R^2 e^(2iθ)) too

faster than the remaining R

[hollow warren]

not necessarily

e^(2iθ) can be negative

in that case e^(r^2) blows up to infinity

[crisp fulcrum]

right

[hollow warren]

im not very advanced in contour integration so i dont know how to get around this

when i solved this integral i used feynmann's trick

[crisp fulcrum]

maybe i'll try that

thanks for the help

[robust cairn]

Help me if my answers correct

[hollow warren]

!occupied

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Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[robust cairn]

oh sorry

[lone heart]

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[primal pewter]

im being stupid anyone got a clue?

.closed

.close

[winter light]

Do you know what the gradient is? Or at least how to calculate $\grad f(x, y, z)$?

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[primal pewter]

yeah yeah i just forgot how to differentiate lol

fixed it now

[ocean seal]

Alberto Z.

[winter light]

Oh ok nice

[primal pewter]

thanks anyway

[winter light]

You're welcome 🤗

.close

[primal pewter]

.close

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[junior quarry]

im not real sure what to do for this one

[gray isle]

look up properties of a parallogram
and properties of a rhombus

[junior quarry]

so i know that a rombus needs to have four congruent sides and opposite sides that are parallel

but i dont know if they are congruent

but its given that all sides are parallel

[gray isle]

what do you know about properties of a parallelogram

[junior quarry]

two diagonals bisect and sides are equal

[gray isle]

which sides

[junior quarry]

all of them?

[gray isle]

no

not all sides of a parallelogram are equal

[junior quarry]

im lost

[gray isle]

look up properties of a parallelogram

[junior quarry]

only opposite sides have to be parallel

[gray isle]

you mean equal?

[junior quarry]

yes

[gray isle]

first based on the first two statements,
do you agree that you have a parallelogram?

[junior quarry]

yes i agree that the shape is a parallelogram

[gray isle]

now applying the property concerning opposite sides,
what would AB be congruent to?

[junior quarry]

bc

[gray isle]

no

pay attention to the wording of my question

[junior quarry]

DC

[gray isle]

yes

now again pplying the property concerning opposite sides,
what would BC be congruent to

[junior quarry]

AD

[gray isle]

yes

[junior quarry]

This means all four sides are congruent which is the definition of a rhombus.?

[gray isle]

yes

[junior quarry]

thank you

.close

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[gray isle]

be careful with which option you select though

[junior quarry]

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[junior quarry]

.close

[fluid epoch]

could i get some help on this question?

gcse level

[crimson field]

Youll need to square both sides twice

[silent stratus]

I remember this from blackpenredpen vid

he used geometry

[fluid epoch]

thanks il do that

my classmate said i needed to do difference of 2 squares and make a simultaneous equation which im not sure how id do

since theres only x no y or any other variables

[crimson field]

Nah what

Don’t overthink it

Just square twice and make sure to consider domain throughout the problem

[crimson field]

Difference of squares would be 9-x^2 for example, which can be factored

[fluid epoch]

Whats domain? I havent learnt that yet in class

my teacher put this question up as a challenge

[crimson field]

Let’s say you’re taking the square root of a

[hollow warren]

just check your solutions with original equation after ur done

[crimson field]

What could a be?

What values can you not take the square root of

[fluid epoch]

do u mean like to get a square root that is an integer?

or just cant be square rooted

[crimson field]

The latter

[fluid epoch]

maybe irrational numbers like pi cant be square rooted?

[crimson field]

They can be, it’ll just be an irrational number

[fluid epoch]

oh ok then im not sure

[crimson field]

-1 cannot be

[hollow warren]

whats the square rooot of -1

[bleak ledge]

Double squaring is the only method I see ig

[fluid epoch]

oh got it yeah i shoulve got that

[crimson field]

So what numbers can be square rooted

[fluid epoch]

positive numbers

[crimson field]

And 0

So at every step of the problem, make sure that what you have under a square root is >=0

Or just solve it normally and then retrace your steps to make sure you never had a negative in there

And also one more thing, make sure you’re never setting a square root to a negative number

[alpine sable]

bro is hella smart

wtf

[crimson field]

For example if sqrt(2-x) = x, you know x>=0 cause otherwise it couldn’t be equal to a square root

[fluid epoch]

oh ok thanks i got it il try solve now

so if what im square rooting is negative what should i do?

i mightve gone wrong but i have x^2 = -305 and negatives cant be square rooted

wait nvm i did go wrong

[thick lynx]

It will be much less work if you get sqrt(9 - x^2) to the RHS first

Then square twice

If you want to, you can also make the substitution z = 9 - x^2 first

Then 16 - x^2 = 7 + z

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[fallen shoal]

someone got me right

[merry iris]

what's different between the graphs?

[fallen shoal]

theres no graph

[merry iris]

i mean equations

[fallen shoal]

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[sterile wyvern]

does anyone have a nice derivation of this? I've searched for a while but I always see this just as an accepted definition. Ik this may seem super basic but this is a formula that has bugged me since high school and I always took it at face value, and now that im at uni the lack of intuition is bugging me more than ever

[lofty gorge]

Variance is intuitively understand as a measure of spread, in the sense of how wide the values you get during observations will be

The obvious thing to do for that

Is I agree, not variance

However those definitions end up not having nice properties, so we use variance

Which is defined using this formula, and doesn't really get more basic than that

As for how to come up with 'this formula will probably do what we want'

X - μ measures how far off our observations are from the center, and the square makes that positive

(And introduces a weighting)

And then expected value so that we get an average

[tight locust]

Also to add on to that

You can use the linearity of expectation

[lofty gorge]

(The 'obvious' thing would be E(|X-mu|), that is just measure the average distance directly, but the absolute value makes it annoying and you can't use calculus, hence why we use a square instead)

[sterile wyvern]

well why didn't we define it using idk absolute values for example

[sterile wyvern]

yeah exactly

so the answer is just, "we did it like this cause its easier"?

[tight locust]

E[(x-mu)^2] = E(x^2 - 2xmu + mu^2)] = E[X^2] - 2E[X]^2 + E[X]^2 = E[X^2] - E[X]^2

So you can also view it as:
The difference between the expected value of the square of the set and the expected value of the set squared

[lofty gorge]

Uncertain, there probably are other reasons for it (e.g. I would expect it to have much nicer properties because of it being smoother) but I don't know them

A stackexchange post I found says that variance is more consistent (if you do variances of say 10 samples, then another 10 samples, etc. variance will vary less) with no source or proof

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[fading abyss]

Hi I’m trying to use ratio test to determine convergence or not but wanted to see if I’m doing the steps right so far?

[subtle light]

yeah looks good

[tight pier]

yea

[fading abyss]

Cool I’m getting stuck at the last line I wrote

[tight pier]

pull out n

looks like a familiar limit

[fading abyss]

Oh then it just is 1/n?

[tight pier]

no

[ocean seal]

𝔸dωn𝓲²s

[tight pier]

right

So you have

[fading abyss]

Oh yeah I see

[ocean seal]

𝔸dωn𝓲²s

[tight pier]

Where would this go if you were to simplify the right inner term

[fading abyss]

Like after plugging in the limit?

[tight pier]

yea

Does the limit of the right term seem familiar?

[fading abyss]

Oh yeah does it become (1+1/n)^n?

[tight pier]

yes!

[fading abyss]

I think we went over something like this and it equals e?

[tight pier]

yes

what about the left term?

[fading abyss]

Infinity I would assume?

[tight pier]

yes

so you basically have infinity times e

[fading abyss]

Oh I see then that is infinite still and that diverges right

[tight pier]

yes

n^n is basically growing much faster than 3^n

technically

[ocean seal]

𝔸dωn𝓲²s

[tight pier]

it's not even a null sequence so it fails the necessary condition

[fading abyss]

Oh yeah I didn’t consider that either

Thank you for the visualizations, really helpful !

[tight pier]

thank you too!

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[fading abyss]

.close

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[tawdry urchin]

What is the circumference of the circle?

[rugged agate]

not sure

is that something i need to find out

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Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[subtle palm]

yes boss

[tawdry urchin]

npnp

[tawdry urchin]

yes

[tight pier]

all help channels occupied

[subtle palm]

i made one

[rugged agate]

It says in my notes that the equation for this is O = s/r

where s is arc length and r is radius

[tawdry urchin]

that is correct

I was merely trying to guide you to find arc length

[rugged agate]

the arc length is not the 2?

im mostly confused on what the 2 is

[sour verge]

The arc length is 2, you're solving for theta. You can plug both known values in that formula you have there

[rugged agate]

its just 2/1?

[rugged agate]

So radian measure is just 2/1, and if i wanted the degrees that would be 180/pie?

Im just tryna review all the material for this quiz bc I dont understand this unit very well

[sour verge]

Yes you would get like 2/(2pi) = d/360, whence d = 2*180/pi.

[rugged agate]

ok and when it asks for radian measure i just stop at the 2 right

[sour verge]

It's also not especially intuitive that when the radius is 1, the arc length is the same as the angle.

So it takes time to learn and understand

[rugged agate]

that makes sense on paper

i just dont really get how theta works right now

or radian measure or wtv

Am i allowed to use this for multiple questions?

or do i need to close it now

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[cosmic rose]

I don't understand how to do this at all lol. Professor never covered this and there's no video going over the problem.

[fallow solar]

if the sum of all the vectors is the zero vector you should be at the same point where you started your first vector

you can think of it as the displacement vector

[cosmic rose]

What is a zero vector?

[fallow solar]

a zero vector is a vector of magnitude zero

it's simply a vector connecting two points which are the same

[cosmic rose]

So am I making lines that form a resultant of 0

So a negative line and a positive line

[fallow solar]

well there's nothing called a negative line and a positive line

do you know vector addition?

geometrically?

[cosmic rose]

Where two vector points form a resultant?

Like a and b give resultant a+b

That's about as much as I know for what I'm being asked

[fallow solar]

but geometrically ?

like what they actually mean

[cosmic rose]

I'm not sure

[fallow solar]

https://youtu.be/ghf2602GFZg?si=liUmRzJePJSSgyYn

[cosmic rose]

Oh I know this yeah

[fallow solar]

so try to add three vectors this way so that their sum is 0

[cosmic rose]

I guess I don't understand how to do that on a graph. The only vector addition I've seen is like in the video where it's using letters.

And I can't connect lines like that on the homework site. It keeps forcing me to move the arrow around if I try to start a line from a vector point.

Oh nvm I got it working. Now I just don't get how to make the sum 0 since I've never done these involving numbers.

[fallow solar]

it has nothing to do with numbers

the sum is zero when the third vector ends at the starting point of the first one

[cosmic rose]

I would need to see an example because I don't get it. Zero vector wasn't taught at all or mentioned in the class ever

And usually these problems have videos to explain what is happening but this one doesn't

I only know the vector addition from the video put. I don't know what zero vector means.

So when I see a graph and the only thing I know is vector addition with letters, I assume I need to form three lines that make vector x + vector y = 0 but that isn't correct

[fallow solar]

for two vectors the sum is zero when the two vectors have the same magnitude but point in the opposite direction

[cosmic rose]

I tried that and it said it was wrong. So am I supposed to make six vector points on the graph for this since it wants three vectors?

[fallow solar]

why did you double it

[cosmic rose]

cause isn't a vector formed with two "lines" and it wants three vectors that equal zero?

maybe I'm misunderstanding what a vector is

[fallow solar]

a vector is not formed with two lines

a vector is a line segment with direction

[cosmic rose]

oh so the line segment itself with a direction is the vector?

[fallow solar]

yup

[cosmic rose]

okay I got it now.

so that triangle that is formed is from line segments with a direction which are vectors

the triangle isn't the vector

it's just made when you do vector addition

[fallow solar]

yup

actually a triangle is made of three vectors that add up to zero

[cosmic rose]

okay I think I understand it better now

one quick question, what's an orthogonal vector?

[fallow solar]

orthogonal=perpendicular

they're synonyms as far as I know

[cosmic rose]

would it still be orthogonal if you had two vectors with the same line segment length but one was negative?

like 2u and -2u

or is that impossible since they wouldn't be in the same spot on a graph

or nvm I'm thinking of parallel

[fallow solar]

2u and -2u are never orthogonal they are antiparallel

[cosmic rose]

alright then. thanks for your help! Imma make notes of this stuff!

.close

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[alpine sable]

Alex drives from point a to point b, then immediately returns to A bob drives from point b to point a then immediately reutns to b. They travel on the same road, starting simultaniously, each maintaining a constant speed. Alex's speed is three times Bob's speed. They pass eachother for the first time 15 minutes after the start. How long after the start will they pass eachother for the second time.

[main path]

Looks like a question about comparing function, try making two and seeing when they cross

[alpine sable]

sorry what how do i do that

[main path]

Wait no Nvrmd, try mapping it out using 15 min intervals on a numberline

[alpine sable]

but i dont know how long the distance from a to b is, and alex goes back to a after going to b and bob goes back to b after going to a

[main path]

You just need to know the ratios, Alex passes Bob 3/4ths of the way to point B, because it’s 3x the speed

[alpine sable]

so they meet again in 20 mins?

[main path]

Close but no, I don’t think so

[alpine sable]

if alex passes bob 3/4ths of the way to point b, and that was 15 min
3/4=15min
1/4=5min
alex time=20
bob time= 60

so

wait

[main path]

Try mesureing via bobs time

So 4 intervals of 15 mins

Alex travelling 3 intervals in 15 mins

[alpine sable]

im lost

[main path]

So Bob travels x distance in 15 mins right?

[alpine sable]

yea

[main path]

And Alex travels 3x distance in 15 mins

[alpine sable]

yes

[main path]

So In total, there is 4x distance between A and B

Now, if you put this into a number line, and count each persons distance per 15 mins, you get your answer

[alpine sable]

they meet again in 15 minutes right

so overall 30 minutes after the start time

[main path]

Yes, great job!

[alpine sable]

oh my god thank you so much

[main path]

Whenever you get to problem like this, the best thing to do is figure what kind of problem it is: Ratios, Comparing Functions, etc..

[alpine sable]

ohhhh got it 👍

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[wise pasture]

Help

[wise pasture]

<@&286206848099549185>

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[wise pasture]

<@&286206848099549185>