help-13

[buoyant latch]

What do you get when you expand it

[upbeat mirage]

a^2+ab-ab-b^2 ?

[buoyant latch]

Which simplifies to a² - b²

Right?

[upbeat mirage]

yes u cancel the middle

[buoyant latch]

Right and if either a or b or both are a square root

They disappear!

Because you squared both of them

Do you see that?

[upbeat mirage]

do u mean for the middle part of it?

[buoyant latch]

No I mean if a = root 2

Instead of root 2 - b

You now have 2 - b²

[upbeat mirage]

how come the b doesn’t turn into 2

[buoyant latch]

If $a=\sqrt{2}\
a-b = \sqrt{2}-b$

[wraith dagger]

Frosst

[buoyant latch]

Do you follow this

[upbeat mirage]

yes i see

u put in root 2 for a

[buoyant latch]

Ok now when we multiply both sides by a+b

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

[wraith dagger]

Frosst

[buoyant latch]

Yeah?

[upbeat mirage]

yess

[buoyant latch]

Ok let’s sub a in

[upbeat mirage]

that makes sense

[buoyant latch]

${\sqrt{2}}^2-b^2$

[wraith dagger]

Frosst

[buoyant latch]

Yeah?

[upbeat mirage]

yes a = the root 2 thing

[buoyant latch]

And that simplifies to $2-b^2$

[wraith dagger]

Frosst

[buoyant latch]

Yeah?

[upbeat mirage]

how’d u get rid of the square root sign

[buoyant latch]

Well if you square root something

And then you square it

You get the original number back

[upbeat mirage]

ohhhh

i rmbr that

that makes sense

cus it’s like cancel out

[buoyant latch]

Right

So we went from having a square root

To not having a square root

[upbeat mirage]

yes

[buoyant latch]

And what we did is multiply by the “conjugate”

[upbeat mirage]

i’ve heard that word before

[buoyant latch]

When we have (a+b), the conjugate is (a-b)

When we have (a-b), the conjugate is (a+b)

[upbeat mirage]

so i go like this

[buoyant latch]

and u understand that makes the BOTTOM have no square roots

while the top makes it so you're really just multiplying by 1

so you're not changing the actual value

[upbeat mirage]

yes

cus anything over itself is 1?

[buoyant latch]

absolutely

except 0 but that's a special case

(cant divide by 0)

point is the bottom goes from being $\sqrt{14}-\sqrt{13}$ to $14-13=1$

[wraith dagger]

Frosst

[buoyant latch]

and boom the bottom has no root signs anymore!

[upbeat mirage]

yes so it’s not as hard as it looks

[buoyant latch]

that's what we call "rationalise the denominator"

rationalise as in remove the square roots

[upbeat mirage]

math is cool when is makes sense

[buoyant latch]

:D just gotta go through it step by step and really understand why what you do works

[upbeat mirage]

is it okay if i have 2 more questions

[buoyant latch]

of course

i was very stunned by this ingenuity when i learned about rationalising the denominator too :p

[signal oracle]

Can somebody help me please

[buoyant latch]

#❓how-to-get-help

[upbeat mirage]

i know how to do the graph part but i don’t get the shading part

[buoyant latch]

im guessing that's not a piecewise function

so we need the region we shade to satisfy both equations

if we shade 1 equation, then shade the other, wherever is shaded twice satisfies both equations

[upbeat mirage]

i don’t get how they figured out what to shade in

[buoyant latch]

let's look at the first equation

y < -3x + 2

[upbeat mirage]

that one ik is dotted line or something like that

i think shaded below

[buoyant latch]

this is the line

now

we want where y is LESS than this line

everything that's shaded here is less than the line

[upbeat mirage]

that makes sense now

[buoyant latch]

as in the y value at those points, is less than the line y = -3x + 2

so you do it for the other equation as well

and where u shade twice is the region you want

[upbeat mirage]

the other one wants where y is greater than?

[buoyant latch]

yeah so let's see the line y = 4x-1

looks like this

we want when y is either equal to this line, or greater

so everything above the line including the line

[upbeat mirage]

that makes sense but i would think if it wants greater than it would go to the positive numbers how come it doesn’t?

[buoyant latch]

ok so what it means is this

when i have the line y = x

this is simple

ok now if we think about the region of y > x

that means the points (x, y) satisfy y > x

so (3, 4) would count

(3, 3.1) would count

(3, 3) would NOT count because y is not greater than x

(-2, -1) would count because -1 is greater than -2

[upbeat mirage]

that makes sense

[buoyant latch]

see how all the points are on the top

[upbeat mirage]

yes

they’re not below the line

[buoyant latch]

if we look at say the point (1, 4)

we can go to y = x, then go up 3

if we look at say the point (-6, 3)

we can go to y = x, then go up 9

if we go on the line and move upwards any amount, that'll be where y is greater than x

[upbeat mirage]

ohh

[buoyant latch]

now if we move downwards

x will be bigger

so we shade in everything on top

[upbeat mirage]

thank you

i have one last question

this one i tried to look in the notes but i don’t understand the (1/b)^m thing

[buoyant latch]

a simple way to do this is just checking points

when x = 0, x^3 = 0, y = 0

so (0, 0) is on the line

of the 3 only 1 of them pass through (0, 0)

another way would be to check big/small numbers

[upbeat mirage]

if it is a fraction does it go top left?

[buoyant latch]

a very big positive number ^3 = a very bigger positive number (as you move right it should go up)

a very big negative number ^3 = a very bigger negative number (as you move left it should go down)

[buoyant latch]

what (1/b)^m thing

[upbeat mirage]

the one in the green square

[buoyant latch]

that's so convoluted lol

[upbeat mirage]

yea it’s called saxon math i hate it

[buoyant latch]

if u have a function (a/b)^x

if a/b is = 1

you have 1^x so it's always 1

if a/b is > 1

then taking it to the power of x means

x is positive = number goes big

x is 0 = (a/b)^0 = 1

x is negative = number gets xth-rooted

so it gets smaller and smaller

(but still positive)

thus an asymptote at y=0

[upbeat mirage]

is range y>0?

i have to go to sleep cus it’s am & i have school & i’ve never stayed up this late but i’m super worried for my quiz & i’ve been doing review since 10pm

[buoyant latch]

https://www.desmos.com/calculator/tdsloh783t

[upbeat mirage]

thank you so much for your help today

now i actually understand & i will get a good grade

[buoyant latch]

alright good luck <3

[cedar kiln]

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[cedar kiln]

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[naive ginkgo]

Let there be a number $n=42^m$ with $m \geq 1$ \
I have to find a function t(m) which calculates the amount of natural divisors (aka, numbers that $42^m$ can be divided by resulting in a natural number)

[wraith dagger]

Zabbx

[naive ginkgo]

so I find all the primes of $42^m$; $P(42^m)=2^m+3^m+7^m$\
for m=1, the amount of divisors would be 6m+2, so ${1, 2, 3, 7, 2\times 3, 2\times 7, 3 \times 7, 42}$

[wraith dagger]

Zabbx

[naive ginkgo]

Am I correct to assume that the amount of divisors depending on m is $(3m)!+2$ ?

[wraith dagger]

Zabbx

[cedar kiln]

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[naive ginkgo]

i was wrong, but solved

[cedar kiln]

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[formal walrus]

anyone mind helping me check my endpoints for my interval of convergence on this

my interval is (-1/2,1/2)

[spare peak]

How to solve this ? I have no clue where to start

[formal walrus]

oop

[spare peak]

Loll

Faye floor is urs

The *

[formal walrus]

you could take everything to the 9th power for all terms in there

[spare peak]

No a quicker method

It’s mcq

[formal walrus]

like [1^9 + sqrt(-3)^9 ]/2^9

oh yeah idk shit about that lmfao

i'd just do it w that way

[south tundra]

Rewrite in polar form?

[formal walrus]

in response to whom

[crimson sedge]

Hmm

Are u sure about that

[formal walrus]

certain

it's the answer given in the book too

[crimson sedge]

For part a?

[spare peak]

I saw some video where they were using moivres theorem or smthing but I have no clue abt tht theorem

[formal walrus]

oh shit dude I forgot to even say the number I was on

I'm on 3 lmfao

[crimson sedge]

Oh okay

Explains more

Okay then yeah radius is 1/2

[formal walrus]

ye r=1/2 and interval is -1/2 to 1/2

but i still dk how to test if an endpoint is included in that interval or not

currently I have [2^n*(-1/2)^n]/n

which I haven't the faintest idea what to do with

[crimson sedge]

Sub in those values I suppose

so like

[formal walrus]

that's what I got after subbing in

now I'm supposed to check for convergence on that x value

ya reckon root test would work for this shit

oh dude wait nevermind I'm a fucking genius

nevermind no I'm not

1/2 can be written as 2^-1 but idk wtf to do with the negative in front

[crimson sedge]

$\sum_{n=1}^{\infty} \frac{(2)^n \cdot (1)^n}{n \cdot 2^n}$

Use alternating series test

Oh wait forgot this was a different question

Well uh

[formal walrus]

ye already got the first one

[wraith dagger]

♡LexQa♡

[crimson sedge]

The two 2's cancel

[formal walrus]

which one is that for

[crimson sedge]

Third

The one u r doing

[formal walrus]

x should be -1/2

so -1/2^n

oh shit

[crimson sedge]

I'm doing 1/2

First

[formal walrus]

ye but isn't it negative too

good idea tho

[crimson sedge]

Why

You have 1/2 and -1/2

[formal walrus]

idk lol I just started with -1/2 first

ig it doesn't matter

k 1/2 diverges

[crimson sedge]

For -1/2 it will be the same with -1 being negative, use alternating series test

[crimson sedge]

Yes

[formal walrus]

sounds good

ty for the help :D

[crimson sedge]

Ofc that should converge

I believe

[formal walrus]

do I have to use the alternating series test actually

cuz no matter what if it's in the absolute value it's gonna be positive

nvm lmao

stupid quesiton

alright that's all

ty :)

[crimson sedge]

Ofc

[formal walrus]

.close

[cedar kiln]

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[cedar kiln]

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[lucid drift]

how do i do q2 part four

[lucid drift]

,rotate

[lucid drift]

<@&286206848099549185>

[cedar kiln]

@lucid drift Has your question been resolved?

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@lucid drift Has your question been resolved?

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[ember plaza]

Is this a valid proof by contradiction? (I wanna proof the green statement)

[balmy apex]

this isn't a valid proof by contradiction

because you're assuming m = 4 and n = 3

also please use words instead of symbols

and don't forget to do a proper $\bN$

[wraith dagger]

i believe in mathemagic

[ember plaza]

How do I fix this?

[ember plaza]

Yes I know, its just a rough sketch

[balmy apex]

don't assume the values of m and n

your only assumption should be "24m - 12n = 1 with m,n in N"

[ember plaza]

Okay

And how do I continue?

Shall I proof that 1 is uneven

[balmy apex]

use the fact that addition/subtraction is closed in Z

[ember plaza]

and the RHS is even?

So it cant be one?

[balmy apex]

yeah that might work

yip

that will work

[ember plaza]

Okay thx mate

I can just say that 2 | 24m - 12n right?

And that not 2 | 1

@balmy apex

[balmy apex]

yes

@ember plaza

[ember plaza]

yes

done

thx :)

.close

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[cold oxide]

how do I solve a polynomial of degree 3/2

[cold oxide]

specifically

[balmy apex]

degree 3/2?

[cold oxide]

specifically answering this question

well i feel quite dumb

as I know for degree 2 and 3 etc there are the quadratic and cubic equations

[dusty hazel]

I'll have to tell you, degree of a polynomial can't be 3/2.

[balmy apex]

beat me to it

[cold oxide]

I assume i have to integrate the function no

to get a function of velocity

which ends up being t^3/2

then I need to solve that function when v=0

[red pumice]

yes because a = dv/dt

[dusty hazel]

Not quite.

You integrate, yes.

But v(t) is not t^3/2

[cold oxide]

of course no but its whatever the integral is

[dusty hazel]

yeah, right.

[cold oxide]

which includes some power of t^3/2

[dusty hazel]

It does.

So, what is it that you're implying?

[cold oxide]

how do I solve it jaja

when v=0

[dusty hazel]

Well you get something of the sort kt-bt^3/2 = 0

Yes?

k and b being some constants.

[cold oxide]

exactly

[dusty hazel]

We can safely divide by t^1. As t isn't zero. Because t is when the body comes to rest again, other than t = 0

so k - bt^1/2 = 0

t = (k/b)^2

Simple?

[cold oxide]

ahh

im stupid

i guess i am used to there being a constant

wwithout t

so when you didvide by t

it doesnt work

thnx so much man

[dusty hazel]

[cedar kiln]

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[grave jungle]

Hi would this be an approach that would work ?

[grave jungle]

ANALYSIS QUESTION

[cedar kiln]

@grave jungle Has your question been resolved?

[cedar kiln]

@grave jungle Has your question been resolved?

[elfin hemlock]

Sometimes it would but here that gives you no information

Its continious for all values of a and b is it not?

[grave jungle]

Well if it doesn’t equate to 0 then the line would be continuous and at x=0 just jump to 0 and the. Jump back to what ever it is

[elfin hemlock]

Did you read what I said?

No matter the choice of a and b you get its continious

If a=168118161718711 and b=617171 its continous

If a=17 and b=1 its continious

Any pair (a,b) makes that function continious

[grave jungle]

Let’s say 0<a<1

[elfin hemlock]

.

[elfin hemlock]

.

[grave jungle]

For example let a=1/2 and b=-1/2
It is NOT equal to 0
There fore it breaks the line and is not continuous ??

[elfin hemlock]

Its continious if limit as x->0 is 0

So try and compute the limit

And you will see it is in fact continious

Then try new values of a,b and you will see the same

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[sinful dragon]

I am asked to verify the following statement:

If $A \in \mathcal{M}{m}(\mathbb{K})$ satisfies $\textrm{im}(A) \subseteq \textrm{ker}(A)$, then for all $x \in \mathbb{K}^m$ it verifies that $A^2x = \theta{\mathbb{R}^m}$

I managed to prove that $\forall x \in \textrm{im}(A)$, $A^2x = \theta$ because all vectors in the kernel verify that by multiplying the matrix by then we get $\theta$, but don't know how to prove or disprove that this would work for every vector in $\mathbb{K}^m$

[wraith dagger]

Spiwocoal

[native crown]

A survey is carried out in a classroom in which 40% of the students play basketball and 60% are in a soccer team, knowing that 85% practice one of the two sports.
Calculate the percentage of students who play both sports.

100%??

[wary scaffold]

Please get your own help channel. But tip draw a venn diagram

[wraith dagger]

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

[cedar kiln]

@sinful dragon Has your question been resolved?

[sinful dragon]

thanks

.close

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[quasi wasp]

How do i know the long run behavior

[quasi wasp]

as x-> +-infinite, y ->

Since it goes up then down

Oh wait isn't it 0

[dull oxide]

Looks like it could be. But having the equation is the only way to know for sure

[quasi wasp]

Yeah i double checked it was right

Which means... with this other function

As x-> infinite, f(x) = 1/7
As x-> -infinite, f(x) = 1/7

Correct?

[dull oxide]

looks good

you got it

[quasi wasp]

How do i know if a function is rational

Like

(4^x + 8)/(2^x - 1)

[dull oxide]

Are you familiar with the definition of a rational function?

[quasi wasp]

a ratio of two polynomials

[dull oxide]

yup

[quasi wasp]

so i would need to know if both of them are polynomials since they are dividing each other

Is 2^x - 1 a polynomial?

[dull oxide]

You just need to know that the numerator and denominator are both polynomials

[quasi wasp]

Nah it isnt

[dull oxide]

This is an exponential function

[quasi wasp]

since its a variable exponent

So its not a rational function

[dull oxide]

2^x is exponential, x^2 is polynomial

[dull oxide]

Exactly right

[quasi wasp]

So my explanation would be

4^x + 8 and 2^x - 1 are not polynomials, and we can't put f(x) in the form of one polynomial divided by another

[dull oxide]

right

That would not be a rational function

[quasi wasp]

So

x+8/x^2 - 36 is not rational

Or wait it is

Since its

x^1 + 8 / x^2 - 36

[dull oxide]

yes

[quasi wasp]

So

x + 8 and x^2 - 36 are both polynomials, so f(x) is a ratio of polynomials, meaning it is rational

[dull oxide]

exactly right

[quasi wasp]

How would I know a minimum degree for a polynomial

I am given this graph

[dull oxide]

hmm

Well I know the answer is 3, but am having trouble explaining why

The first test to get the minimum degree is to see how many times the polynomial goes through the x-axis

However, here, there are only two points at -7 and 7

but 7 hits the x-axis at a local maximum, so it would technically be hitting that point twice.

Another way to be sure it's 3 is to mentally move the graph up a little bit. Then it would cross the x-axis 3 times.

[quasi wasp]

So it's 3 but we just don't know why its 3

Ok, I'll check it out some more

.close

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[atomic wedge]

I need help guys

[atomic wedge]

When taking moments around B

for the normal contact force at A vertical

what angle would I use

[cedar kiln]

@atomic wedge Has your question been resolved?

[fallen heath]

(180 - 45)° ?

well, you see... here, if you're talking about torque about B I think? so, the force will be perpendicular to the wall, and the r vector would be along BA, so you want the angle between the two vectors

[atomic wedge]

OKAY

thank you

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[cedar kiln]

@timid hill Has your question been resolved?

[ornate stratus]

From Start to where you finish on path C is over 2 blocks and up 1 block.

The triangle made there is a 30/60/90 triangle.

[timid hill]

If the black dot is the start, and the tip of the arrow is the finish point, isn't it just to the east 1 block?

[ornate stratus]

I'm sorry, I thought start was "start"

[timid hill]

no worries, I think what I have written makes sense. I was just hoping to confirm it c:

[cedar kiln]

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[crystal mason]

Now when I read linear mapping, i think linear in this context means additive and homogeneous with respect to integer scalars, but I'm not sure how to apply it to this function.

And for the kernel I'm only familiar with finding it for matrices.

[crystal mason]

and ig with additive its f(x+y) = f(x) + f(y) and for homogenous its f(kx) = kf(x)

[crystal raptor]

yes this is what you need to check, you are explicitly told what f does so it shouldnt be too hard to check

[crystal mason]

im not quite sure how elements in the input and output set would look like

im having a hard time visualising

[crystal raptor]

ditch the summation notation and just write out what an element of this looks like

not that it really matters here, but its not clear what n is supposed to be

[crystal mason]

ig cause of the second part we can just write n=3 and ditch the second part? is that legal

[crystal raptor]

yeah

[crystal mason]

so $a_1 t$ for i=1, $a_2 t^2$ for i=2, and $a_3 t^3$ for i=3?

[wraith dagger]

Levens

[crystal raptor]

well for one youre missing the i=0 term, and second this is not yet an element of the set

[crystal mason]

why not

[crystal raptor]

well you've just listed terms, the guys in that set are sums of things

[crystal mason]

oh right

so $a_0 + a_1 t + a_2 t^2 + a_3 t^3$

[wraith dagger]

Levens

[crystal raptor]

ya

[crystal mason]

ok let me rewrite this one to x and y

so lets say $a_0 + a_1 t + a_2 t^2 + a_3 t^3$ is our x

[wraith dagger]

Levens

[crystal mason]

is our y just $b_0 + b_1 t + b_2 t^2 + b_3 t^3$ or what?

[wraith dagger]

Levens

[crystal raptor]

sure

can use what ever letters for the coefficients as you want

[crystal mason]

damn ok

yeah just making sure thats allowed

[crystal raptor]

just another element, but you want to distinguish the two

[crystal mason]

yeah okay

ok lets see ill do the homogenous one first

so let $x = a_0 + a_1 t + a_2 t^2 + a_3 t^3$, then
$$
f(kx) = ka_0 + ka_1 t + ka_2 t^2 + ka_3 t^3 \text{ and we have to get to }
$$
$$
kf(x) = k[2]_3 a_2 - ka_0, ka_1 - ka_2
$$
ok lol im already lost

[wraith dagger]

Levens

[crystal raptor]

write out f(kx) using the defintion of f

you basically already wrote it but you shouldn't have wrote kf(x) yet

[cedar kiln]

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[crimson sedge]

How would I determine the zeros of this

[dull oxide]

Looks like you successfully factored it

The zeros of the polynomial equation are the value for x where the polynomial will be zero.

Since you factored it, if either of the factors are zero, then the whole thing will be 0

So you need to find the values of x where 4x²+9=0 or 4x²-9=0

[crimson sedge]

so what would be teh best way to go about doing this?

[dull oxide]

You familiar with algebra?

Solve for x?

[crimson sedge]

oh

okay

one sec lmao

[ionic harness]

Somebody help me with C and D

[crimson sedge]

this channel is taken

[crimson sedge]

ohh my dumbass misread this 💀

i thought u were talking about something else

ok thanks i figured it out

[dull oxide]

We've all been there

[crimson sedge]

.close

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[crimson sedge]

Julies 9 friends invested 10,000 $ for her company to start. She needed 100,000$. Her company becomes sucessful so how much ownership will each friend get in percentage?

[crimson sedge]

Help

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[outer iron]

What is the summation form of this sum-of-series?

[foggy merlin]

you mean, writing this as a power series ?

[outer iron]

Suppose, I'm not great with the terminology

This is for a generating function question. I need to express it in terms of a sigma expression so I can extract a coefficient

Is it this?

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[uneven kayak]

Could someone help me fill in the blank for this question? 'The ordered pair for the y-intercept is ___________'

[vagrant elbow]

Uh

(0, c)

I suppose

[uneven kayak]

yeah i was a bit confused

its a weird question

[vagrant elbow]

Assuming the equation is y = mx + c

[uneven kayak]

well in my case the formula is

y = mx + b

why are they different

[vagrant elbow]

Yeah alright put in a b instead of a c

[vagrant elbow]

Doesn't matter

[uneven kayak]

alr

im not in like 7th grade btt

btw

[vagrant elbow]

Depends on the education system of your country :P

[uneven kayak]

we just circling back to this stuff from a couple years ago

idk why

[vagrant elbow]

Lmao

[uneven kayak]

what is a point slope form?

for a linear equation

actually nvm

[vagrant elbow]

It's a method of representation of a line

It uses one point on the line and it's slope

[uneven kayak]

alright

so the next couple questions give me two points that a line passes through

and it wants me to find the slope of those lines

dont remember how to do that lmao 😂

[vagrant elbow]

Yeah so the difference in y coordinates over the difference in x coordinates

[obsidian coral]

You can google the slope formula

[vagrant elbow]

Rise over run

[uneven kayak]

oh

so i just use the slope formula

okay lol

@vagrant elbow The next questions are 'Write an equation in slope-intercept form for each line described below'

1. slope of 8 and b =5

would it just be

y = 8x + 5?

[obsidian coral]

Yes

[uneven kayak]

How would you change an equation from point-slope form to slope-intercept form?

[obsidian coral]

You have $y - y_1 = m(x - x_1)$ and you need to manipulate it to be in the form of $y = mx + b$, what do you think you should do first?

[wraith dagger]

dldh06

[uneven kayak]

distribute?

[obsidian coral]

Yes

[uneven kayak]

alright

so in point slope form it was y-6=9 (x-4)

so it would be y = 9x - 30

[obsidian coral]

Yes

[uneven kayak]

Also, I have another question.

This one gives me (8,3) and (2,5) as points a line passes through

and i found the slope as 2/-6

it wants me to put it into both point-slope form and slope-intercept form

for point-slope form, which coordinate do I use?

[obsidian coral]

Any one you want

It doesn't matter

[uneven kayak]

alright

y - 3= -0.3333333333333333(x - 8)

i got that

idk what to do

if thats right

[obsidian coral]

Leave that as a fraction

Not a decimal

[uneven kayak]

ok

so would it just be

y - 3 = 1/-3 (x-8)

[obsidian coral]

Yes

Now do the same process you did on the last question to get that from point slope to slope intercept form

[uneven kayak]

alright i did

if the points are

-3,3

and -4,4

would there be no y int?

[obsidian coral]

I don't get your question

[uneven kayak]

the coordinate points that the line passes through are -3,3 and -4,4

if you solve that, the equation is just

y = -1x

no y int?\

[obsidian coral]

Yes that's fine

[cedar kiln]

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[winged pewter]

im overthiknig

[winged pewter]

so csc = 1/sin

so im thinking csc (pi/4) as 1 / srt2 / 2

[gleaming epoch]

correct

[winged pewter]

then im thinking multpy reciprocal bc divison by fraction so 2 / sqrt 2

is that right

[gleaming epoch]

yes

imagine the 1 is over 1 like 1/1

then you have 1/1/sqrt2/2

[winged pewter]

but when. i do csc pi/4 on calcualtor

it comes up as sqrt 2

[gleaming epoch]

yes

same thing

[winged pewter]

how are they getting it in that form

[gleaming epoch]

(2/sqrt2)*(sqrt2/sqrt2)

[obsidian coral]

Rationalizing

[gleaming epoch]

its like (2/sqrt2)*1

[winged pewter]

i got it now

ty

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[rugged obsidian]

how does one draw this?

[cedar kiln]

@rugged obsidian Has your question been resolved?

[cedar kiln]

@rugged obsidian Has your question been resolved?

[rugged obsidian]

<@&286206848099549185>

[cedar kiln]

@rugged obsidian Has your question been resolved?

[cedar kiln]

@rugged obsidian Has your question been resolved?

[still thorn]

Usually you plot four points and interpolate

I think you could use eccentricity here

[cedar kiln]

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[maiden relic]

How exactly do I figure this out?

[fallen heath]

well, the shortest possible value for b exists when you throw a perpendicular from A to BC.
Hmm, that should be good enough for a hint. You should draw a figure of this, and think of how you'd get no triangle/exactly 2 or exactly 1

@maiden relic

[maiden relic]

Oh I believe I remember now!

If the height is less than the side, there is no triangle

If the height is greater than, then there is 1

and if the height is in between, there is a possibility of 2 if the given angle is acute which it is

[maiden relic]

Are you also able to help me with this problem?

I do not know how to solve this as only 2 values are given so I cannot use law of sines or cosines

[cedar kiln]

@maiden relic Has your question been resolved?

[maiden relic]

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[floral lance]

I just wanted to confirm whether or not this would have no solution since I'm pretty sure a square root can not equal a negative number

[fallen heath]

Good job noticing that

Yes, you're right

[floral lance]

ok, thanks we have been doing so many things in math recently I was just making sure I wasnt being thrown some curveball lol

[fallen heath]

you'll be fine, dw ^^" just take things one at a time and practice

[floral lance]

thanks again

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[stone lion]

Why (a) isn't correct? When I put it in calculator, i do get -√3/2

[cerulean sail]

Is that $p$ supposed to be $\pi$?

[stone lion]

Oh yes

[wraith dagger]

chartbit

[stone lion]

Printing mistake

[cerulean sail]

Ahh, I was so lost, I was like wtf is this

But "...while $\frac{\pi}{2} < \alpha < \pi$"

[wraith dagger]

chartbit

[stone lion]

While 5π/6 does fall under that condition

[stone lion]

I was too at first

Took me a while to get, that instead of the symbol, they straight used 'p' of pie💀

[cerulean sail]

Which one did they say is correct? Yeah I get the same too

[stone lion]

c

[cerulean sail]

Bet it's absolute 🧢

[stone lion]

Absolute 🤔

Blue cap

Does it mean it's a joke

[cerulean sail]

Hahah, cap is a way of saying basically lie, sorry 😂

[stone lion]

I thought it is for jokes

Well ik the right thing now

[cerulean sail]

But I don't get how they came to that conclusion that it's c), I'm going with a)

[stone lion]

c is also correct

But i wanna know why it isn't a

[cerulean sail]

Because that's outside the interval, no?

[stone lion]

No

[cerulean sail]

Ah, what I mean is that $\frac{7\pi}{6} > \pi$, so it doesn't satisfy the "...while..." thing they stated

[wraith dagger]

chartbit

[stone lion]

It is supposed to be less than π

Look again

[cerulean sail]

Yea but how can it be c) then? Am I missing something at this hour

[stone lion]

That img was a mistake

[stone lion]

I'm fkin dumb

I missedmit

My headdddd

Arrrrrrr

i took π as 360 instead of 180

Anyway thanks for your help

.close

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Please don't occupy multiple help channels.

[fallen heath]

.close

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[crimson sedge]

How is it that an immediate consequence of the axiom of extension is that the axiom of specification determines the set B uniquely?

[crystal raptor]

Well the axiom of extension says if there are two sets B and B' whose elements are those elements of A such that S(x) holds, (I.e. they have the same elements) then B and B' are equal

So B is unique

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[slow apex]

How do I solve this? I currently have 32000 = x^3 and s = 5x^2

[silent tide]

Solve for x in 32000 = x³

[slow apex]

x=31.75

[silent tide]

Now substitute that

[slow apex]

s=5040.31

[slow apex]

doesnt that = 0 if i take the derivative?

[silent tide]

sorry i haven't reached that level of maths yet

[slow apex]

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[sand kayak]

Help my solutions doesnt work!

[jagged marlin]

please translate the problem to english

[valid yacht]

this is like

macroeconomics

why ask in a maths server lol

and also without translations

(on an unrelated note, although these questions/formulae were quite simple, I still hated them for some reason)

[cedar kiln]

@sand kayak Has your question been resolved?

[sand kayak]

Because i dont get the rule that is used when solving this

I think its a general maths rule

[jagged marlin]

translate the problem bruh

[cedar kiln]

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[crimson sedge]

i understood upto the x2 <= a2 part, how did they get the domain tho

as -a <= x <= a

[silent tide]

They applied square root on all sides

[crimson sedge]

if u root it, it becomes x <= a

[silent tide]

x² ≤ a²
x ≤ ±a

[crimson sedge]

how did they reach the conclusion?

ohh

[sand kayak]

.reopen

Illustrate and show the easy keynesian model (45 degree diagram) with the help of the following data

Private consumption c, public consumption g, investments i, savings as part of the disposable income s, taxes t

Bnp = c+ i + g

And c= 1000 + (1-s)(1-t) * Y

I put all that data in

That was given to me above

And am left with Bnp= 2000+0,063Y which is a problem

Bnp is supposed to be just a number without any variables like Y

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[lavish plank]

how would i find du/dx

2nd part unaffected as no x term

[vagrant elbow]

It won't be unaffected exactly

Because you'll have to apply the chain rule

You'll get a bunch of dt/dx terms

[lavish plank]

vdu +udv

[sacred grail]

well if dt/dx = 0...

[vagrant elbow]

yes I know but time would also be a function of velocity

[vagrant elbow]

Can't be sure of that :/

[sacred grail]

[lavish plank]

might plug into derivative calc

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@lavish plank Has your question been resolved?

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[thin anchor]

i'm trying to solve this problem "d/dx [ln(√x + 3x - 9)]/3"

i first used an easier problem "9[ln(x^5 - 2x)]" to see how mathway would differentiate that one, and then mathway gave me back this [(5x^4 - 2)×9]/(x^5 - 2x).

so i tried to make a step-by-step process to try to understand what might have had happened.

i assumed that it substituted (x^5 - 2x) with a dummy variable (u), then took the 9 from the outside and placed it inside of the ln to make it "d/dx ln(u^9)."

and here is where i stopped understanding. i don't know how it went from "d/dx ln(u^9)" to "[(5x^4 - 2)×9]/(x^5 - 2x)."

the closest thing i could guess is that it differentiated (u^9) to get u × 9, and then differentiated u(which was originally x^5 - 2x), and then it divided all that by u(which was originally x^5 - 2x) to get [(5x^4 - 2)×9]/(x^5 - 2x).

and i used that to solve for "d/dx [ln(√x + 3x - 9)]/3" and inevitably got it wrong.

so....how do you actually solve "d/dx [ln(√x + 3x - 9)]/3?"

[scarlet garnet]

$d/dx \frac{ln(√x + 3x - 9)}{3}$

[wraith dagger]

Jigglyproff

[crystal raptor]

by the sounds of things you have not yet come across the chain rule?

[pseudo galleon]

$\frac{d}{dx}\left(\frac{ln(√x + 3x - 9)}{3}\right)$

[wraith dagger]

Asagao 朝顔

[thin anchor]

i have, but my memory

dead

[crystal raptor]

good job there are plenty of resources online to refresh your memory

[thin anchor]

i've watched like, 3 youtube videos the past week refreshing and refreshing

something's wrong with my memory

[scarlet garnet]

here

start by finding your g(x) and h(x) for the function (let h(x) = f(x) from the example)

[thin anchor]

i have not seen this before

thank you

[scarlet garnet]

h

[thin anchor]

i'll take the night trying to solve it

[scarlet garnet]

that's good to hear!

[thin anchor]

two hours past my bedtime

so good

thanks

[cedar kiln]

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[fervent depot]

i need help with this

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[narrow geyser]

I don't get this problem 🤔

[honest saddle]

What is the question

[narrow geyser]

I need to figure out how much soil this thing needs to fill it

[snow grail]

The first step is to determine if it is a right triangle using the Pythagorean theorem

If you know it’s a right triangle, it becomes very easy to find the area of it, and from there the volume is base times height

[narrow geyser]

Ooh

ty

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[crimson sedge]

let $\Omega={0,1,2,\dots}$, given that $P({n})=\frac{c}{6^n}$ for every $n \in \Omega$, find $c$
hint: use the sum of an infinite convergent series

[wraith dagger]

metnal

[crimson sedge]

im not sure where to start with this

isnt the probability for any given subset basically zero because the sample space is infinite?

[crimson delta]

you need that $\sum_{n\in\Omega} P({n}) = 1$

[crimson sedge]

wait no thats a dumb assumption

[wraith dagger]

Denascite

[crimson sedge]

what does {n} refer to here

a subset of omega?

or just a singleton?

containing n

[crimson delta]

{n} is the singleton containing n. and this is a subset of omega

[cedar kiln]

@crimson sedge Has your question been resolved?

[cedar kiln]

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[winged pewter]

want clarificaiton

[winged pewter]

so 2nd derivative is d^2y/dx^2

right?

[dull oxide]

Yes

[winged pewter]

also i presume finding second derivative has to deal with finding concave upward?

like first derivative is speed, second derivative acceleration

but like whats the connection in finding second derivaive

with concave upward

i get how to solve just dont understand the connection

[dull oxide]

Do you remember first and second derivative tests?

[winged pewter]

first derivaitve tells us if fn is inc or dec

second deriv tells us if first derive is inc or dec

oh second deriv deals with ccup ccdw

[dull oxide]

Yep. That's right

Also, be sure you know how to get second derivative of parametric equation

https://www.nagwa.com/en/explainers/613184581056/

It's a little more tricky than first derivative

[winged pewter]

ty

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[iron flume]

Hi, is there a way to get the y-intercept to be 4 instead of 16?
I've gotten the roots correctly, but the y-intercept is off

[sage forge]

You can multiply everything by 1/4. The roots won't change but the y-intercept will

[iron flume]

thanks!

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[strange fulcrum]

$y''+2y'+y=e^x+e^{-x}$

[wraith dagger]

Yousssef

[strange fulcrum]

can someone help me

[cedar kiln]

@strange fulcrum Has your question been resolved?

[dire geode]

Have you done anything?

[strange fulcrum]

yes

[strange fulcrum]

i did my homogenous solution

so im already done with that

[dire geode]

Your particular solutions* should be linear combinations of the right side.

[strange fulcrum]

i need help with my particular solution

[cedar kiln]

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[winged pewter]

they said if we cos(theta) = pi/2 + n*pi

cna someone explain why

im not seeing it

[crimson sedge]

When is cos(theta) = 0

[winged pewter]

pi/2 3pi/2

[dire geode]

3pi/2 = pi/2 + pi

[winged pewter]

oh ok im starting to get it

what does the n mean

in npi

[dire geode]

n is an integer ..., -3, -2, -1, 0, 1, 2, 3, ....

,w solve cos(theta) = 0

[wraith dagger]

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Requested by riemann#2529

[dire geode]

same thing here. except instead of pi/2, wolfram starts with -pi/2

but since -pi/2 = pi/2 - pi, they're equivalent

[winged pewter]

👍

ty

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[keen berry]

One antiderivative of p(t)=3t^(2) is P(t)=t^(3) if Q(t) is another antiderivative with Q(3)=50,find Q(t).

[keen berry]

is it simply 3Q or am I botching it completely?

[crimson sedge]

u are finding C

such that

P(3) = 50 = t^3 + C

[keen berry]

ok so I am way off

[crimson sedge]

yes

[keen berry]

yea I'm stuck thx for the help though much appreciated

[crimson sedge]

how are you stuck?

[crimson sedge]

you are just solving this for C

[keen berry]

yea i came up with 23 not sure its right

[crimson sedge]

yeah so whats the full specific anti derivative its asking for

Q(t) =

[keen berry]

Q(t)=1/3t^4? prob should watch a video or two and brush up I am beyond rusty

x(t^3+c)+c is my last attempt

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[quartz frost]

hey

[quartz frost]

i have this recursive defined sequence

i have to show that it converges and also get the value to which it converges

i never did that for recursive sequences though

wolframalpha tells me that it falls
thus i would try to get a lower limit and show that it falls monotonously

any tips how i would do that in this case?

[dire geode]

try calculating the differences

!help

[cedar kiln]

Please read #❓how-to-get-help

[quartz frost]

alright so, as far as i know i can't simplify further

via graphing i found out, that the -sqrt(2)a_n will always be greater than the root on the left

ohhh

nvm

a_n can be negative, then this is not true anymore

[dire geode]

$\sqrt{a^2} = |a|$

[wraith dagger]

riemann

[dire geode]

the first term can be negative. all terms afterward are positive

[quartz frost]

alright, but sqrt(2)*a_n can be positive or negative

[dire geode]

negative only for the first n

[quartz frost]

ohhhhh

yeah true

[dire geode]

you should be able to come up with a bounding argument that sqrt(1+a_n^2) - sqrt(2) * a_n is always negative after the first term

oh wait it's not true for small a_n?

[quartz frost]

ok looks like i forgot something
it is given that a_0>=1

[dire geode]

(╯°□°)╯︵ ┻━┻

[quartz frost]

ok

i showed that a_(n+1)-a_n<0

thus the sequence is monotonously decreasing

alright i found a lower limit (1)
thus the sequence converges

what remains is to show that 1 is also the value it converges to

[cedar kiln]

@quartz frost Has your question been resolved?

[quartz frost]

.close

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[placid skiff]

is it possible to factor even more?

[gusty vine]

,w factorise x^2 -8x +17

[wraith dagger]

Results provided by WolframAlpha

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Requested by swaggofishballs#1529

[warm bone]

,w is 17 prime

[placid skiff]

lmao

[wraith dagger]

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Requested by 草w#3875

[warm bone]

This will answer the question.

[hollow minnow]

The other roots are complex…

[warm bone]

Ah that makes sense

[gusty vine]

oop- didn’t see the first part

[placid skiff]

so what do i do

[hollow minnow]

You can write x(x-(4+i))(x-(4-i))

[placid skiff]

im scared it wont take it

imma just do it at this point

i've spent too long on this one question

its right

thank you

[sacred grail]

[placid skiff]

I should've done it from the start

[hollow minnow]

[sacred grail]

you had the complex roots in the first part

you were so close

[placid skiff]

didn't know I was supposed to submit an answer with them, thought i was missing something.

thanks guys

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[north otter]

hel p ples

[violet night]

f(x)=y

[north otter]

?

[sonic thistle]

instead of f(x) write 0

and solve for x

that's how you find the x intercepts

[cedar kiln]

@north otter Has your question been resolved?

[north otter]

yo mb

im back

alr bet

[sonic thistle]

ok