help-0

[narrow vector]

x-3/-3/4 =2^y-5

[worthy verge]

Ye

[ocean seal]

Chai T. Rex

[narrow vector]

we turned it into -4/3 to cancel out correct

forgot to do that

[oak chasm]

Yes, dividing both sides by something is multiplying both sides by the reciprocal of whatever.

[alpine sable]

i think next step is to take log

[narrow vector]

to get rid of the 2 would make it log2?

[ocean seal]

Chai T. Rex

Chai T. Rex

[narrow vector]

last add 5 to the equation

[oak chasm]

$$x = -\frac{3}{4}\left(2^{y - 5}\right) + 3$$
$$x - 3 = -\frac{3}{4}\left(2^{y - 5}\right)$$
$$-\frac{4}{3} (x - 3) = 2^{y - 5}$$
$$\log_2\left(-\frac{4}{3} (x - 3)\right) = y - 5$$
$$\log_2\left(-\frac{4}{3} (x - 3)\right) + 5 = y$$

[ocean seal]

Chai T. Rex

[oak chasm]

So, that's the inverse.

[narrow vector]

thanks for the help

[ruby current]

why is this the expression for the coefficient of the n^k-1 term?

[oak chasm]

No problem.

[alpine sable]

vieta's formulas

[ruby current]

oh

is there a conceptual reason why we have to sum from j=1 to k-1 for -j?

it seems like we have to choose n from all of the factors except for one to give n^k-1

i just don't see how that translates into the expression there

[alpine sable]

basically to generate an n^(k-1) we need to "pick" an n from all of the factors except for 1

so f(n)=1/k! * n(n-1)(n-2)...(n-(k-2))(n-(k-1))

lets look at some of the ways to get an n^(k-1) term

[ruby current]

ok sure

[alpine sable]

so here we "pick" the constant term in the (n-1) factor and "pick" the n in all the other factors

[ruby current]

ohhh

it's starting to make sense now

[alpine sable]

same thing, but for the (n-2) factor we pick the constant

list out the n^(k-1) terms you get when you pick the constant from (n-1), (n-2), ..., (n-(k-1))

find the pattern

[ruby current]

so in the final product you will have picked every constant term as a coefficient for n^k-1 and that's why you sum them

[alpine sable]

yes

[ruby current]

ok thank you so much for the help even going through and making those diagrams

that was awesome

[alpine sable]

np, happy to help

[upbeat isle]

how do i go about solving arctan 1/3

i've only gotten to theta=arctan1/3

[alpine sable]

tan(theta)=1/sqrt3

you forgot to take the square root

[upbeat isle]

ahh

[alpine sable]

this is a special value of tan(theta)

draw a right triangle if that helps

[foggy onyx]

What is Long Division?

I need a lighter answer than google.

[charred flint]

when you divide the long way

[alpine sable]

in arithmetic?

[foggy onyx]

mhm

and what is synthetic division

[oak chasm]

@foggy onyx See https://www.mathsisfun.com/long_division.html.

[foggy onyx]

Okie thanks @oak chasm

[oak chasm]

@foggy onyx For synthetic: https://www.purplemath.com/modules/synthdiv.htm.

No problem.

[foggy onyx]

If sin θ is equal to opposite over hypotenuse, then __ is equal to hypotenuse over opposite. *

A. tan
B. cot
C. sec
D. csc

*Its a homework not a exam btw

[woeful pulsar]

do you remember what each of them are?

[foggy onyx]

nvm got it

its tan

[ionic jewel]

it is not tan

[ornate raptor]

i thot tan was opposite over adjacent

[woeful pulsar]

^^

[ionic jewel]

it is

tan is not the right answer here

[foggy onyx]

OH ITS CSC

[ionic jewel]

yep

[ornate raptor]

i only know tan, sin and the other one

[foggy onyx]

its csc nvm lol

[woeful pulsar]

so cot is 1/tan

[oak chasm]

cot is the reciprocal of tan

csc and sin

[ionic jewel]

1/sin = csc

[woeful pulsar]

and sec is 1/cos

[foggy onyx]

got another one

[oak chasm]

sec and cos

[ornate raptor]

the heck

[oak chasm]

Notice how the co switches. Cosine and secant, sine and cosecant.

[foggy onyx]

Given an illustration below, _______ is the geometric mean of AD and DC.

[ionic jewel]

is geometric mean the middle?

average length of the two?

[alpine sable]

by similar triangles a/b=b/c

b^2=ac

b=sqrt(ac)

so b is the geometric mean of a and c

[alpine sable]

Given an illustration below, BD is the geometric mean of AD and DC.

[vale wigeon]

given the illustration below...

[foggy onyx]

You are standing 15 feet away from a tree, and you measure the angle of elevation to be 42 degrees. How tall is the Acacia tree?

its 13.52 right?

wait no 13.50

am i right?

or im gonna round off the number

then it will be 13.51?

[vale wigeon]

how far does it say to round?

nearest centimeter?

er

wait the units are feet not meters

[foggy onyx]

finding opposite

[vale wigeon]

no dbrandonx that's not what im asking you

what instructions were you given as far as rounding your answer?

like, just to get it out of the way, your value is correct (ish)

we just need to figure out how it's meant to be rounded

well

i need to know how it's meant to be rounded, 'cause so far you haven't told me.

are you gonna tell me or no

[foggy onyx]

nvm got it

reciprocal of cos θ.

?

reciprocal of cos θ?

is it tan, cot, sec or csc

[vale wigeon]

1/cos(θ)...

[alpine sable]

do you have the reciprocal identities memorized?

[vale wigeon]

sec(θ) = 1/cos(θ) isn't really an identity it's the definition of sec

[slender marten]

A way to remember which it is, is to remember it matches the third letter. For example, cosec(x) = 1/sin(x) =csc(x).

[alpine sable]

yeah that's true i was just taught those definitions as being the "reciprocal identities"

[pale isle]

I'm having a hard time here because I know nothing on how to resolve this exact things

[alpine sable]

so you have a relation, and you need to find the equation of the tangent line to the curve that goes through the point (3.2) on the curve?

looks like a job for implicit differentiation

equation of the tangent line to the curve that goes through the point (3.2) on the curve
can be found if we know

• the slope of the line (dy/dx)
• a point it goes through (3,2)

@pale isle are you good from there?

or do you need help with the process of implicit differentiation and solving for dy/dx

[jagged imp]

Probably looking for the equation of the normal

La normal de la curve

Idk spanish but i assume thats what it is

[alpine sable]

oh yeah you're right the question asks for the equations of the tangent and normal lines to the curve at the point (3,2)

[ionic jewel]

"get the angles of inclination and the equations of the tangent and the normal to the curve... at the point..."

[alpine sable]

ohhh

The angle inclination of a line is the angle formed by the intersection of the line and the x-axis.

so we only need to find the slopes of the lines

[fossil shoal]

hi

Claim: If two operators commute, we can find a mutual basis.

Proof:
Let $[\hat{A}, \hat{B}]=0$ i.e. $\hat{A}\hat{B}=\hat{B}\hat{A}$

Let $\varphi_i$ be the eigenfunctions of the operator $\hat{B}$

[
\hat{B}\varphi_i = b_i \varphi_i \tag{1}
]

We multiply $\hat{A}$ onto (1) from the left and get

[
\hat{A}\hat{B}\varphi_i = \hat{A} b_i \varphi_i \tag{2}
]

since both operators commute we can rewrite (2) to

[
\hat{B}\hat{A}\varphi_i = b_i \hat{A} \varphi_i \tag{2}
]

We see that $(\hat{A}\varphi_i)$ is a eigenfunction of $\hat{B}$ with the same eigenvalue $b_i$.

This can only be the case, if

[
\hat{A}\varphi_i = s_i \varphi_i \tag{3}
]

i.e. if $\varphi_i$ is an eigenfunction of $\hat{A}$.

Assuming $\varphi_i$ is not degenerate (3) is true for all $i$. q.e.d

Question: I can't see how we follow (3). Why does (3) need to hold, how do we know that?

[ocean seal]

balance

[fossil shoal]

I see that (A\varphi_I) is an eigenfunction of B but why does (3) follow from that?

[vale wigeon]

$A\varphi_i$ must lie in the $b_i$-eigenspace of $B$

[ocean seal]

Ann

[vale wigeon]

hats on A and B but w/e

[dusk thorn]

how will I go about dividing 1/20 in my head to arrive at "0.05"?

[vale wigeon]

this channel is busy please move

[dusk thorn]

sorry

[vale wigeon]

okay so i... think you may run into trouble if B has repeated eigenvalues

maybe?

[fossil shoal]

yeah we don't care about degenerate cases

i.e. 1 Eigenvalue per eigenfunction

[vale wigeon]

i mean if you assume all eigenvalues have multiplicity 1 then

yeah itll work

so A*phi_i must be in the eigenspace spanned by phi_i

[fossil shoal]

yeah we only do that here but I can't see why we follow(3)

[vale wigeon]

thus it's a multiple of phi_i

thats it rly

[fossil shoal]

I get that Aphi_i is in the b_i eigenspace of B i.e. is a eigenvector/function but so?

[vale wigeon]

you just said all eigenspaces we're working with are 1-dimensional

[fossil shoal]

how does it follow that it's also a element of the eigenspace of A for that aeigenvalue?

[vale wigeon]

we defined phi_i as an eigenfunction of B with eigenvalue b_i

B's b_i eigenspace is span(phi_i)

Aphi_i ∈ span(phi_i)

[pale isle]

in ended with -4x+y/-x+2y

for the slope dy/dx

[fossil shoal]

@vale wigeon thanks I think I see my issue now

[alpine sable]

correct

so are you good from there?

[fossil shoal]

@vale wigeon thanks, now I also see why we had to assume the multiplicity of 1. nice have a good day

[pale isle]

so...
now that I got the derivate, I should replace with my point (3,2)

[alpine sable]

yes sub in x=3 and y=2

[pale isle]

ok, so what I got was the slope of tangent which is -10

[alpine sable]

correct

now find angle of inclination for a line with a slope of -10

The angle inclination of a line is the angle formed by the intersection of the line and the x-axis.

the slope of the normal line is just -1/(-10)=1/10

[pale isle]

then my tangent equation is 10x+y+28

[alpine sable]

you mean the equation of the tangent line going through (3,2)?

[pale isle]

yes

[alpine sable]

incorrect

equation of a line should be in slope intercept form

y=mx+b

as we found the slope of the tangent line is -10

y=-10x+b and goes through the point (3,2)

2=-10(3)+b

b=32

equation of the line tangent to the curve at p=(3,2) is y=-10x+32

[pale isle]

Ok

So I have already 2 of 3 things

I just need the normal line equation

[alpine sable]

so the normal like goes through the point (3,2) and is perpendicular to the tangent line

find slope then find the line given the slope and the point (3,2)

[pale isle]

I already know the reciprocal of the tangent slope its equal to the normal slope

now the equation

[alpine sable]

the reciprocal of the tangent slope its equal to the normal slope
the negative reciprocal of the tangent slope its equal to the normal slope

slope of tangent = -10
slope of normal = 1/10

find the equation of the line with slope 1/10 that goes through the point (3,2)

[pale isle]

-1/10+y+1 (?

[alpine sable]

no

that's not an equation of a line

think back to alg 1 we'll use slope intercept form

y=mx+b

plug in m=1/10

y=(1/10)x+b

and we can plug in x=3 and y=2 to solve for b

[pale isle]

Im still using this rule of y-y1 = m (x-x1)

so I get a Ax+by+c=0

[alpine sable]

well in this you don't even have an equation

there's no "="

[pale isle]

ah thats right

[alpine sable]

just use slope-intercept form and plug in the slope you found

easiest way

then plug in the point

[pale isle]

alright

[pine lake]

Hello

[pale isle]

b = 32

[alpine sable]

this is the y-intercept of the equation of the tangent line

not the normal line

[pine lake]

Is this channel free

[alpine sable]

nope

[pine lake]

Ok

[pale isle]

I'm stuck here

[alpine sable]

what step are you stuck with?

we have a line y=(1/10)x+b that goes through the point (3,2)

the goal is to find the value of b

the line goes through the point (3,2), so this means when x=3, y=2

so we plug that in

(2)=(1/10)(3)+b

b=17/10

[pale isle]

ok

now that I have both, those are my points for normal line?

[alpine sable]

now you have the equation for the normal line, y=(1/10)x+17/10

is that all you need?

[pale isle]

I need the inclination degree

[alpine sable]

ok let's start with the degree of inclination for the tangent line, which has a slope of -10

its just right triangle trig

[pale isle]

84.28°

[alpine sable]

correct

[pale isle]

So I have everything now

[alpine sable]

did they also ask for the angle of inclination of the normal line?

[pale isle]

hmmm

I'll do it just to be sure

its 5.70° for the normal line

[alpine sable]

5.71

you got the right answer just the rounding was off

[pale isle]

alright

then its done

enough today, I finished 37/40 problems of my homework, what a productive day

I'll finish tomorrow when I wake the last three, I left the hardest until the end so I could concentrate.

Thank you so much!

[alpine sable]

np

[fossil shoal]

In this solution, they heavily used the fact, that for a scalar a we have a=transpose(a) right?

I mean, then everything makes sense but then we also have matrix calculus here which is, in my "experience" a pain.

E.g. if it wasn't for the fact that R hat is a map into R i.e. scalars, then I couldn't see how the above works.

[graceful lance]

(ii) and (iv)

with steps

[alpine sable]

help

[woeful pulsar]

try multiplying out the denominators

then you get a symmetric polynomial

[graceful lance]

??

[woeful pulsar]

a symmetric polynomial is a polynomial which is symmetric in the variables

which means it remains the same after exchanging variables

[celest void]

How do i go about finding the inverse of this matrix?

[woeful pulsar]

diagonal matrix

how do they multiply?

[graceful lance]

with ?

[woeful pulsar]

alpha*beta

[celest void]

just multiply the diagonals?

i thought that's for finding the determinant

[woeful pulsar]

what happens when you multiply two diagonal matrices together?

[graceful lance]

I got the (ii) one

[woeful pulsar]

a similar process can be used for (iv), clear the denominators

[celest void]

I'm not so sure...

[graceful lance]

like this right

[woeful pulsar]

yeah

[graceful lance]

then

[woeful pulsar]

then you have alpha^2+beta^2 which you should be able to deal with

[graceful lance]

alpha + beta = 2

I am getting it wrong after that

[celest void]

is the inverse just the reciprocal of each value or something?

[graceful lance]

ok, I got it

done

[eager spade]

I get most of the proof of cauchy's integral formula but the last part. First im not entirely sure what they mean by "the right-hand side of 8 is bounded so that its integral goes to 0", and additionally im confused as to where the -z when in the denominator after switching to a parameterized integral in the second line of the integral equation on page 47

closer pic of the part im not sure about

so i guess the first part youd say $\left|\frac{f(\zeta)-f(z)}{\zeta - z}\right|\leq\phi(\epsilon)+f'(z)$ with $\phi(\epsilon)\to 0$ as $\epsilon \to 0$ so that $\left|\int_{C_\epsilon}\frac{f(\zeta)-f(z)}{\zeta - z}d\zeta\right| \leq \left|\phi(\epsilon)+f'(z)\right|\cdot 2\pi\epsilon^2$ and the RHS goes to 0 as $\epsilon\to 0$?

[ocean seal]

Little Narwhal

[eager spade]

but the second part i still dont get why the -z disappears from the denom

i wouldve thought it would need to be $-f(z)\int_0^{2\pi}\frac{\epsilon ie^{-it}}{\epsilon e^{-it} - z}dt$

[ocean seal]

Little Narwhal

[eager spade]

<@&286206848099549185>

[graceful lance]

<@&286206848099549185>

[alpine sable]

rewrite everything in terms of the sum and product of A+B

[graceful lance]

aight

[alpine sable]

so for that first one, square it and then take the square root

[graceful lance]

I have reached - [-b/a]×√a/√c

[alpine sable]

for problem 1?

[graceful lance]

yea

[alpine sable]

oh wait this channel is occupied

little narwhal had an unanswered question

we should move

[graceful lance]

ok

[quick surge]

what does this mean

im learning statistics

[noble sinew]

Tabulate the data just means make a table that says 4 ppl like pineapple, 5 ppl like orange...

[quick surge]

wat

oh wait

does it mean

i just write the fruit name and i write the numbers

Pineapple | 4

like this?

[woeful pulsar]

yeah put in a table

[quick surge]

oh okk'

ty

[harsh acorn]

how can i simpify this

all i did are tan^2= sin^2/cos^2
so 2 cos^2+sin^4a+cos^4a

and answers are integers

[gray gorge]

Isn't $2\cos^4(a)\tan^2(a)=2\cos^4(a)\frac{\sin^2(a)}{\cos^2(a)}=2\sin^2(a)\cos^2(a)$?

[ocean seal]

(R / I) / (J / I)ttgenshark

[harsh acorn]

oh i forgot sin^2 :s

[gray gorge]

yeah

Do you think you can take it from there?

[harsh acorn]

yep

[gray gorge]

[eager spade]

it's okay no one had answered for 30m that's generally enough time to post yours afterwards

i can live with an unanswered question lmao

[versed osprey]

Guys I still can’t understand how to convert a whole number into a radical square root

[alpine sable]

5 = sqrt(25)

3=sqrt(9)

[upbeat gorge]

do you know sum and product formulas

[versed osprey]

Nonon not like that I mean square root like this

[upbeat gorge]

$5=\sqrt{25}$

[ocean seal]

CST

[upbeat gorge]

?

is this what you mean

[versed osprey]

OH WAIT OH YES I GOT IT NVMNVM

[ocean seal]

Chai T. Rex

[graceful lance]

in (iii)

what will be alpha + beta ?

<@&286206848099549185>

[gray gorge]

If you know the roots $\alpha$ and $\beta$ of a quadratic polynomial you can always factor it as $(x-\alpha)(x-\beta)$

[ocean seal]

(R / I) / (J / I)ttgenshark

[graceful lance]

(-1/root2 ) + (root2/3)

= ?

[gray gorge]

Do you know how to add fractions?

[graceful lance]

yea

LCM

3 root2

[gray gorge]

Yeah it's kinda like that

[graceful lance]

I;m getting numerators wrong

that's why I'm asking what are they

[gray gorge]

First make the denominators common $-\frac1{\sqrt{2}}+\frac{\sqrt{2}}3=-\frac3{3\sqrt{2}}+\frac{2}{3\sqrt{2}}$

[graceful lance]

I think I got it

Done

[ocean seal]

(R / I) / (J / I)ttgenshark

[gray gorge]

ok

[graceful lance]

In this it will be as = -1/3 root2

which isn't that answer

how do you run these command ??

[gray gorge]

TeXit, which is a $\LaTeX$ bot for Discord

[ocean seal]

(R / I) / (J / I)ttgenshark

[graceful lance]

$\LaTeX$

??

[ocean seal]

Yuv

[graceful lance]

oh

[gray gorge]

If you want to discover its functionalities, feel free to do so in #bots

Try to run a ,help to get a manual dm'd to you

[red fog]

Hi everyone, just wondering, does anyone knows some clue about the history of insertion sort or the origin of its popularity ? Couldn't really find the source of information for those, so I thought of asking you all. Thanks 🙂

[oak chasm]

@red fog Insertion sort is easy to implement and, for very small lists, is very, very fast.

A lot of fast sorts will use some other sort for large lists and then, once they're dealing with a very small sublist, they'll switch to insertion sort because it's faster for very small lists.

[steady wave]

Hey, can someone help me figure out how to get the parabola equation by looking at the graph?

[manic quail]

@steady wave if you posted a picture of the problem, sure.

[steady wave]

second.

i need to find an equation corresponding to this graph.

i know how parabola's are defined.

but i don't know how i can get the focus or directrix from just looking at it.

[manic quail]

So you know how to shift it downwards, right?

[steady wave]

yeah y=x^2+- some value

[manic quail]

Great.

[steady wave]

i know also horizontally but it's not intuitive.

at all

[manic quail]

Why not?

[verbal ermine]

yeah - is right and + is left

[manic quail]

Yes, but that seems logical to me.

[steady wave]

i don't quite grasp how subtracting from x makes it move left

and adding to x makes it move right

sor

sorry other way round

[manic quail]

because you would get the same as plugging in a lower x value

[steady wave]

same what?

okay so plugging lower x value makes it shift right?

[manic quail]

f(x)=x^2, g(x)=(x-1)^2
So, f(1)=g(2)

[steady wave]

wait why? if f(1)=1^2 = 1, if g(1-1)=0^2

[lavish grove]

hi there is someone here good in microeconomy ?

[manic quail]

I said g(2).

[steady wave]

okay i guess they are equivalent.

[manic quail]

Maybe just draw a couple graphs, maybe it makes sense then.

But do you still need help with that problem?

[steady wave]

yeah

[manic quail]

So maybe, make a parabola shifted a to the left first.

[steady wave]

so y=(x+a)^2-1?

[manic quail]

that is shifted down.

But yes, that is correct too xD Just seek a value for "a" now.

[steady wave]

rearrange it so that a is the subject?

[manic quail]

No, just plug in a value for a, you are fine then.

Leaving it like that makes it easier.

[steady wave]

i guess based on the graph it's 1.

[manic quail]

Yep.

[steady wave]

oh wow

that was easier than i thought lmao

[manic quail]

Now, watch the slope.

I am not sure if it is 1, because my eyes are bad xD

Seems like it though.

[forest oriole]

Bro tip, you can use the factored form
$y = a (x - x_1) (x - x_2)& where $x_1$ and $x_2$ are the roots of the function

[steady wave]

yeah, but ideally i want to be able to do this somehow systematically.

[forest oriole]

Fuck

[manic quail]

We just did?

[ocean seal]

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

[steady wave]

i mean can i derive some sort of formula for this?

[manic quail]

Count the units shifted left, count the units shifted down, watch for the slope.

I think Dieguinho posted another way, yet you still have to find the slope by checking the values.

[forest oriole]

Yes

[steady wave]

i sort of want to figure this out in the way that it is taught in the book

[forest oriole]

You still need to have another point to find out the value of a, which is the first coefficient

[steady wave]

because otherwise i won't retain any of this information.

it seems random at best.

[manic quail]

If it does, do you have some more examples?

[steady wave]

sure.

this makes a lot of sense to me.

cuz i can see how it all fits the definitions

2.5

but then i go online and i see no mention of directrix or focus

and i'm left more confused.

[manic quail]

Focus is more of a physics thingy that every mathematician detests.

[versed osprey]

How do I do 22(I’m asking for both parts) I’m so confused

[forest oriole]

These concepts you see more in analitic (idk what is the name) geometry

[steady wave]

iv'e been told that focus and directrix is a more formal maths thing.

[manic quail]

Yeee, but we don't need that here xD

Only for ellipses, then.

[steady wave]

yeah, but i don't understand it otherwise.

[alpine sable]

i think you should calculate x first

[steady wave]

cuz i can't work with definitions

[alpine sable]

and then put it there

[steady wave]

it's just random equations and plugging

[forest oriole]

Send -2 to the right side, and 1/3x to the left side, and square both sides

[steady wave]

i want to have a solid base so to speak

i tried definining the formula's where focus is not F=(0,p) but something F=(h,k)

and i do get different things, because in the book it seems to be dealing with a special case.

[manic quail]

That just means the focus shifted left and right, which makes the parabolas annoying.

If you want to go with that technique, doesn't the book give examples?

[versed osprey]

No don’t think so, the chapter is expansions

[versed osprey]

I think you might be looking at 21, I meant 22.

[steady wave]

that's all i can find about them.

in the book at least.

[forest oriole]

It's an analog way

[manic quail]

the directix defines at which y value your parabola lies, and the focus defines the x value.

Actually, the focus should define the slope too.

[steady wave]

i learn about slopes etc when looking at lines.

but here it seems completely absent.

[manic quail]

With slope I mean "steepness"

Do you know how to do the tasks in the book? I mean, if you just want to find the equation of a parabola, I would recommend doing it like we just did.

[steady wave]

uhmm idk if i gained any understanding.

okay, so i'm doing an online course on calculus

or introduction to one.

and i consulted a completely different book for the parabola section.

and i see it being defined in 2 different ways.

[manic quail]

Both are completely fine.

The one without foci is way more intuitive though.

[steady wave]

okay but it doesn't exactly help me find the coefficient for x

[manic quail]

you find that by counting xD That isn't that difficult xD

Let me repost.

[steady wave]

this is my answer for this one

[manic quail]

,w simplify (x+1)^2 -1

[ocean seal]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by veryhappyperson#1152

[manic quail]

Wolfram forgot the 1, but yes, you are right.

[steady wave]

yeah cuz the one above can't be right cuz it

it's shifting to the right

i now need to figure out horizontal shifting intuitively.

[manic quail]

I might have some intuitive explaination...

If you imagine f(0) is some height on the graph, and if you have a function like f(x+1), your x has to be 1 less than previously to give you the same height on the graph.

Does that somewhat make sense? xD @steady wave

[steady wave]

i wanna say yes, but honestly no ;D

[manic quail]

xD

[steady wave]

okay i get it now

so if y=f(x) then for the f(x) = g(x) the x for g has to be x-1?

f(x)=x^2, g(x)=(x-1)^2
So, f(1)=g(2)

actually this example makes sense now.

for the above to be true you have to send x+1 to g function right?

and you're sort of working backwards

from the definition and not the argument

[manic quail]

Yes.

f(x)=g(x+1)

[steady wave]

okay cool, thank you veryniceperson 😉

[hasty gale]

I want to find what this converges to

and am stuck on finding a partial sum formula

could I get some hints?

[alpine sable]

Can someone pls explain me how to compute this?

[alpine sable]

have you tried partial fraction decomposition?

[dry ridge]

Hi, I'm working on a shader, and I've got two 2D points p1 and p2 (black dots) and then a midpoint mp (the red dot):

[alpine sable]

Helpp

[dry ridge]

I want to draw another line that's perpendicular to the black line, centered on the red line

Any ideas on how to efficiently/simply get two points along the perpendicular line?

[hasty gale]

@alpine sable yep I have tried

[shell widget]

well first you can find the midpoint easily

[dry ridge]

Yeah, I've got it already

[shell widget]

Ok

[dry ridge]

Just did mix( p1, p2, .5 )

[shell widget]

Find the slope of the black line

[alpine sable]

Umm sorry but am I invisible to y’all? 😭

[dry ridge]

You're visible

[shell widget]

Now take the negative reciprocal of it and you get the slope of the line perpendicular to the black line

[dry ridge]

I see, and then just multiply it with some value and add that to the midpoint to get a second point on the line?

[shell widget]

use the slope formula

whats the slope of the perpendicular line?

[dry ridge]

One moment, lemme make some changes to code

[shell widget]

@alpine sable try #probability-statistics

[harsh acorn]

is this channel occupied

anyway

$tana=2,\frac{3sin^2a+2}{4+5cos^2a}=x$

[ocean seal]

Hyl1s

[harsh acorn]

need to find what x equals to

[shell widget]

you have tan(a) = 2 = 2/1

Make a right angled triangle, find the missing side

From there, you get sin(a) and cos(a)

then just plug in their values

[harsh acorn]

but how can we know other sides are 2 and 1 they can be 4 and 2 too right ? im confused

[shell widget]

Doesnt matter

2 and 1, 4 and 2, 8 and 4

wont matter

[harsh acorn]

oh true

[dry ridge]

Nice, it works!

[shell widget]

what is that? xd

[vivid pagoda]

Hello, i am unsure about how to solve for x.

Does anyone know how?

[shell widget]

yes

Find the angle between the sides 10 and 8 first

then just use cosine rule on the triangle of sides 10, 8, and x

[vivid pagoda]

how do i find the angle?

[shell widget]

use the definitions of sin,cos,tan (any 1 will do)

it's a right angled triangle

[vivid pagoda]

the whole hypotenuse is 10 units

you do realise yes?

[shell widget]

oh

LOL

[harsh acorn]

is x area

[shell widget]

no x i think is the length of the diagonal

[vivid pagoda]

yes

[harsh acorn]

oh

[alpine sable]

check the first few terms of the series i think it might telescope

[manic quail]

That still doesn't change anything, use trigonometry, @vivid pagoda .

[vivid pagoda]

i still dont know how

oh yeah

i think i've figured it out

[manic quail]

You could try calculating both angles and setting up some equation with the laws of sine/cosine

[manic quail]

Nice.

[red fog]

Thanks for the response, but what I'm trying to find is its background or historical clue for the context of it as widely used introductory sorting algorithm. Couldn't really find much info of such meta thing. That's why I try my luck here.

[alpine sable]

so see if you can find the partial sum formula, then take the limit as n -> infinity

Help

[vale wigeon]

looks like a voronoi diagram to me

[hasty gale]

So I tried that

but I couldnt see a pattern

[alpine sable]

first 3 terms and leftover terms are circled

so if we have n terms of the sequence the 1st 3 terms will be

[alpine sable]

so here we went from n=2 to n=7

[hasty gale]

yea

[alpine sable]

so if we go from n=2 to n=w the last 3 terms will be

so as we let w approach infinity these last 3 terms will approach 0

everything in the middle will cancel

[alpine sable]

and the infinite series will approach the value of just these first 3 bits

[hasty gale]

wait before that, just to make sure we are on the same page

[dry ridge]

A fragment shader for visualizing animated Voronoi diagrams :)

[hasty gale]

$\frac{1}{2(n+1)} + \frac{3}{2(n-1)} - \frac{2}{n}$

[ocean seal]

Mako

[hasty gale]

is this what you got for the partial fraction?

[alpine sable]

yes, then i rewrote it as
(3/2)/(n-1)-2/n+(1/2)/(n+1)

[hasty gale]

yep yep

[hasty gale]

how did you get that all of them cancel out?

[alpine sable]

first i wrote the n=2 term, then i wrote the n=3 term below it, then the n=4 term below that, etc

(1/2)-2+(3/2)=0

[hasty gale]

The term starts when n = 3

[alpine sable]

oh you're right, my bad

just do the same thing i did but start from n=3

[polar glade]

Okay, I need to find the domain and limits of this function.

I figured

x-1 != 0
x != 1

So the domain should be something like
D = R - {1}?

However, I have no clue on how to start on the limits.

[alpine sable]

how tf do i use a sigma

[lavish oriole]

would this be iscoclees?

[versed osprey]

Sorry for asking again but how do I do the 16th one.

Ping me if u respond

[gaunt hedge]

can someone help?

[alpine sable]

square the first equation , plug in ab=4, and solve for the quantity 9a^2+16b^2

[versed osprey]

OH YES I FORGOT ABOUT THAT PART

LESS GO

[gaunt hedge]

@alpine sable can u help?

[alpine sable]

are you sure this isn't a test/quiz?

[gaunt hedge]

its classwork

[alpine sable]

so you're in class rn?

can you send a screenshot of the whole page?

and the title of the google form

[lavish oriole]

hes been real quiet

[gaunt hedge]

?

wdym

by that

[lavish oriole]

send a picture of your entire screen

The name of the quiz

[gaunt hedge]

like this

?

[lavish oriole]

I guess

i'm not the one asking lol

[gaunt hedge]

@alpine sable

wat does he mean by that

[lavish oriole]

I mean it doesn't say quiz so i think its fine

[gaunt hedge]

oh okay

[lavish oriole]

he just wanted to make sure it wasen't a quiz

[alpine sable]

the inequality says x is strictly greater than -2, so x=-2 is not in the solution set. we need an open circle above x=-2 to indicate that -2 is not a solution

[pine lake]

Is this channel free

[lavish oriole]

i belive

[pine lake]

Uh

[ocean seal]

The following error occured while calculating:
Error: Syntax error in part "\sum _{n=1}^5n" (char 1)

Malum

[alpine sable]

\sum _{n=1}^{5}

[distant bay]

Hi

Can a domain

Have multiple ranges

[alpine sable]

like a range doesnt belong to a certain domain

[frail viper]

yes

[1,3]u[5,9]

@distant bay

@alpine sable

[distant bay]

Okk but a range can only have one domain

Nvm

I meant to say

[alpine sable]

why are you pinging me

[digital rivet]

Hello

[frail viper]

@alpine sable to correct u

[alpine sable]

I didnt say anything wrong

[distant bay]

Domain can only have one range but that range can also be for another domain like 5.7, 6.7

[alpine sable]

think about (-inf,+inf)

[distant bay]

Correct

[alpine sable]

how many functions map to that?

[digital rivet]

Please help me

[alpine sable]

infinitely many

[distant bay]

Infinite Okk so that range can have an infinite amount of domains correct

[digital rivet]

Above photos are the work given by our professor.
Please help me in that please 🙏

[alpine sable]

mmh I dont like the wording

[distant bay]

Should I say outputs can have an infinite number of inputs

[digital rivet]

Anyone?

Guys please

[alpine sable]

\sum _{n=1}^{5} n

[alpine sable]

it's not your turn, wait

[digital rivet]

Okay

[alpine sable]

how

\sum_{n=1}^{5}n

there was a space

[ocean seal]

The following error occured while calculating:
Error: Syntax error in part "\sum_{n=1}^{5}n" (char 1)

[alpine sable]

it works in wolfram

[gray isle]

calc only does simple stuff

[distant bay]

Hello, can anyone explain the vertical line test to me to check for a function

[gray isle]

try ,w

[alpine sable]

ram is there a good way to answer this

[honest glacier]

How many terms of the series must you add to approximate the sum to within 0.005?

[gray isle]

sets?

[honest glacier]

alternating series

[alpine sable]

not sure if anyone has done Cauchy's theorem but some help here would be appreciated

[native temple]

hello

can I ask a question?

or is there still a lingering one?

[fallow igloo]

Don't ask to ask, just ask @native temple

How can I compute this summation of by hand?

There should be some kind of trick or technique which I'm missing

[native temple]

logs by hand inpossible

anyway

[alpine sable]

this is a telescoping series

[fallow igloo]

Ahh.. true

[alpine sable]

Guys

Can u make x the subject of the formula for this equation

10/(3x+3)+5/(x+3)=1/3

Pls

[kindred anchor]

(10/3)/(x+3) + 5/(x+3) = 1/3

Now can you simplify it further?

[surreal sierra]

Hew everybody, I already learned this but I keep getting confused with it, does anybody know some quick tips about how to graph quadratic functions like y=4x^2 +8x + 6.

[alpine sable]

well

I think so

[kindred anchor]

Okay, so how would you do that?

[alpine sable]

well

I am doing it wait

[kindred anchor]

Sure, take your time

[alpine sable]

well

you can do this (10/3)+3

I mean

5

[kindred anchor]

Yes

You got it

[alpine sable]

In order to do it

as a fraction since the denominator is the same

[kindred anchor]

Yes

[alpine sable]

and then after I think we can x+3 at the other side

times it

[kindred anchor]

Yep

And you can get x from it

[alpine sable]

I found this answer

(10/3-5)3-3=x

[kindred anchor]

Yep

[alpine sable]

But quick question how did u made 10/(3x+3)+5/(x+3)=1/3
to dis (10/3)/(x+3) + 5/(x+3) = 1/3

[kindred anchor]

Make it +

(10/3-5)3-3=x
@alpine sable (10/3 +5)3-3

[alpine sable]

Oh yea forgot dat

true

But how did u formulate the equation to dis (10/3)/(x+3) + 5/(x+3) = 1/3

[kindred anchor]

Okay

Wait that's wrong

[alpine sable]

yea 10/(3x+3)+5/(x+3)=1/3

[kindred anchor]

I don't know what's happening to my brain today

Sorry

[alpine sable]

it's ok don't worry

[kindred anchor]

Let me explain how to do that

[alpine sable]

ye It's pretty hard

[kindred anchor]

10/(3x+3) + 5/(x+3) = (10(x+3) + 5(3x+3))/ ((3x+3)(x+3))

[alpine sable]

how doe

[kindred anchor]

Same concept of making denominator equal for both the fractions

[alpine sable]

ohhh k

[kindred anchor]

For the first fraction, multiply and divide by (x+3) because in denominator of 1st fraction (x+3) is lacking

[alpine sable]

Yea

[kindred anchor]

Now for the second fraction, multiply and divide with (3x+3)

10/(3x+3) + 5/(x+3) = (10(x+3) + 5(3x+3))/ ((3x+3)(x+3))
@kindred anchor Then it will result in this expression

[alpine sable]

OHHHHHHH

now I got it

[kindred anchor]

So now you can just add the numerators

[alpine sable]

yea thx

Which class are u in?

I'm in year 8

[kindred anchor]

I'm a uni student

[alpine sable]

damnnn

[kindred anchor]

Fml

[alpine sable]

Dat's epic

[kindred anchor]

I made the mistake

[alpine sable]

it's ok

Also, from where do u come from

I come from Cyprus

[obsidian pike]

yo i need help

[safe island]

ok

[blazing rose]

im confused here

they gave us A'(t)

and are asking for the equation of a tangent line of A

but they already gave the formula for the tangent line aka derivative

[safe island]

no

[slender marten]

It wants you to use A(t) ≈ A(10) + A'(10)(t-10).

[safe island]

they asking for the tangent at that specific point

[blazing rose]

hmm

[blazing rose]

can you explain

where does this formula come from

[slender marten]

At t = 12, Δt = 2 so the approximate change will be the starting point A(10) + the change estimated based on the tangent approximation A'(t)Δt giving you A(t) ≈ A(10) + A'(10)(t-10) .

Does this help?

[lavish idol]

anyone have the answer to this

[lime valve]

I'm paying to the person what does my homework.

(GEOMETRY)

[teal cosmos]

@lavish idol as a general thought, a line of symmetry has to pass through a vertex or the midpoint of a line segment and nothing else

[lavish idol]

so d?

[teal cosmos]

thats up to you to think of 🙂

[blazing rose]

yep, thanks. what is this formula known as?

[slender marten]

I think they just call it the linear approximation.

[fallow laurel]

logab B = 2/3

I need help ^^

should I use

loga x = y <=> x= a^y ?

[blazing rose]

thanks

[slender marten]

No problem.

[blazing rose]

can u send a picture i dont think i understand this

is it $log (ab)$

[ocean seal]

Josh.

[fallow laurel]

no

@blazing rose log_ab

[blazing rose]

oh

[ocean seal]

Chai T. Rex

[blazing rose]

ab^(2/3) = b

i think

[oak chasm]

$(ab)^{\log_{ab}(b)} = (ab)^{\frac{2}{3}}$

[ocean seal]

Chai T. Rex

[oak chasm]

$b = (ab)^{\frac{2}{3}}$

[ocean seal]

Chai T. Rex

[oak chasm]

Need the parentheses at the end.

[blazing rose]

o sheee i got it right

lol

[oak chasm]

Yeah, except you needed parentheses.

[blazing rose]

oh yeah true

[fallow laurel]

what is that rule?

[oak chasm]

It's the rule that says $b^{\log_b(x)} = x$

[ocean seal]

Chai T. Rex

[blazing rose]

here ya go mate

a bunch of log rules

[fallow laurel]

oh!

[oak chasm]

Rule 7 on that.

[blazing rose]

the one ur lookin for its #7

yeah

[oak chasm]

Rule 6 and rule 7 are using log to undo an exponential and an exponential to undo log.

They're opposites like addition and subtraction or like multiplication and division.

They can undo each other.

[fallow laurel]

But shouldn't be

I don't understand

It's a question*

[ocean seal]

Chai T. Rex

[fallow laurel]

Yeah

I'm confused

[oak chasm]

@clever tapir Sorry, channel busy.

[clever tapir]

ok ssry

[pallid horizon]

Can someone help me in 8

[ocean seal]

Chai T. Rex

[oak chasm]

You have to do the same thing to both sides, right?

[ocean seal]

Chai T. Rex

[oak chasm]

Does that make sense?

[fallow laurel]

Is that this rule 7?

[oak chasm]

Yes, it is.

[fallow laurel]

Damn this is confusing, because I don't understand why K is ab^2/3

[oak chasm]

What's K?

[fallow laurel]

Here in the rule, K

[oak chasm]

Yes, that's the argument to log.

Oh, I see the confusion.

The rules above are simplification rules.

When you take (ab) to the power of the left and right side, you get $$(ab)^{\log_{ab}(b)} = (ab)^{\frac{2}{3}}$$

[ocean seal]

Chai T. Rex

[oak chasm]

Do you see how the same thing was done to both sides?

[fallow laurel]

So, if I ever use that rule. I need to do it both sides?

[oak chasm]

No.

We're not to using that rule yet.

First, we do the same thing to both sides.

Do you see that I've done that?

[fallow laurel]

Yes , I see both things where done to both sides. But I just don't understand how you got there in first place

I hope my question makes sense

[oak chasm]

$$\log_{ab}(b) = \frac{2}{3}$$
$$(ab)^{\log_{ab}(b)} = (ab)^{\frac{2}{3}}$$
$$b = (ab)^{\frac{2}{3}}$$

[ocean seal]

Chai T. Rex

[oak chasm]

See how we start with what you started with?

We do the same thing to both sides on the second line?

And then we use rule 7 on the left to give the third line?

[fallow laurel]

Yep. But doing the same thing to both sides, is that a rule? I'm confused with the same thing to both sides

[oak chasm]

That's just keeping both sides equal.

It's a basic rule of algebra, like when you subtract the same thing from both sides.

[fallow laurel]

I think I'm confused there.

I don't understand that still

[ocean seal]

Chai T. Rex

[fallow laurel]

x/y = 3 ?

Sorry, I'm noob

[oak chasm]

No, you shouldn't be dealing with logarithms if you don't know this.

[fallow laurel]

So, should I go with basic algebra first?

[alpine sable]

Yo which channel is best to discuss logic like temporal logic?

[oak chasm]

Yes, definitely.

[fallow laurel]

Do you recommend me any resources?

[oak chasm]

@fallow laurel There's some good stuff on Khan Academy online or Schaum's Outlines "Elementary Algebra" if you like a paperback book with explanations followed by lots of problems to solve and their answers so you can check whether you got it.

[fallow laurel]

Thanks

[oak chasm]

@alpine sable Probably #proofs-and-logic.

@fallow laurel You're welcome.

[junior yacht]

need help with some explanation here

is (i) or (ii) true, or are they both true, or are they both not true?

at first glance, both statements should not be true as they are the same right?

I can't see them being different

[oak chasm]

They can be. $f(A) \cap f(B)$ first does the function, then the intersection. $f(A \cap B)$ first does the intersection, then the function.

[ocean seal]

Chai T. Rex

[oak chasm]

For example, A = {1}, B = {2}. f(1) = f(2) = 10.

[junior yacht]

so, which would be the subset of the other?

[oak chasm]

Well, look at that example.

What's f(A) and f(B)?

[ocean seal]

Chai T. Rex

Chai T. Rex

[junior yacht]

no idea about this one

[oak chasm]

Well, what's A?

[junior yacht]

{1}

[oak chasm]

What's {f(1)}?

[junior yacht]

idk? i don't know what the function is

[oak chasm]

The linked one.

[junior yacht]

hmm 10?

[oak chasm]

{f(1)} = {10}

So, f(A) = {10}.

What's f(B)?

[junior yacht]

10 as well?

[oak chasm]

f(B) = {10}

[ocean seal]

Chai T. Rex

[junior yacht]

{10}

[oak chasm]

Right.

[ocean seal]

Chai T. Rex

[junior yacht]

since A={1} and B={2}, I can't see what A N B is, is it nothing?

so f(A N B) is nothing

[manic quail]

||$\emptyset$||

[oak chasm]

Which elements are both in A and in B?

That's the intersection.

[real briar]

yep

[junior yacht]

oh does every set have an empty set even if its not explicitly stated?

[oak chasm]

@junior yacht Well, intersection is defined as the set of all elements that are in both sets.

If no elements are in both sets, the intersection set is empty.

Sets don't really have an empty set usually.

[junior yacht]

i see

[oak chasm]

So, we can see that i is wrong.

[junior yacht]

so (ii) will be correct?

[oak chasm]

Well, let's see.

[ocean seal]

Chai T. Rex

Chai T. Rex

[oak chasm]

So, let's think.

(ii) has all elements of the first as elements of the second.

So, are all results of inputs that are in both A and B also results of inputs in A that are also results of inputs in B?

[junior yacht]

my answer is yes

[oak chasm]

OK, why?

[junior yacht]

not exactly sure why, but my guts lean me towards that answer 😂

[oak chasm]

Each input is in A and in B. So their result will be a result of an input of A and a result of an input of B. Because the input is in both A and B.

There's probably a nicer math notation to write this all in, but I'm too tired right now.

But yes, (ii) is true.

(i) is false.

[junior yacht]

alright cool!

[junior yacht]

tell me abt it, its 2am in the morning for me and my brain is half awake

[viral owl]

@surreal meadow BRO AYUDA NECESITO QUE ME EXPLIQUES

[junior osprey]

how to calculate spearman’s coefficient on desmos

like the formula

i can get r1 and r2

but not rs

[alpine sable]

hey

anyone know how to solve for this trig function

[junior osprey]

<@&286206848099549185>

[versed osprey]

how do you convert whole number into radical square root

Like this

[versed osprey]

This is the radical square root for 32

No, but I got it now, thanks for replying

But one more thing

How do I do the 23rd one

This is my working for 22nd one

Ping me if you respond

[muted raft]

where did you get 2 from?

[versed osprey]

Wdym

There is no two

[muted raft]

for your q22

[versed osprey]

Ye?

The “4 multiplied by sqrt(2)”?

[hard thorn]

try to multipy everything by x to get a quadratic formula

[muted raft]

that does not look correct

[hard thorn]

then find x using it

[muted raft]

yes

[versed osprey]

Oh ok

Thanks

Wait what

I haven’t started that yet, don’t think I will for a few years

This is the expansions chapter

With identities

[hard thorn]

$x^2/x+1/x=6x/x$

[ocean seal]

Galaxy

[alpine sable]

@hard thorn Can you help me with a question

[hard thorn]

sure @alpine sable

[versed osprey]

thanks I’ll use this formula, TYSMM

[hard thorn]

calculus?

[alpine sable]

not calc trig

really tired my bad

she gave us some review homework for upcoming exam and i forgot everything

[hard thorn]

we haven’t started that yet but i’ll try to help u

[muted raft]

what is the question? Create an equation in terms of cosine?

[eager fern]

im confused as to how i will no when there will be a y intercept or no y intercept

[alpine sable]

@muted raft yes

f

[pastel bane]

anyone know how to do this stuff?

[alpine sable]

For this graph

[muted raft]

First determine its amplitude.

[alpine sable]

2

[muted raft]

now find the vertical displacement

[alpine sable]

4

[muted raft]

you sure?

Check again.

[alpine sable]

I am not sure.

[muted raft]

how can we determine displacement?

$\frac{max + min}{2}$

[ocean seal]

zshyn

[alpine sable]

6

[muted raft]

what is the period?

[alpine sable]

3pi/4

@muted raft

[muted raft]

yeah ok now what is the phase shift

[alpine sable]

pi/12

@muted raft

[muted raft]

So then finally, putting it all together, we get: $2 \cos \left( 3x - \frac{\pi}{4}\right) +6$

[ocean seal]

zshyn

[muted raft]

which is the same as $2 \cos \left( 3\left(x - \frac{\pi}{12}\right) \right) +6$

[ocean seal]

zshyn

[alpine sable]

thanks!

[muted raft]

Yw

[surreal sierra]

Hey everyone! I am learning quadratic functions right now, and am having a bit of trouble graphing them, does anybody have any tips to quickly graph a quadratic equation? Graphing from any form is fine.

[pale basin]

i think you left out a part of the question
i assume that these triangles are proportional

if so, your answer would be 90-74

[feral crag]

Can you help me next?