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[feral summit]

So, this is a Taylor series problem I have.

What I believe you have to do is make the taylor formula equal to some term in the taylor series?

But I'm not sure how one would go about doing that.

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[chrome pebble]

can someone expound this part for me, how come is

[vale wigeon]

the intersection of a family of sets consists of all points which are in every set in the family at once.

the only such point in your case is 1

[chrome pebble]

how come 1 appears in each set

[vale wigeon]

do you think there is a set of the form {1, 2, ..., i} of which 1 isn't a member?

[chrome pebble]

so its like A1 = {1} ; A2 = {1,2} ; A3 = {1,2,3} etc. ?

[vale wigeon]

yes that is the definition of your A_i

[chrome pebble]

ok thanks

close.

.close

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[carmine lion]

how do i do this?

[solar merlin]

try cross multiplying first, or doing something similar

[alpine sable]

More attention should be paid to signs than that suggests

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[sly coral]

hi i was wondering, how do you solve this? the answer is 9 If a 4096k x 8 SRAM uses a dual core arrangement with 256 8-bit words per word line, how many bits are required to address the column lines?

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[scenic wing]

why did -3p become squared ?

[solar merlin]

3p times -p

[scenic wing]

equals -3p right ?

[solar merlin]

no, it's squared because p times p is p^2

and that applies to negatives too

$p^2$

[ocean seal]

alias

[scenic wing]

but there's a 3 next to it

[solar merlin]

yeah, so you do 3p times - p

that's equal to $-3p^2$

[scenic wing]

so anytime i times one number(letter) times a singular letter it comes out to squared ?

[ocean seal]

alias

[solar merlin]

only if the letters are the same. example:

we call it squared because squaring is the way we find the area of a square

squaring is the same value times itself

S x S

[scenic wing]

and in this problem why dont i turn the minus symbols into plus symbols and turn the number into a negative ?

[solar merlin]

they are:

[scenic wing]

like this

[solar merlin]

well, they are doing that, but they're not showing that part

in the picture above i show how you can do it

$-p * 3p = -1 * 3 * p * p = -1 * 3 * p^2 = -3*p^2 = -3p^2$

[ocean seal]

alias

[scenic wing]

ok it makes sense now

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[teal ridge]

whoops i forgot to close sorry i did it

.close

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[muted locust]

How to simplify this?

[novel breach]

You'll have to group like terms together

[muted locust]

But where do you put them together in the parenthesis

[rare folio]

just break it out

in additions, they don't matter

[tardy rapids]

Open the paranthesis

[rare folio]

just keep the positives and negatives

[muted locust]

So in the end just one parenthesis?

[rare folio]

none

[muted locust]

Ok thanks also do i need to put it in an order

Descending or ascending?

[rare folio]

it depends

[tiny minnow]

example ax+bx=(a+b)x

[tardy rapids]

Generally we write higher order terms first

[rare folio]

yea

[tardy rapids]

So n³ term first, then n² and then n

[muted locust]

Oh thanks

Now i do type do i type close?

.close

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[peak iron]

Hi, I know what is a derivation but I have no idea with this question

Does anyone have an idea? Thanks

[vale wigeon]

you have modus ponens (R1) and some rules for working with implications at your disposal

i would probably start by deriving A -> (B -> A) by R2

oh wait no hold on

...nvm we don't have double negation elimination

[peak iron]

yeah...

btw thanks Ann for always being here for questions 😄

[vale wigeon]

oh wait

1. A [given]
2. !A [given]
3. !A -> (!B -> !A) [R2]
4. !B -> !A [R1: 3, 2]
5. (!B -> !A) -> (A -> B) [R3]
6. A -> B [R1: 5, 4]
7. B [R1: 6, 1]

[peak iron]

I'm looking at this, thanks a lot!

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[scenic wing]

Do i multiply both sides by 8 or 6 or do I multiply the left side by 8 and the right side by 6 ?

[zealous bloom]

There are many ways to go about it - however I would recommend moving 1/6 to the RHS of the equation - that way your variable terms is isolated on one side of the equation.

[scenic wing]

RHS ?

[zealous bloom]

Right hand side

[scenic wing]

normally all the denominators are the same but here there are 3 different ones I just dont know how to start

[zealous bloom]

Like I said, I would start by moving 1/6 to the RHS. That would be an easy way to start tackling this.

Do you know how to go about that?

[scenic wing]

ok what then

[zealous bloom]

Then, you have your variable term nearly isolated, so isolate it further by moving the -3/8 to the RHS as well.

That will leave you with x = answer

[zealous bloom]

Do you know how to approach it now?

[scenic wing]

that makes zero sense

they are already both on the left side opposite of the answer

what difference does inverting it make

the answer has to be a singular number

im sorry but can someone that knows what they're talking about help me ?

[zealous bloom]

Wow....I'm not talking about inverting at all. I'm telling you how to solve for x.

[scenic wing]

re writing it so x = the entire equation isn't actually solving for the variable

[zealous bloom]

It literally is.

I'm not telling you to rewrite it - I'm telling you to isolate x.

Moving 1/6 to the RHS means you must subtract it from both sides of the equation.

That means with that step you'll have (-3/8)x = -3/4

Then you just keep going.

Since I don't know what I'm talking about, though, maybe someone else will be willing to help you and your attitude.

[scenic wing]

how am i suppose to times x by 6

when i dont know what it is

because it will be "isolated"

on the far left side

[zealous bloom]

Like I said, maybe someone else willing to deal with your attitude will help you.

[lone heart]

@scenic wing Has your question been resolved?

[rough spear]

Multiply both sides with 8 and 6 or just 48

[scenic wing]

thats what i just started doing

[rough spear]

That will eliminate all denominators and makes it simpler

[rough spear]

So did you get it done

Brr

[scenic wing]

now i have 1/48 and -7/96

after multpilying both sides by 8

[rough spear]

How did you get 1/48

[scenic wing]

i have to times 1/6 by 8 aswell

[rough spear]

1/6 times 8 gives you 8/6

[scenic wing]

no just the bottom 6

[rough spear]

3/8 times 8 gives you (8×3)/8

[scenic wing]

im just multiplying the denominators

[rough spear]

It's the top (numerator) that you're multiplying with

[scenic wing]

the 8 is the denominator

and thats what im multiplying with

[rough spear]

To times denominators you would be doing 1/8 × the fraction

[rough spear]

Ok relax, you're only concerned with the numerators for now, okay?

$a × \frac{b}{c} = \frac{a}{1} × \frac{b}{c}$

[ocean seal]

rael rael

[scenic wing]

ok i put in the wrong answer and this is the work

where tf did they get 24 from ?

[rough spear]

The lcm of 8 6 and 12

[scenic wing]

that sums up to 26

[merry haven]

8×6 is?

[scenic wing]

48

[rough spear]

Lcm doesn't mean sum

[merry haven]

Do you know how to sum this 3/8 +1/6+7/12?

@scenic wing

[scenic wing]

0.375 ?

[scenic wing]

.......

[merry haven]

In fraction form

[scenic wing]

no

[raven rover]

LCM = least common multiple

[merry haven]

So you need to know this to solve these equation

[raven rover]

The smallest number that is a multiple of 6, 8, and 12 in this case

[merry haven]

Do you know what equivalent fractions means?

[scenic wing]

just assume i dont

are you trying to showcase everything i dont know ? Why do you think im here right now ?

[merry haven]

We need to know what you know to see how far we need to go

[scenic wing]

all I want to know is where they got 24 from

and how to multiply fractions by a whole number

[merry haven]

To know this you need to know equivalent fractions

[scenic wing]

wow thats great

[merry haven]

Yeah

Equivalent fractions means fractions that have value but is different fraction

Like 1/2 and 2/4

It's the same value .5 but different denominators And numerator

[scenic wing]

ok im aware

[merry haven]

Note that 2/4 is 1/2 with numerator and denominator multiple by 2

Now to add 3/8 + 1/6

They have different denominators

[scenic wing]

i dont need to add them in the equation

i just need to multiply them individually by 24

which i still dont know where they got 24 from

[merry haven]

My goal here is to teach you this

But first you need to learn how to add fractions with different denominators

Because that knowledge will help you to understand what you want

So how to add 3/8 + 1/6

You need to first find a equivalent fraction that put all these two in the same denominator

Do you how to find it?

[scenic wing]

no

[merry haven]

There's a candidate for you 8×6

48

How many times I need to multiple 8 to 48?

[scenic wing]

6

[merry haven]

Ok

So 3/8 is equivalent to 18/48

Check it for me

This is true?

[scenic wing]

yes

[merry haven]

Ok

Now do the same for 6

How many times I need to multiple 6 to get 48?

[scenic wing]

6/36

[merry haven]

???

[scenic wing]

8

[merry haven]

So 1/6 is equivalent to?

?/48

[scenic wing]

8

[merry haven]

Yeah 1/6 is equivalent to 8/48

Now I have those fractions on the same denominator

And can add those

3/8 + 1/6 = 18/48 + 8/48

26/48

Note that 26/48 is equivalent to 13/24

Note that 8 and 6 divides 24

What we did is find a common denominator

So to the original problem

[scenic wing]

where is this going...

[merry haven]

What is the common denominator here?

A number that 8 6 and 12 divides

[scenic wing]

24

[merry haven]

Can you find the equivalent fractions to 24 for each fraction?

How much times you need to multiple 8 to 24?

[scenic wing]

3

[merry haven]

So write the equivalent fraction of -3/8

[scenic wing]

4/6

[merry haven]

???

[scenic wing]

2/12

[merry haven]

-3/8 = ?/24

[scenic wing]

8

[merry haven]

?

-3/8 is equivalent to ?/24

How much times you need to multiple 8 to 24?

[scenic wing]

3

[merry haven]

So how many time we need to multiple -3 to get the equivalent fractions?

[scenic wing]

8

[merry haven]

Is -3/8 the same of -24/24?

[scenic wing]

omg this is going no where

[merry haven]

They're not equivalent.

[scenic wing]

i learn stuff by it being clear, quick and concise this endless rambling

[raw shard]

what is confusing you

i can try to help

[merry haven]

https://www.cuemath.com/numbers/equivalent-fractions/

[scenic wing]

Where did they get 24 from and how do i multiply a fraction by a whole number

i guess no one here understands/can explain this simple stuff rip

The only fking example they gave with more than one fraction both the fractions have the same denominator so its easy

yet on the test there is a question with 3 fractions with 3 different denominators with no explanation on how to solve for that im fking done

How does 3 and 6 reduce to 2 and 1 ??????????????????

this makes no sense

where did one come from ?

no where

Wow so instead of going on a giant drawn out charade about nothing making me answer pointless questions @merry haven could have just said

"They got 24 by finding the lowest common denominator this is how you find the lowest common denominator"

"what is the lowest number all 3 of the denominators can fit into ?"

some people just have zero clue how to teach i guess it is what it is

like I know you spent the time trying to teach me and i appreciate it but this can get seriously frustrating

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[gilded vale]

I understand what quadrant this is in, and the side lengths are cos=0.9063 and sin=-0.4226

[gilded vale]

But how do you find that angle?

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[gilded vale]

<@&286206848099549185>

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@gilded vale Has your question been resolved?

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@gilded vale Has your question been resolved?

[lone heart]

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[wind hare]

I wanted to find the ratio of x to y that gave the steepest raise in value for this function: y * x^2, so I differentiated the function to x^2 +2yx, and then solved for x like a quadratic function

[wind hare]

The solutions I got was 0 and -2y

x = -2y does seem like the right answer for the steepest climb, however I get the same climb for x = 2y

how do i mathematically arrive at the answer x = 2y?

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@wind hare Has your question been resolved?

[urban pine]

how did you get 0 and -2y

from 0 = x^2 + 2yx?

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[fallen frost]

How does the information about the triangle help here? I'm not sure how to calculate the area of the triangle with α, β, and γ.

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[spark latch]

Find the zeros of (-2sin(x)+1)/(sin(x)+2)^2 for intervals pi <= x <= 2pi

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[spark latch]

.

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[coral geyser]

how do i solve this?

[ivory zodiac]

from what i understand in the question, x=0 y=10 for the minimum

and max x=20 y=10

wait no nvm

[urban pine]

first set up an equation for what you want to optimize

[coral geyser]

how can I do that? im really bad at this lol

[urban pine]

so you want the value y right?

[coral geyser]

yes

[urban pine]

how can we express y with what's given

[coral geyser]

oh

y = 10^2+ (20-x)^2

y^2*

[urban pine]

great!

we can differentiate that now (to look for mins and maxes)

are you familiar with implicit differentiation?

that's prob the easier way to diff that

[coral geyser]

implicit?

what do you mean

[urban pine]

like

[ocean seal]

citrusmunch

$\frac{d}{dx} ( x + y^2) = 1 + 2y \cdot \frac{dy}{dx}$

[coral geyser]

but here y = sqrt(100+(20+x)^2)

[urban pine]

ahh that's what i mean

we can avoid the nasty root and just keep y^2

doesn't really matter tho

[coral geyser]

so this is how you derivate y^2 right

[urban pine]

yes

[coral geyser]

i got -4xsqrt(100+(20-x)^2)

+1

which is probably wrong lol

[urban pine]

hmm

[coral geyser]

anyway i used an online calculator and the solution is x = 20

[urban pine]

well implicitly it's easier; ill show you

\begin{align*}
y^2 &= 100 + (20 - x)^2 \
2y \cdot \frac{dy}{dx} &= -2(20 - x)
\end{align*}

[ocean seal]

citrusmunch

[urban pine]

right, so since dy/dx is zero, the whole lhs is gone 😄

[coral geyser]

looks easy

[urban pine]

well anyway we solve for x

[coral geyser]

i got x = 20

[urban pine]

great

now don't forget that the bounds should be included as critical points for optimization problems

[coral geyser]

yes

thank you for your help

[urban pine]

glad to help 😄

[coral geyser]

.close

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[static copper]

I found the determinant to be 0 for that first question because I was able to come up with a transformation matrix that applied the cross product,

when I do it for x cross v I also get a 0 determinant but it tells me it's wrong, what does x being orthogonal to v have to do with the determinant of the transformation?

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@static copper Has your question been resolved?

[static copper]

.close

[lone heart]

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[coral geyser]

can someone explain this to me

[arctic wren]

this is the expect value f a function

i know it is defined like that

but can't argue/demonstrate that

wikipedia can help you here https://en.wikipedia.org/wiki/Expected_value

[coral geyser]

which one is the correct answer?

[urban pine]

gonna have to disagree

just a definition of riemann sum problem
https://en.wikipedia.org/wiki/Riemann_sum

[buoyant kayak]

juust summation notation for a definite integral

[urban pine]

see the "Connection with Integration" section

[arctic wren]

it isn't expect value ?

[urban pine]

essentially you match the integrand and summand

[coral geyser]

isnt it the answer i selected

[urban pine]

the expected value seems to use that specific function as the integrand but it's unrelated here

[arctic wren]

ty

always learning 🙏

[urban pine]

same 😅

[urban pine]

pretty sure it's B

[coral geyser]

thanks

.close

[lone heart]

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[rich basin]

what is that ||| mean?

all sides are parallel?

[bleak ridge]

Typically parallel would be ||

Idk what else it would mean

[urban pine]

i've never seen that lol

is there a symbol glossary in the book?

[rich basin]

no

i looked through the previous sections of the bool

[buoyant kayak]

i'd assume it's a typo

[rich basin]

and is like this

so it means it similar right?

[buoyant kayak]

neeeevermind

[rich basin]

what is kind of misleading is <P = <R

[bleak ridge]

Oh yeah

[rich basin]

what angle of P is it referring to?

[bleak ridge]

Its just what they defined right after

In the parentheses

[rich basin]

so ||| means similar triangle right?

[bleak ridge]

Yeah

Or maybe some special case of it

[urban pine]

oh there yeah

[rich basin]

so weird, i don't think this even is a illegimate syhmbol

[urban pine]

any symbol is legitimate if you define it well enough 🙂

[bleak ridge]

I think ~ is usually used to notate similarity

[rich basin]

they sdidn't even define it properly

and when i was readi ng the question i didn't know what <A could be refering to

it could be eaything

[bleak ridge]

I mean thatd just be the angle at point A prob

[rich basin]

the angle of inclination for A?

or is it the triangle

[bleak ridge]

The triangle

[urban pine]

which book is this?

out of curiosity

[rich basin]

@urban pine this is the advanced mathematics boo k

[urban pine]

sorry that's real vague, is there an author?

[rich basin]

terry lee

[urban pine]

extension hsc stuff?

[rich basin]

yeah

[urban pine]

sweet thanks

[rich basin]

what happened?

[lone heart]

@rich basin Has your question been resolved?

[lone heart]

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[maiden path]

I don't really know how to start this question

[lone heart]

@maiden path Has your question been resolved?

[devout summit]

Consider vectors x,y as (1,1,2) and (1,1,-3)

[maiden path]

are thoe supposed to be the answer?

but how would i get to them?

[devout summit]

You require an example of two vectors which have the same projection onto vector (1,1,0)

[maiden path]

yeah

but how would I find both vectors? Like solving for them algebraically

Or am I looking at this the wrong way

The thing is ive no idea how projection works

[devout summit]

Algebra isn't necessary. We just want an example of what works

[maiden path]

you showed a picture earlier

is that how you get the example?

[devout summit]

That was just to visualise

[maiden path]

Could you explain to me how projection works? All I know is an equation for it

[devout summit]

Here, a1 is a projection of a on b. You can think of it as component of a along b

[maiden path]

okay

[devout summit]

Yes. An equation can be derived by this diagram

[maiden path]

so a is (1,1,0)

and were looking for a1 and a2?

[bleak ridge]

Dot product 😳

Epic

[devout summit]

We are looking for a1 and a2 which have the same dot product with a

[maiden path]

what's dot product?

[devout summit]

$\vec{A}\cdot\vec{B}=|\vec{A}||\vec{B}|\cos(\theta)$

[ocean seal]

Euclid31415

[devout summit]

Where theta is the angle between vectors A and B

[maiden path]

okay

[bleak ridge]

Theres also a cooler definition

[devout summit]

Which one?

[bleak ridge]

The sum one

[maiden path]

Im kinda lost

[devout summit]

Notice that the determining factor in vector a1 is the length that it has along b. That length can be written as acos(theta)

[maiden path]

so a1 is acos(theta)?

[devout summit]

You said here you know an equation for it. What is it?

[devout summit]

a1 is a vector. Length of a1 is acos(theta)

[maiden path]

I dont think its relevant to the question

[devout summit]

This is full of dot products though

[maiden path]

yeah

[maiden path]

do you need to know the cosine law for this?

[devout summit]

It is by definition of cosine

[maiden path]

so you also use the cosine law to find a2?

[devout summit]

Length of a2 is a*sin(theta)

[maiden path]

ohhh

okay so you just use sohcahtoa?

and use the angle between a and a1 as the reference?

what then after finding the lengths of a1 and a2?

[devout summit]

Yes

[maiden path]

what do I need to do after finding both the lengths of a1 and a2?

and what would a be?

is it the vector?

you mentioned that the length and the vector are different

[devout summit]

Yes

Do you know of the notion of dot product being written as the sum of product of respective components?

[maiden path]

Tthis?

[devout summit]

Yes. But in 3D here with an extra z component

[maiden path]

yeah

so you just distribute costheta and sinetheta to a?

which would get you a1 and a2?

[devout summit]

Look, there are infinitely many pairs of vectors x,y which satisfy this property.

[maiden path]

I thought that when I read the question and it only asked for one example

[ocean seal]

Euclid31415

[maiden path]

when someone asks for the projection of vector x on vector a, is it an equation or like a diagram youre supposed to draw?

[devout summit]

It is a vector quantity. Specifically, the vector along a such that it's dot product with a is the same as dot product of x with a

[devout summit]

This means $(\vec{x}-\vec{y})\cdot \vec{a}=0$

[ocean seal]

Euclid31415

[devout summit]

It is the same as saying vector x-y is perpendicular to vector a

[maiden path]

Is this relevant to the question?

[devout summit]

Well, there are some analogies but i think it is a bit far-fetched

[maiden path]

yeah that one is dealing with points

[devout summit]

If we choose point P as a point lying along vector a, and the normal vector along the direction of a, every position vector of point on the plane has the same dot product with a.

[devout summit]

Can you find a vector perpendicular to (1,1,0)?

[maiden path]

its just a vector thats when multipied to (1,1,0) equals to 0 right?

but yeah I dont really know how I would find that vector

[devout summit]

Yes it's dot product with (1,1,0) is 0

[devout summit]

Again, there are infinitely many of them. One of them would be (0,0,1)

Since 0×1+0×1+1×0=0

[maiden path]

For the main question that I had, do I just need both vectors that equal to zero?

their dot product would be 0

[devout summit]

We need two "distinct" vectors x,y

[maiden path]

yeah, so different vectors with a dot product of 0?

[devout summit]

That works

[maiden path]

But I still don't know how I would actually find those vectors

[devout summit]

We found this as an equivalent condition to our question.

[devout summit]

Can you find any two vectors x,y such that x-y=(0,0,1)?

[maiden path]

(1+1,1+1,1+0)

so (1,1,1) and (1,1,0)

[devout summit]

These two vectors satisfy the conditions given in the question!

[maiden path]

So its that simple

i think its just that I have no idea what projax=projay means

[devout summit]

Concept is more important than answers

[maiden path]

Well the problem is that its an assignment that I need to hand in and as much as I want to understand the concept, I only needed the answer.

well you did give me an answer tho

but I cant just slap them onto my paper and show no work at all

also I missed a lot of the previous lessons

and me skipping to this without knowing the previous stuff made it so much worse

I didnt even know what dot product was

[devout summit]

Try reading more sources on projection, dot product

[maiden path]

yeah

well thanks for the help

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[wraith flax]

does anyone know how to get the answer to this question. the formula is rise over run...

[fallen tartan]

I think you should just choose whatever two points you like and then solve the system of two equations

[wraith flax]

so would it be 3/4

i think that may be the answer

[fallen tartan]

let me check

I think it's -3/4

[wraith flax]

why?

[fallen tartan]

hold on I'm trying to recheck

[wraith flax]

okay thank you 🙂

[fallen tartan]

so

let's say you take the first two points: 5,4 and 1,7

the linear function formula is y=ax+b

you plug in the values for y and x

4=5a+b

7=a+b

multiply the first one by -1 and add it to the second one

then you have 3=-4a

so a=-3/4

[wraith flax]

okay. thank you soooooooo much

.close

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[onyx hatch]

just a quick simple question: is this a viable step:
$$\sum_{j=0}^{\infty} a_k = \sum_{j=0}^n a_k + \sum_{j=n+1}^{\infty} a_k$$

[ocean seal]

lewis

[onyx hatch]

and if so, what is the explanation

[glass lichen]

yes

a1+a2+a3+a4
(a1+a2)+(a3+a4)

that's what happened

[onyx hatch]

so that implies that infinite series do not create a problem here?

[glass lichen]

nope

[onyx hatch]

okay, thank you

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[frosty sparrow]

assume A,B,C are 2d vectors. can B + (|B-A| + |B-C|) be simplified further

[glass lichen]

no

cause that's complete nonsense

[frosty sparrow]

u can't combine the |B-A| + |B-C| into anything?

i thought maybe you would be able to because they are both of the form |B-x|

but idk

[glass lichen]

you can, it's a scalar

so B+scalar is nonsense

[frosty sparrow]

i meant for |x| to mean normalized x

not length of x

should have wrote ||x||

[glass lichen]

that's also length/norm

[frosty sparrow]

k let me rewrite it

[glass lichen]

but that im aware, nothing meaningful can be done to simplify that

[frosty sparrow]

$$B+((B-A)/|B-A| + (B-C)/|B-C|)$$

[ocean seal]

cxii

[frosty sparrow]

shit thats ugly

but yeah thats what i meant

[placid zinc]

Is ugly, doesn't simplify very well

Mind you, can write it as
aA + bB + cC
For some scalar a,b,c

[frosty sparrow]

computing the scalars would take more operations tho

and i'm trying to cut down on those

lets say B-A = x and B-C = y

then the part in brackets is

x/|x| + y/|y|

which expands to

x/sqrt(x.x^2+x.y^2) + y/sqrt(y.x^2+y.y^2)

and then uh

[placid zinc]

Yes, expressing this as normalized vectors is as easy to express as possible, imo

[frosty sparrow]

actually i dont think theres any way to reduce this to only having 1 square root

ugh

that sucks

sqrt is expensive on small computers

[placid zinc]

Okay so you're looking to make this computationally efficient

[frosty sparrow]

yes

ill write it as pseudocode

function foo(a,b,c){
  return b+(norm(b-a)+norm(b-c))*t;
}

t is just a constant

doesnt depend on a,b,c

[placid zinc]

How much do you care about accuracy?

[frosty sparrow]

well

[devout summit]

What is the program about?

[frosty sparrow]

i want as much as sqrt or rsqrt can give me

ill show. u

here are three points ABC

they form an angle

if you give that angle a thickness and wish to rasterize the resultant shape using triangles

without overlaps

you need to find the points X Y

which are the intersections of the lines (start at i, direction AB) and (start at O, direction CB) for X

and similar process for Y

i was using two parametric ray equations to begin with

but then realized i could just do the B + (|B-A| + |B-C|) thing

and arrive at X,Y like that instead

the good thing about this is that |B-A| and |B-C|are actually already computed in order to find the points I,J,P,O

but if it can be made even simpler than it would be good

prob not but i wanted to ask since this kind of operation will be done for thousands of angles 60 times per second

[devout summit]

So for example, JY and IX are the same distance from AB?

[frosty sparrow]

ya

or u might say I and J are same distance from A

and Y and X are same distance from B

P and O same distance from C

call the distance t

ykw ill make this nice and math rendered

[devout summit]

But, distance of I,J from A isn't equal to distance of X,Y from B right?

[frosty sparrow]

yeah

those aren't equal

don't ask how i figured these out

i was just trying random stuff and it somehow worked

(doesn't work when ABC are in a straight line btw)

[devout summit]

Yes that should work otherwise because XY is along the angle bisector of ABC

[frosty sparrow]

so like u can't see any way to simplify this right

ykw. i think it's good it shouldn't be a big problem having like two square roots

ok thanks guys

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[alpine sable]

dont understand how to move on from where im stuck

very similar problem but i dont know how to get the Sn equation from what i have

[lone heart]

@alpine sable Has your question been resolved?

[lone heart]

@alpine sable Has your question been resolved?

[visual iris]

,,S_n=-\frac{1}{2} - \frac{n}{12}

[ocean seal]

@visual iris

[visual iris]

Does this work?

$\begin{aligned}&S_3 = -\frac{1}{2}-\frac{1}{6}+(\frac{1}{4}-\frac{1}{3})\&=-\frac{1}{2}-\frac{1}{6}-\frac{1}{12}\&=-\frac{1}{2}-\frac{3}{12}\end{aligned}$

[ocean seal]

@visual iris

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@alpine sable Has your question been resolved?

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[alpine sable]

are these valid steps?

im unsure that i can just divide by e^2t

[buoyant kayak]

no can do

[alpine sable]

y not

[buoyant kayak]

you'd be dividing the entire integral by e^(2t)

you can't divide into the integral

[alpine sable]

damn that sucks

ty ill keep that in mind 🙂

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@alpine sable Has your question been resolved?

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[harsh ocean]

I'd like to know how to integrate sin^2 (x) via DI method or tabular integration

[urban pine]

doesn't seem to lend itself to that method...

just regular chain rule there

[harsh ocean]

:/

Ok thx then

!close

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[viral spruce]

What will be the area of rhombus

[viral spruce]

<@&286206848099549185> please help

[buoyant kayak]

don't ping helpers right away and that is a very simple question you can ask google

[viral spruce]

Dude

But I want a step by step explaination

[buoyant kayak]

of the area of a rhombus?

[viral spruce]

Yes

[buoyant kayak]

1.) multiply number by number
2.) divide by two
3.) be happy ;)

[viral spruce]

Which number

.close

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[viral spruce]

.close

[lone heart]

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[rugged turtle]

i need an equation to transform 6000 into the number 11, 7000 into the number 12
therefore
11: 6000
12: 7000
13: 8000
14: 9000
so when i plug in 6000 i should get 11

[stiff forge]

do you need a consistent equation for n < 11 and n > 14 as well?

so like, 10: 5000, 9: 4000, 8: 3000, etc.

[rugged turtle]

no

it needs to start at 11 and the value has to be 6000

because im using a different equation for the numbers before that

but yeah i need an infinite one for n > 5000

so 15: 10000

etc

[severe sluice]

right
let's use a function f(x)

where f(11)=6000, f(12) = 7000, f(13) = 8000, f(14) = 9000, ...

[rugged turtle]

yes

[severe sluice]

then, f should be a linear function

right?

[rugged turtle]

yes

[severe sluice]

so f(x) is of the form ax+b

[rugged turtle]

yeah

$\mathbb{Kanga Gang Advocate Adavocowana}$

[ocean seal]

Hex GT

[rugged turtle]

lol

had to see what it does

[severe sluice]

f(x) = ax+b, right?

if x increases by 1, f(x) increases by 1000

so what should a be?

[rugged turtle]

uh

1?

[stiff forge]

not quite

you want 'a' to be the value that your function is increasing by. Since you want your y to increase with every 1000 x, your a should be 1000

[rugged turtle]

oh

okay

[stiff forge]

do you know what b would be?

[rugged turtle]

10?

[severe sluice]

no

think of it this way

[rugged turtle]

because i want it to statrt at 11

[severe sluice]

f(x) = ax+b

f(x+1) = a(x+1) + b = ax+b+1000

[rugged turtle]

what

confusion

[severe sluice]

i got ax+b+1000 since when x increases by 1, f(x) increases by 1000

[rugged turtle]

okay

yes

[severe sluice]

and f(x) = ax+b

[rugged turtle]

so why would a be 1000

if b = 1000

im so confused rn

[severe sluice]

let me start from the beginning

[rugged turtle]

okay

[severe sluice]

you want a function f(x) so that f(11) = 6000, f(12) = 7000, and so on

[rugged turtle]

yes

[severe sluice]

and the function is linear, meaning that f(x) = ax+b for a given a and b

[rugged turtle]

a is gradient right?

b is y intercept?

[severe sluice]

yes

now

if x increases by 1

then f(x) increases by 1000

therefore f(x+1) = f(x) + 1000

amirite?

[rugged turtle]

yes

[severe sluice]

now

f(x+1) = a(x+1) + b

f(x) = ax+b

f(x+1) = f(x) + 1000 => a(x+1) + b = ax+b+1000

[rugged turtle]

nope you lost me

[rugged turtle]

i get this part

and every thing before that

[severe sluice]

f(x+1) = f(x) + 1000

[rugged turtle]

yes

[severe sluice]

replace f(x+1) with a(x+1) + b

replace f(x) with ax+b

[rugged turtle]

okay

so

[stiff forge]

alright Hex here is a different approach if this seems confusing

I know you said you needed the function starting at 11, but because it's linear it's gonna have an x intercept at zero

[rugged turtle]

okay

[stiff forge]

that means you can find out what b is if you find out the value of f(x) when x = 0

so like this: 10: 5000

9: 4000

...

[rugged turtle]

-5000

[stiff forge]

yes!

[rugged turtle]

:)

[stiff forge]

that's your b

[rugged turtle]

okay

[stiff forge]

so if you plug into the equation your slope

you'll have:

?

[rugged turtle]

y = mx - 5000

[stiff forge]

correct

and m is?

[rugged turtle]

gradient

[stiff forge]

well yes

but do you remember it's value?

(what are you increasing by?)

[rugged turtle]

1000

[stiff forge]

exactly

[rugged turtle]

so y = 100x -5000

[stiff forge]

that's your equation but if you don't want the values before 11 you need to specify that your bound is from [11, infinity)

[stiff forge]

1000x but I'm guessing this is a typo

[rugged turtle]

yeah it was

oops

i need to do this in code

can i set bounds in code

or is there another way

like how do i set bounds

[stiff forge]

hmm what language?

Also if this is for a class I don't know how exactly they'd want you to do that regardless

[rugged turtle]

its not for a class lol

python

brb

how do i rearrange it so that i plug in the value and i get like 13

so i plug in like 8000

and i get 13

[stiff forge]

do you now how to find an inverse of a function?

[rugged turtle]

i think

1 sec

im stuck at y/x = -4000

im trying to make x the subject

[alpine sable]

y/-4000 = x

[rugged turtle]

oh

lol

i forgot about that

[stiff forge]

wait did you change the equation? where's the -4000 coming from?

[rugged turtle]

i rearranged

idk

probsably did smt wrong

[stiff forge]

the way I remember learning how to find inverse functions in school was to try to isolate the x variable and then switch x and y

so for y = 1000x - 5000 you would start by adding 5000 to y and then dividing both sides by 1000

@rugged turtle does this make sense?

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@rugged turtle Has your question been resolved?

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[ripe zenith]

hey Im a bit confused by the integral test

[ripe zenith]

so as far as I understand it I'm supposed to integrate then from there evaluate at 1 and infinity and if its a number then it doesn't diverge

but some reason when I had the sum of (cosn-2)/n I had to take the derivative?

[vale wigeon]

???

what are you talking about

do you have a worked solution that tells you this?

[ripe zenith]

I did but I accidentally clicked the next problem on khan.

it was the sum of (cos(n)+2)/n when I clicked the explanation it showed taking the derivative.

but it didn't show taking the integral

[vale wigeon]

i am almost sure you misread or misunderstood something tbh

[ripe zenith]

I found a problem kind of similar

[vale wigeon]

To find whether it's always decreasing for x ≥ 2, let's consider its derivative.

[ripe zenith]

okay so this is only if the integral test applies?

[vale wigeon]

??

to apply the integral test you need the function under the integral to be decreasing on some ray

otherwise applying the integral test w/ that function is invalid

i can explain why this decreasingness requirement is necessary if you want

[ripe zenith]

Yes please, I'm not sure why im doing it.

[vale wigeon]

if you drop the requirement that the function under the integral should be decreasing, it's possible to prove that the series 1 + 1 + 1 + 1 + ... is convergent (which it very obviously is not)

[ripe zenith]

is the integral test proof that it is convergent?

or just not divergent?

[vale wigeon]

...let me be more clear

[torpid delta]

Being convergent == not being divergent

Series that loop values are divergent

[vale wigeon]

this is the \textbf{real} integral test:

Let $\sum_{n=0}^{\infty} a_n$ be a series and let $f: [0, +\infty) \to \bR$ satisfying the following:
\begin{enumerate}
\item $f(n) = a_n$ for every $n \in \bN$.
\item $f$ is continuous.
\item $f$ is decreasing on $[x_0, +\infty)$ for some $x_0$.
\end{enumerate}

Then $\sum_{n=0}^{\infty} a_n$ converges if and only if $\int_0^{\infty} f(x) \dd{x}$ does.

[ocean seal]

Kanga Gang Annihilator Ann

[vale wigeon]

and this is the \textbf{phony} integral test:

Let $\sum_{n=0}^{\infty} a_n$ be a series and let $f: [0, +\infty) \to \bR$ satisfying the following:
\begin{enumerate}
\item $f(n) = a_n$ for every $n \in \bN$.
\item $f$ is continuous.
\end{enumerate}

Then $\sum_{n=0}^{\infty} a_n$ converges if and only if $\int_0^{\infty} f(x) \dd{x}$ does.

[ocean seal]

Kanga Gang Annihilator Ann

[vale wigeon]

the phony integral test differs from the real one only in one place: it is missing condition 3

[ripe zenith]

why do they use that one?

[vale wigeon]

who where

[ripe zenith]

the phony one?

[vale wigeon]

you're implying "they" use the phony test.

who are "they"?

[ripe zenith]

I looked again they use the real one

[vale wigeon]

you're not letting me continue my point...

what i was going to say is, it's possible to make an example of a series that is obviously divergent but the phony integral test says it's convergent.

[ripe zenith]

would that be the 1+1+1 series?

[vale wigeon]

yes, and i was going to demonstrate the function that i was going to use with it.

but you didn't let me

[ripe zenith]

Sorry what would the function be?

[vale wigeon]

it's easier to show its graph than to describe it with a formula

basically around every integer there is a triangular spike that goes up to 1 to match the value of the term in our series, and the widths of the triangles (and the total area under the graph, which is proportional to their sum) shrink quickly enough for int[0,+∞] f(x)dx to converge

in this case i picked the width of the triangle around x=n to be 1/2^(n-1)

since geometric series are very easy to analyze - we not only know exactly when they converge and diverge, we even know exactly what a geometric series converges to.

[ripe zenith]

so this series converges?

[vale wigeon]

the phony integral test says the series 1+1+1+1+1+... converges

but you see that's obviously not true, right?

[ripe zenith]

Yeah because it goes between 0 and 1.

But the area would converge?

[vale wigeon]

yes, that's the point

$\int_0^{\infty} f(x) \dd{x}$ converges yet $\sum_{n=0}^{\infty} f(n)$ diverges.

[ocean seal]

Kanga Gang Annihilator Ann

[vale wigeon]

my point is: if you don't check f(x) for decreasingness, you get a test that can lie to you.

are we understood?

[ripe zenith]

so the f(n) is not true?

[vale wigeon]

f(n) is a function

it isnt true and it isnt false its a FUNCTION

[ripe zenith]

or f(n) is seeing if the point converges vs f(x) is seeing if the area converges?

[vale wigeon]

aaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaaa

aaaaaaaa

please

please i am begging you

please

please read and understand what im saying

right now it feels like youre just mashing words together

and nothing else

and you dont understand what the words you're saying mean

[ripe zenith]

I think I am close. Because with the integral it has the actual area of the geometric series so it has a real change. The summation notation it goes up by 1 each time so it doesnt have a change in area because each time it goes up to the point perfectly landing on 1 so each time the area is 1.

[vale wigeon]

you're overthinking it.

you're really overthinking it.

[ripe zenith]

I think I get it you are checking that it decreases because if it decreases then it converges. this image is showing what it is if you check for decreasing vs if you dont

I will practice this to get better

thank you for the help

.close

[lone heart]

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[vale wigeon]

I think I get it you are checking that it decreases because if it decreases then it converges.
no

you've missed my point

[lone heart]

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[tall marsh]

i did question 4 but when i go to question 5 im confused

[supple tundra]

You know what x^3 looks like?

[tall marsh]

yeah

[supple tundra]

and 2x + 0.5?

[tall marsh]

yes

but the thing is i dont know what this means ≪

[supple tundra]

$\leq$

[tall marsh]

do i do from -2 to 2 or -1 to 1

[ocean seal]

azeem321

[supple tundra]

less than or equal to

[tall marsh]

oh so its just the same

[supple tundra]

Yup. It means sketch the graphs for the values of x between -2 and 2 including the end points. (-2,2)

and then to find intersection points equate them

[tall marsh]

okay thank you

you really helped

.close

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[drowsy fable]

$\sqrt{((x+y) + 2\sqrt{xy})} = \sqrt {((\sqrt {x} + \sqrt {y})^{2})}$

[ocean seal]

lol funny child

[drowsy fable]

How'd this happen

[severe sluice]

try expanding $(\sqrt{x}+\sqrt{y})^2$

[ocean seal]

Kanga Gang Advocate Adavocowana

[drowsy fable]

$(\sqrt {x} + \sqrt {y})(\sqrt {x} + \sqrt {y}) = \sqrt {x}(\sqrt {x}) + \sqrt {x}(\sqrt {y}) + \sqrt {y}(\sqrt {x}) + \sqrt {y}(\sqrt {y})$

[ocean seal]

lol funny child

[severe sluice]

that's right

[drowsy fable]

x+y + 2\sqrt{xy}

Ooh

Thanks

.close

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[spring mountain]

Math Circles help pleaseee

[severe sluice]

send your question

@spring mountain

[spring mountain]

I have my answers, I just want someone to like check or add something to it ToT

[severe sluice]

argh
i don't like it when people ask me to check answers

but yeah

send them

[spring mountain]

A.

1. circle A
2. Line segment: AT, AB, AN
3. Line segment BN
4. Angle TAB, angle TAN
5. Angle BNC
6. Arc TN
7. Arc BC
8. Line segment NC
   B.
9. a) 100 degrees b) 80 degrees
10. 80 degrees
11. 40 degrees
12. 100 degrees

[lone heart]

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• Show your work, and if possible, explain where you are stuck.
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[severe sluice]

hey bot this isn't my help channel i'm just continuing keso's question

[severe sluice]

1. yep

2. i feel like AC should be included?

3. yep

4. yep

5. yep

6. maybe specify that it's a minor arc?

same with 7

[spring mountain]

ok okkk I will

[severe sluice]

8. yep

[severe sluice]

i'm unsure though

like
maybe they haven't connected the 2 points for a reason

but AC is technically a radius

b1. yep

2. yep

3. yep

4. seems right

[spring mountain]

SHOOT, THANKS A LOT!!!

that was an actual relief tbh

[severe sluice]

@spring mountain any questions?

because the bot made this my help channel

and i dont like having a help channel

[spring mountain]

none, I guess aaaaa. would it be fine to ask for help later if ever i find something hard again?

[severe sluice]

yeah

fell free to ping/dm me

[spring mountain]

thank youuuu!

[severe sluice]

.close

[lone heart]

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Remember:
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• Show your work, and if possible, explain where you are stuck.
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• Be polite and have a nice day!

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[fallen tartan]

I just need some quick translation help. What's the name of convergence where every point of the function converges to the same value?

[fallen tartan]

Not to be mistaken for uniform convergence

[vale wigeon]

do you mean like... a sequence of functions f_n such that lim f_n(x) is the same at every x?

[fallen tartan]

yes

what is it called?

[vale wigeon]

f_n converges pointwise to a constant

¯_(ツ)_/¯

[fallen tartan]

thank you. I knew it had to do something with points 😄