help-23

[slate quail]

So the limit does not exist?

[rain bear]

what do you think about part e

i want to hear what you are thinking (or read for that matter : P)

[slate quail]

False, because of that

X approaching from both sides

[rain bear]

but do you need the point when you are searching for the limit?

let me ask you this then

lets say we have f(x) = 1/x

very simple

what is the limit when x -> inf

will you say that limit also doesnt exist since there is no inf point?

[slate quail]

No, I’d call it infin

[rain bear]

the point itself is of no concern in limits, what you need is continuous convergence

both sides of the point 0 are going to 0

in a continuous matter

and thus limit exists

and is 0

limit is as of dictionary definition

its the limit, not the value

do you understand? i can explain in another way if you want

[slate quail]

Somewhat

I figured it was zero if it did exist, I must have made up a rule in my head regarding the open circle being indefinite = not existing

[rain bear]

u dont need the function to be continuous at the point

just before and after the point suffice

cuz the way of seeing if a limit exist is to see if
$lim_{x\Rightarrow n^-}f(x) ?= lim_{x\Rightarrow n^+}f(x)$

[slate quail]

Ok so I am reading f as lim > 0 = f(x) = 0

G says the same, except f(x) = 1

I see a solid point above 0 at y=1 which in my head means > definite > G Is true

[flat frigate]

Rootsyl

[rain bear]

you go to the point from right and find the limit

then go from left and find the limit

if the limits are the same

and not divergent

the limit exists

[slate quail]

AHHH ok, I remember that rule.

G is untrue

There are two points

[slate quail]

.

[rain bear]

yes

[slate quail]

Next I am reading h

I see from the left, open circle at 1 the line confused me a bit but we only read the y values ig

[rain bear]

look from right, look from left

see if they equal

[slate quail]

Approaching 1 I see it as an endpoint

Are you also counting the line along y=0 1-2 as the right side?

[rain bear]

yes

left and right do not converge to the same point

thus that limit is invalid

[slate quail]

Ok in that case I see 1 and 0

The real limit is DNE?

[slate quail]

Back 2 this rule

[rain bear]

the left and right side are not same

you dont even need to look for divergence

oh you asking for general function?

like all of it

[slate quail]

No I’m asking if it was a question, would that be DNE?

[slate quail]

Since there’s not the same

[rain bear]

of which point

[slate quail]

H

[rain bear]

yea the limit is dne

[slate quail]

I see i as true? There is a definite point at 0…I also see a open point at 1 but I’m choosing to define it with 1

[rain bear]

lim x goes to 1?

[slate quail]

On I now yes

[rain bear]

that limit also dne

cuz its literally the same with h

[slate quail]

What if there are two definite points, like for k?

Is the lim still one? @rain bear

[rain bear]

k is true since there is no f(x) in that region

there cannot be a limit if the function do not contain the points region

if it was -1^+ you could say its converging to that point

points region = infinitely small area around the point

which is used to determine the limit for a point

[slate quail]

Because for 1+, you can lead up from the right side

[rain bear]

i might be using the wrong term to define point region

i think there was another name for it

[slate quail]

And that’s where the point 1 I came up with comes from

But there’s no points that can be made from the -1-, since x is only approaching it

Not on it

@rain bear am I understanding correctly…?

I am also ready to move on to the next set of questions

(I can reopen for someone else. If you might not have time)

.close

[safe radish]

Channel closed

Closed by @slate quail

Use .reopen if this was a mistake.

[safe radish]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[golden bluff]

Hello, im not sure how to approach this

[golden bluff]

calculate the determinant of:

[icy lance]

try doing some row operations, you could probably make the matrix look a lot simpler

[frank horizon]

consider factoring out n in the last row

[golden bluff]

wdym

[frank horizon]

you're trying to calculate the determinant of A, it will be n times A with the last row divided by n

[golden bluff]

ohhh right

[frank horizon]

then you could try subtracting the last row from each of the other rows n times

[golden bluff]

wait could i just factor out n from all of the rows below 1?

[frank horizon]

how would that help

[golden bluff]

everything below the diagonal would be 0

oh nevermind brainfart

[frank horizon]

you can first cancel everything out besides the diagonal and the last row

[golden bluff]

right but then i cant calculate it based on the last row

[frank horizon]

and then cancel out the last row minus the last entry

[golden bluff]

i dont konw if n is even or odd

[golden bluff]

wdym

[frank horizon]

if A is almost diagonal with just the last row, then i can subtract each of the rows times a constant from the last row to cancel out the last row, except for the last entry

[golden bluff]

right

[frank horizon]

so the resulting matrix is diagonal

[golden bluff]

oh wait

can i subtract that last row from all of the other rows?

[frank horizon]

in the beginning?

yes

[golden bluff]

yeah

then everything above the diagonal would be 0

or am i missing something

then the determinant would be the product of the diagonal

[frank horizon]

yeah

forgot that you didn't need to cancel out the other entries

[golden bluff]

yeah

thanks for the help

.close

[safe radish]

Channel closed

Closed by @golden bluff

Use .reopen if this was a mistake.

[safe radish]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[mighty mango]

So two questions, if x=y does that mean log(x)=log(y) (respecting all the stuff abt logs yk how they cant be negative and smth cant be zero )

[mighty mango]

And if log(x)=log(y)

What does that have to say abt x and y

Do we conclude another releationship?

[left gyro]

are x and y real numbers

[mighty mango]

Yes

Absolutly

[left gyro]

log(x) by itself will say that x > 0

so log(x) = log(y) also says that x and y are positive

you can e^ both sides to see that x = y

so log(x) = log(y) => x = y > 0

similarly you can log both sides by only if theyre positive to begin with

x = y > 0 => log(x) = log(y)

[mighty mango]

Oh! How imteresting.

[median vigil]

log(x) = log(y) implies x = y because log is a one-to-one/injective function

[mighty mango]

Does that mean that it doesnt have an inverse or- wth is an injective function

[median vigil]

an injective (aka one-to-one) function is a function that has an inverse (every output is produced by only one input)

[mighty mango]

Kk!

(Trying to remember a question i wanted to ask)

[median vigil]

for a log with base $b$, $\log_b(x)$, its inverse function is the exponential function $b^x$

[flat frigate]

cloud

[mighty mango]

Wait? So ^e is the inverse for logs? Just like ^2 and sqrt?

[left gyro]

e^

not ^e

e^x

doing e^ to 2 means e^2

[median vigil]

e^x is the inverse function for the natural logarithm (base e), also notated as ln(x)

[mighty mango]

Im trying to comprehend this

So log(x) base b

Is like b^(smth)=x

So- what does the inverse function do to this

How do you apply it

[median vigil]

if you apply one function to a number, and then apply the inverse function to the output, then you get back the original number

you can do this on both sides of an equation, for example

[mighty mango]

Are you thinking of an example or are you waiting for me to respond 🙃

[mighty mango]

Yes! Ik that but im a bit confused on how to apply that to logs

[median vigil]

let's say that we know
[ 3^x = 81]
we can solve for x by applying the function $\log_3(x)$ to both sides:
[\log_3(3^x) = x = \log_3(81) = 4]
similarly we can solve the equation
[\log_2(x) = 6]
by applying the inverse function $2^x$ on both sides:
[2^{\log_2(x)} = x = 2^6 = 64 ]

[flat frigate]

cloud

[mighty mango]

sorry for being slow but- how does 2^log_2(x) translate to x

I mean yes its an inverse function

But im asking abt the algebra

[median vigil]

in general if $f^{-1}(x)$ is the inverse function of $f(x)$ then [ f(f^{-1}(x)) = f^{-1}(f(x)) = x ]

[flat frigate]

cloud

[mighty mango]

Ty miss cloud

.close

[safe radish]

Channel closed

Closed by @mighty mango

Use .reopen if this was a mistake.

[safe radish]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[rugged cargo]

Can i write it like this

What?@rugged marsh

[rugged marsh]

sorry

[rugged cargo]

<@&286206848099549185>

I guess i am right

Cofficent will be0

[gritty pond]

Yes

[rugged cargo]

But power of x will be 1?@gritty pond

Scaler multiplication to matrices

[safe radish]

@rugged cargo Has your question been resolved?

[rugged cargo]

.close

[safe radish]

Channel closed

Closed by @rugged cargo

Use .reopen if this was a mistake.

[safe radish]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[lilac osprey]

i have a question with interpreting my solution to an old question

[lilac osprey]

i took calc 2 a few months ago then dropped it so im a bit unsure what i did for this telescoping series

im not sure how i got from the 4th last line to the 3rd last line

can anyone explain

[drowsy moss]

looks like you added and subtracted 1/n term by term and then split the series.... which is hella sus imo

[lilac osprey]

yeah i agree its sus i had no idea what i did

i have the same question for a new assignment ill try again

.close

[safe radish]

Channel closed

Closed by @lilac osprey

Use .reopen if this was a mistake.

[safe radish]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[kindred fjord]

How do I do #3?

[obsidian oracle]

well quadratic formula

[real canyon]

just solve it normally using formula

[main mural]

same way you did the others?..

[kindred fjord]

ik but when I try to do it I keep getting the square root of a negative number so im not sure if im doing it correct

[lone hornet]

negative determinant init?

[kindred fjord]

I get no solution

[lone hornet]

no solution is correct then

[real canyon]

then show answer in terms of i

[kindred fjord]

idk if i can use imaginary numbers

[real canyon]

I mean its your only option

[kindred fjord]

true

[obsidian oracle]

I don't think they've seen that yet

you can just say "no real solutions"

[lone hornet]

you can also say no solution in reals

[kindred fjord]

yea ill prob do both show solution with imaginary numbers and say no solution in reals

thanks guys

[safe radish]

@kindred fjord Has your question been resolved?

[safe radish]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[hushed onyx]

i need help 😦

[lean otter]

x=1/2 and y=3 is that correct?

[hushed onyx]

yea but lemme show smt

[lean otter]

ahh i did that intuitively lol

[hushed onyx]

For the second equation is this correct 😅

Is doing it these ways the same or wrong ☠️

[safe radish]

@hushed onyx Has your question been resolved?

[wet pike]

just use basic substitution

2x= 10 - 3y
x = (10-3y)/2
now substitute this in 2nd equation

and you should get your answer

[hushed onyx]

yes yes i was able to get the ans

but im not sure if the way im doing the fractions is corret :/

is the way im deriving right?

[safe radish]

@hushed onyx Has your question been resolved?

[hushed onyx]

.close

[safe radish]

Channel closed

Closed by @hushed onyx

Use .reopen if this was a mistake.

[safe radish]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[rustic cloud]

what did i do wrong here

[finite igloo]

wait

[flat frigate]

:D

[finite igloo]

You are

integrating with respect to u, no?

[rustic cloud]

Yeah

[finite igloo]

but the functions inside

are of x

[flat frigate]

:D

[finite igloo]

use this sub instead

but if u insist on using u = cos3x, then

[rustic cloud]

well it seems like u=3x is the correct way based on calculators, but i'm trying to figure out where i went wrong because it looks like i got to some answer

wanna know where i messed up

[finite igloo]

u integrate with function of x with respect to u

we should transform everything into u before integrating with respect to u

[flat frigate]

:D

[finite igloo]

forgot the 1/3

wait

[flat frigate]

:D

:D

:D

:D

[rustic cloud]

Well that sounds nightmarish

[flat frigate]

:D

[finite igloo]

${-\frac{2}{3}\sin(3x) + \frac{\sin^3(3x)}{9} + C}$

[flat frigate]

:D

[finite igloo]

interesting...

i did something wrong

[rustic cloud]

alright well yeah i guess cos3x is not u LMFOA

[finite igloo]

this

i messed this up

i mean

[rustic cloud]

can i maybe do int (cos3x)(cos3x)dx and then ibp?

[finite igloo]

the thing works

ibp?

[rustic cloud]

integration by parts

[finite igloo]

hmm

would take lots of turn

[finite igloo]

This is actually equal to:

nvm

this approach works

but is very complicated

so lets do ibp

wait

but

to answer ur initial question

[finite igloo]

.

that's where it went wrong

[finite igloo]

this sub is wrong

that's wh

y

[rustic cloud]

am trying to ibp rn

its long

[finite igloo]

${|\sin3x| \neq \sin3x}$

[flat frigate]

:D

[finite igloo]

@rustic cloud tbh, i dont think ibp will work

the functions are periodic

uv

nvm

[rustic cloud]

yeah this problem is ghoulish

[finite igloo]

wait

it does

[rustic cloud]

i got to here, i'm sure i got something wrong on the way

but i don't know where

[finite igloo]

no like

after the sub

u use trig identity

[flat frigate]

:D

[finite igloo]

butttt

let's let idp

cuz hell yeah

[rustic cloud]

if it makes any sense i'm trying to do the brute force methods to make sure my fundamentals work, so incase i don't know a particular identity i can still make it to the correct answer

[finite igloo]

${\int \cos(x)\cos(x) dx = \cos(x)\sin(x) + \int \sin(x)\sin(x)dx}$

[flat frigate]

:D

[finite igloo]

${\cos(x)\sin(x) + \int \sin(x)\sin(x)dx = \cos(x)\sin(x) + x - \int \cos^2 (x) dx}$

[flat frigate]

:D

[finite igloo]

${\int \cos(x)\cos(x) dx= \cos(x)\sin(x) + x - \int \cos^2 (x) dx}$

[flat frigate]

:D

[finite igloo]

${\int \cos(x)\cos(x) dx= \frac{1}{2}\cos(x)\sin(x) + \frac{1}{2}x + C}$

${\text{Actutal integral }=\frac{1}{3}\cdot I }$

[flat frigate]

:D

:D

[finite igloo]

${\therefore \int \cos^2 (3x) dx =\frac{1}{6}\cos(3x)\sin(3x) + \frac{1}{2}x + C}$

[flat frigate]

:D

[finite igloo]

@rustic cloud

it worksx

LMAO

[rustic cloud]

where did x come from here

cos(x)sin(x) + x

[flat frigate]

:D

[finite igloo]

this is a nice way

to approach this problem tho

[rustic cloud]

is doing ibp on sinxsinxdx bad then?

[finite igloo]

Don’t think so

We’ll have to try

[rustic cloud]

yo brah this is my math struggle channel

[finite igloo]

@rustic cloud it will

But I recommend

[rustic cloud]

it will what

[finite igloo]

Transforming sinxsinx to 1 - cosxcosx

[rustic cloud]

Oh okay thanks

[finite igloo]

It think it will work tho

Let’s give it a shot

I’m too lazy for latex now

[rustic cloud]

for latex?

[finite igloo]

It does work

[rustic cloud]

there’s really no way outta this without identities…..

So cooked

Toast, even

[finite igloo]

There are only a few identities u need

[rustic cloud]

For this problem of just in general?

[finite igloo]

For calc

If u’re not doing some competitions

[rustic cloud]

Oh okay

I guess I’ll just get to memorizing some identities

If you could drop a few that you deem essential that’d be great

But you’ve been helping a lot already

Thank you!

[finite igloo]

Okiee

Wait

It’s actually mentioned in me textbook

1. Trig iden and their derivations
2. Double angle
3. Half angle

[safe radish]

@rustic cloud Has your question been resolved?

Channel closed

Closed by @rustic cloud

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[tawny charm]

What is x to the power of 2?

[finite igloo]

Wait

Find area

[tawny charm]

Yeah

[finite igloo]

Lemme try

[tawny charm]

It needs some geometry theorem that I don't know

[finite igloo]

Nvm

I’m too lazy

If there is an unknown theorem

I’m dead 💀

Cuz I was some algebra

[tawny charm]

I have the answer but don't know how to solve that

[finite igloo]

The sum of all triangles inside = x^2

[tawny charm]

4 triangles

[finite igloo]

Yes 💀

Some terms cancelled

But was still long

[tawny charm]

Only one of them can be caculate

[finite igloo]

Is that thing a square tho?

[tawny charm]

Yeah

[finite igloo]

So

U can find areas of all triangles?

[tawny charm]

I tried to name the parts and solve by equations but it didn't work

[tawny charm]

Only one of them

[finite igloo]

Wait a second

U don’t need the area

U can just equate

[tawny charm]

Is that 16 times x to the power of 2?

[finite igloo]

Nvm

Couldn’t do this 😞

[tawny charm]

I'm cooked bro
Been waiting for 2 weeks☠️

[finite igloo]

I’ll try in the morning

It’s 2 am here

[tawny charm]

Tnx

Sorry to bother you

[finite igloo]

Cannot use calculator, right?

[tawny charm]

I don't think it has a hard calculation but if you want you're free to go

[safe radish]

@tawny charm Has your question been resolved?

[safe radish]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[wicked veldt]

For you to better understand my question I think it would be best for me to work through the example problem I have and explain the part I’m confused at after 12 t - 42 + 60 I don’t add 60 to -42 because addition shortcut theorem if f (x) = K where K is some integer the f ‘ (x) = 0, so would the derivative of 60 be 0?

[fathom adder]

The derivative of a constant is always zero

But i dont understand what you call "addition shortcut"

Like to get faster and to not simplify, since youll derivate ?

[wicked veldt]

So anytime I’m asked to solve for the derivative of a problem and I see a number with no variable attached to it the derivative of that number is 0 ? and No I think my professor just calls it that when he is referring to the constant function rule

[fathom adder]

Anytime yes

[wicked veldt]

Ok thank you!

.close

[safe radish]

Channel closed

Closed by @wicked veldt

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[versed kernel]

In a linear system of equations, does the order matter? i.e. instead of a row say being x - y + 0z + 0w = 0, would it be wrong to write it as -y + x + 0w + 0z = 0, or are they equivalant? Because if we convert it into a row in the matrix, it seems wrong. But at the same time, in equations, variables can be written in any order

[safe radish]

@versed kernel Has your question been resolved?

[versed kernel]

.close

[safe radish]

Channel closed

Closed by @versed kernel

Use .reopen if this was a mistake.

[safe radish]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[sinful iris]

this came out in my algebra exam and i had no idea

[long oracle]

What did you try?

and is m here a real number? integer?

[sinful iris]

variable

[long oracle]

Okay, but could m be any number?

[sinful iris]

yeah

real

[long oracle]

So a good start is to determine for what m this is even defined for

i.e since we know sqrt(x) is defined for x>=0

and since we have an equality, the LHS must also be >=0

since the RHS is non-negative

would you agree?

[sinful iris]

what is RHS

[long oracle]

right hand side

while LHS means left hand side

[sinful iris]

yeah i agree

[long oracle]

i.e which side of the equality we're talking about

alright, so for the LHS to be non-negative, we have that 1-m >= 0

try and see if you can do something similar for the expression on the RHS

[sinful iris]

so i separate the RHS and LHS into inequations?

[long oracle]

well as a start, we're for the moment just trying to figure out which m here is allowed

for example, we know that sqrt(x) is only defined for x >=0

[sinful iris]

yes

[long oracle]

so try doing a similar analysis for the RHS

my tip would be to start in the inner sqaure root and work your way out

[sinful iris]

i dont get it

what we're doing is

figuring out for what values, m is allowed

in the equation

?

[long oracle]

are you asking why or how?

[sinful iris]

if thats what we're doing

[long oracle]

yeah as i explained above thats exactly what we're doing

[sinful iris]

doesn't the RHS have to be x >=0 as well theb?

[long oracle]

yup

in which case we know that the LHS also has to be >=0

which is why i said that 1 - m>=0

[sinful iris]

yeah i get that

[long oracle]

Makes sense?

Is something still confusing?

[sinful iris]

so far no

[long oracle]

Alright cool, so try on your own as a start and see which m are allowed in the square roots in the RHS

[sinful iris]

i dont understand what you’re asking there

do I have to solve an inequality in the RHS)

?

[long oracle]

So the square root function is only defined for certain m, we'd like to reduce the number of possible solutions down to something actually possible before we move on, as for example squaring both sides may wrongly introduce solutions

so for example as a start, working from the inside to out

when is $\sqrt{4m^2-7m^4}$ defined?

[flat frigate]

Aslan

[sinful iris]

when its higher or equal than zero

[long oracle]

what exactly?

[sinful iris]

m

[long oracle]

m?

are you sure

[sinful iris]

orrr

[long oracle]

that would be correct if it said $\sqrt{m}$

[flat frigate]

Aslan

[sinful iris]

mmm

idk what it changes then

exponents are still even

[long oracle]

so the key here is that sqrt(m) is more of placeholder, if we already know that sqrt(m) is defined for m>=0

then whatever we replace m with should also be true

so in our case we replace m with 4m^2-7m^4

i.e when $4m^2-7m^4 \geq 0$

[flat frigate]

Aslan

[sinful iris]

oh

yeah

i get that

[long oracle]

nice!

i can cut the chase here and move on

as this just a starting step i'd like to do

but you can check your solutions afterwards eitherway

So going back to the equation

[flat frigate]

Aslan

[long oracle]

what did you try?

oops

meant m^4 on the 7m^2

[sinful iris]

i expent little time on it

but my first idea was

factorizing

the inner most sqrt

[long oracle]

I think we can also notice atleast one solution here by inspection

[sinful iris]

0?

[long oracle]

yup!

so let's keep that one in mind, incase we lose it somehow

[long oracle]

anyways, how did it go after that?

[sinful iris]

i got

sqrt(m^2(4-7m^2))

[long oracle]

right

[sinful iris]

then i thought about separating the roots

and m^2 would simplify to m

uhh following sqrt(a.b) = sqrt(a) x sqrt(b)

[long oracle]

well in general it would be |m|, but in this case it wont matter (by luck tbh)

[sinful iris]

ok ok

but is that useful?

what im doing

i mean

[long oracle]

Yeah it might

before factorizing

did you try to square both sides?

[sinful iris]

yeah thats i thought i had but didnt try

(1-m)^4

in LHS?

[long oracle]

how?

[sinful iris]

squaring both sides

to remove the sqrt

[long oracle]

squaring means ^2

[sinful iris]

i did it twice

[long oracle]

well, why?

[sinful iris]

to remove the other one

ig

[long oracle]

maybe theres a step u can do before that

[sinful iris]

so its best to square once first

?

[long oracle]

yeah just to be safe

maybe u can do something

afterwards

so u got

[sinful iris]

ohhh

now its starting to look easier

but ok

[long oracle]

$(1-m)^2 = 1 -\sqrt{4m^2-7m^4}$?

[sinful iris]

(1-m)^2 = 1 - sqrt(4m^2 - 7m^4)?

[flat frigate]

Aslan

[sinful iris]

yeah

[long oracle]

i would move the 1

[sinful iris]

without factorizing the stuff inside the sqrt

me too

[long oracle]

and to be on the safe side multiply by -1 to get a positive square root on the RHS

[sinful iris]

so

m^2 - 2m - 2

in LHS

before multiplying by -1

[long oracle]

woulnd it be m^2 - 2m?

[sinful iris]

yes

wait

yes

it would

m^2 - 2m

[long oracle]

neat

so then we have $2m - m^2 = \sqrt{4m^2-7m^4}$

[flat frigate]

Aslan

[long oracle]

and now might be a good idea to factorize the RHS

[sinful iris]

yeah

m^2(4-7m^2)

right?

[long oracle]

yup

[sinful iris]

inside the sqrt

then we separate m^2

[long oracle]

yup

[sinful iris]

and (4-7m^2)

then i have

m^2 - 2m = m . sqrt(4-7m^2)

[long oracle]

in pratice, youd need to have |m| after sqrt(m^2)

[sinful iris]

do i divide both sides by m?

[sinful iris]

yeah you're right

[long oracle]

but i can let it slide this time since it wont matter, but this is very important to remember, okay?

[sinful iris]

yes

[long oracle]

it will in fact matter in most cases

[sinful iris]

i get it

[long oracle]

anyways, yes we can divide by m

since we already know m = 0 was a solution

[sinful iris]

ok

m - 2 = sqrt(4-7m^2)?

[long oracle]

oh be careful

we had 2 - m

remember we mult. by -1

[sinful iris]

-2+m

then?

oh wait

[long oracle]

no, it should be 2 - m

[sinful iris]

the result of multiplying by -1 was

2-m

okok

[long oracle]

yeah

from before

[sinful iris]

do we square again

[long oracle]

okay so now that we have $2-m = \sqrt{4-7m^2}$

[flat frigate]

Aslan

[long oracle]

it seems fair to square it again

[sinful iris]

ok

bruh when its like this (2-m)^2 its hard to do the factorizing for me idk why

so i move it to (-m+2)

[long oracle]

sure, thats fine!

in fact notice that (-m + 2)^2 = (m -2)^2

[sinful iris]

4 cancels out?

[long oracle]

seems so!

[sinful iris]

alright then

8m^2 - 4m = 0

[long oracle]

we dont care about m=0

so just divide by m here

[sinful iris]

yes

bc we already found that answer

right

8m - 4 = 0

[long oracle]

and so m = ?

[sinful iris]

1/2

[long oracle]

correct!

Now as we didnt do the first step properly, let's check this one just incase

[sinful iris]

we check the answer?

[long oracle]

so just plug it back into the original equation to see if its a valid solution

you could skip this if you had done the first step (sort of)

[sinful iris]

seems to be a solution

1/2

[long oracle]

dope!

then thats that

m=0 or m=1/2

[sinful iris]

damn

[long oracle]

let me check

,w 1-m = sqrt(1-sqrt(4m^2-7m^4))

[flat frigate]

Results provided by WolframAlpha

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

Requested by .4slan

[long oracle]

dope!

[sinful iris]

yeah

tysm

[long oracle]

ur welcome!

[sinful iris]

great help

[long oracle]

:)

if you feel done, just write .closed or .solved

[sinful iris]

.solved

[safe radish]

Channel closed

Closed by @sinful iris

Use .reopen if this was a mistake.

[safe radish]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[fleet valve]

What does q35 mean?if the domain is symmetric about Origin, does it mean domain of f is x belongs to R?

[gusty trench]

not necessarily that x belongs to R, just a symmetry around 0. for example, the domain could be -3 <= x <= 3

[fleet valve]

Ty sir

.close

.close

[safe radish]

Channel closed

Closed by @fleet valve

Use .reopen if this was a mistake.

[safe radish]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[mossy gyro]

im having a bit of trouble trying to simplify the first equation at the top, can someone help point me in the right direction? I keep ending up at the same spot.

[exotic bolt]

may not be what youre looking for

but its factorable

[mossy gyro]

yea thats what i meant sorry !!

[exotic bolt]

don't try to write in terms of sine and cosine

rather try to make it one common term

with that being said

try rewriting

[mossy gyro]

should i restart from the original equation?

[exotic bolt]

yes

[mossy gyro]

okay doky

[exotic bolt]

what do you think you should rewrite?

[exotic seal]

Hello, good afternoon, any channel in Spanish for algebra help?

[mossy gyro]

maybe split secant into two terms?

since its exponent

[exotic bolt]

yes

which would be what

if you remember your pythag identites

[mossy gyro]

oh 1 + tan squared

[exotic bolt]

ye

so you have this

[mossy gyro]

ya

[exotic bolt]

which is

looks intimidating to factor at first

[mossy gyro]

hmm

theres so much stuff to take into account for !!!

[exotic bolt]

now factor this

easy right?

[mossy gyro]

(x+2)(x+1)

ya

[exotic bolt]

same principle can be applied

[mossy gyro]

omg

wait

[exotic bolt]

you could say something like u=tanx

[mossy gyro]

you are right

[exotic bolt]

which is just this

[mossy gyro]

WOOO

[exotic bolt]

so this

[exotic bolt]

ye

[mossy gyro]

thank you!!!

[exotic bolt]

anytime

it looks hard

but if you just think about it

then its just normal factoring

[mossy gyro]

haha yea

i forgot all about that stuff

i just started trig last week 😔

[exotic bolt]

all about applying what you already know

[mossy gyro]

got it !!!!

tyt!!!!

[exotic bolt]

anytime

[mossy gyro]

.close

[safe radish]

Channel closed

Closed by @mossy gyro

Use .reopen if this was a mistake.

[safe radish]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[west hedge]

Hello

[west hedge]

Switching the x and y in terms of their positions gains us the inverse of a function

what if a function is not invertible

[tardy mango]

Then you'll get the inverse relation

[west hedge]

Is there any difference?

[tardy mango]

all functions are relations

not all relations are functions

[west hedge]

I see

[tardy mango]

ex. x = y^2 is a relation, but y isn't a function of x

[west hedge]

Could you give me an example?

About inverse relation

[tardy mango]

uh

inverse of y = x^2

[west hedge]

It is x^2 = y

And the twos are symmetric with respect to the line x=y

That’s the geometric property of inverse functions!

Therefore y=x^2 and x=y^2 are inverse

Agree?

[tardy mango]

inverse relations, yes

[west hedge]

Function

x=y^2 is not a function

[tardy mango]

y is not a function of x, yes

[west hedge]

I love you

.close

[safe radish]

Channel closed

Closed by @west hedge

Use .reopen if this was a mistake.

[safe radish]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[bold zodiac]

~help

[bold zodiac]

yo i got 0 clue how to do this shit

im not gonna lie

someone tryna help :3

[finite igloo]

Okie

Apply this formula

Also, make sure u evaluate the right value ^^

[bold zodiac]

what is a and b

yeah highkey 0 clue how to use this formula lolz

[finite igloo]

Okie

So, f-1 (2) = f(1), yes?

So the derivative is 1/(f’(1))

In other words, b = 2, a = 1

Cuz inverses

[bold zodiac]

lemme just thug it out rq

[safe radish]

@bold zodiac Has your question been resolved?

Channel closed

Closed by @bold zodiac

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[pastel whale]

how is this not 8.2

[quiet plume]

Why would you expect it to be so?

[austere brook]

yeah

[timber moss]

8.2 would be the answer if the angle was 50 radians

this is 50 degrees

[austere brook]

radian mode

prob calc error

[pastel whale]

ohhh im using google calculator thats why

[austere brook]

always check ur calc mode

[bold zodiac]

shiz tragic

[bold zodiac]

this is so real

[austere brook]

most online calcs use rad

almost never degrees

[pastel whale]

i lost my calculator

[bold zodiac]

n never forget if u use a ti84 to do some exponents and you try running a calm -2^2 its gon give u -4 not 4

[austere brook]

get a new one

[bold zodiac]

lost like an hour of my life over a problem because i forgot about this n i wanted to die

[austere brook]

thats cause - is outside exponent

in calc

[bold zodiac]

yeha

use parenthesis

[austere brook]

you gotta bracket it

[bold zodiac]

howevr u spell that

[austere brook]

i am a calculator god

literally my fidget toy in class

[pastel whale]

how do i do inverse trig things on the google calculator

[austere brook]

atan

type atan

also use desmos

just type ur calculations there

https://www.desmos.com/calculator

for desmos do arctan

[pastel whale]

.close

[safe radish]

Channel closed

Closed by @pastel whale

Use .reopen if this was a mistake.

[pastel whale]

.clsose

[safe radish]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[lean otter]

is this true?

[obsidian oracle]

2|4

what about 3=2+1 and 5=4+1?

[safe radish]

@lean otter Has your question been resolved?

[lean otter]

fair enough

thanks 👍

.close

[safe radish]

Channel closed

Closed by @raven helm

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[fallen oriole]

Determine the smallest area an ellipse disc $x^2 / a^2+y^2 / b^2 \leq 1$ can have, if it is to contain the rectangle $-2 \leq x \leq 2,-1 \ leq y \leq 1$

[flat frigate]

dghf

[safe radish]

What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[fallen oriole]

1

we have two boundaries for x and y

but i don't know what to do with that information, to find the smallest area that the ellipse disc should be coontaine in the bounded region

[tardy mango]

1. What is the formula for the area of an ellipse?
2. What can you say about the vertices of the rectangle? (Especially in light of the fact that the centre of the ellipse is at the origin)

[fallen oriole]

So we have a horizontal line at y = -1, 1 and a vertical line at x = -2, 2

The formula for the area of an eclipse, isn’t that pi * r^2?

Or is there a major and minor radius?

I gave you the correct formula

It is with the major and minor radius

[tardy mango]

The way you wrote it implies those two are equal

Take that as you will

[fallen oriole]

How do I find each of these radiuses

[tardy mango]

I mean

They legit give you the equation of the ellipse

[fallen oriole]

So the coefficient is the radius of each

[tardy mango]

Recall that it’s a^2 and b^2

[fallen oriole]

So if we look for the offset from the center

Wrt a and b

We get a = +- sqrt2, and b = +- sqrt1?

[tardy mango]

The vertices of the ellipse aren’t the endpoints of the axes tho

Try drawing a diagram

[fallen oriole]

If I stretch a to equal 2, it will expand horizontally

To meet the borders of x

@tardy mango

[tardy mango]

“Contain” means the rectangle must be inscribed in the ellipse to achieve minimal area

[fallen oriole]

So what should I set a and b equal to?

[tardy mango]

Just draw a rough sketch by hand for visualization purposes

[fallen oriole]

Ok let’s say this is my rough sketch

What should I do next

[tardy mango]

“Rough sketch” as in not to scale but still capturing the main idea

Not “rough sketch” as in barely representative of the situation

[fallen oriole]

You said that the rectangle should be contained within the ellipse

This is what the sketch is showing here

[tardy mango]

….

I did also say “inscribed”

[fallen oriole]

As far as I can see, the rectangle is inside of the ellipse

If there is a difference between inscribe and inside, then I would like to know

You know what fuck this I’m not drawing a fucking sketch

.close

[safe radish]

Channel closed

Closed by @fallen oriole

Use .reopen if this was a mistake.

[safe radish]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[normal nacelle]

help

[spice furnace]

What do you not understand?

[safe radish]

@normal nacelle Has your question been resolved?

[normal nacelle]

is iii correct answer?

[spice furnace]

iii) is correct

[safe radish]

@normal nacelle Has your question been resolved?

[safe radish]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[tulip yoke]

am I able to use the second derivative to find stationary points?

[tulip yoke]

I think I misunderstood this

i’ll rewrite it

[atomic pebble]

why second derivative tho?

just bcs why not?

[tulip yoke]

• You use the second derivative to work out wether it is a local maximum or minimum

is this right?

[atomic pebble]

yeah so second derivative tells you how it curves

[tulip yoke]

I see

[tulip yoke]

and you do this when you want the coordinates?

[atomic pebble]

yes

that looks good

[drifting knot]

You can find its factors directly

Can you?

[safe radish]

@tulip yoke Has your question been resolved?

Channel closed

Closed by @tulip yoke

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[lilac delta]

How could I manipulate the inequality $n>\ln(n)$ to prove it in terms of real numbers. note $n\geq2$

[flat frigate]

lILi

[marsh walrus]

what math do you know

calculus?

[lilac delta]

I'm in calc 2 this is for a comparison test between a series of the reciprocal of a natural log of n, compared to a generic harmonic series of n

$\frac{1}{\ln(n)}>\frac{1}{n}$

[flat frigate]

lILi

[marsh walrus]

and you want to show that it diverges

sure

[lilac delta]

Yeah

[marsh walrus]

so examine f(x) = x-lnx

[lilac delta]

this is one of the steps for me finding the radius and interval of convergence for a larger series but i'm basically at the end, this is just checking the lower interval

[marsh walrus]

we want this to be positive for x>2 according to what you said

can you think of ways to do that?

[lilac delta]

oh

derivative

[marsh walrus]

sure, yea

[lilac delta]

wait

sorry i didn't get any sleep so it's kinda hampering me

so I just need to show x-lnx>0 for x>=2

[marsh walrus]

,w plot x-ln(x)

[flat frigate]

Results provided by WolframAlpha

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

Requested by janniku

[lilac delta]

yeah I was going to say you could just show that algebraically

prove that it doesn't have any roots past x=2

[marsh walrus]

yea, because if x-lnx>0, we can just add lnx to both sides

[marsh walrus]

i would say, argue f(2) is postive, and f is increasing for x>2

[lilac delta]

I see

[marsh walrus]

then f cant be negative on the interval

so, it must be positive

[lilac delta]

f(2)>0 and f'(x)>0 on x>=2

yeah

[marsh walrus]

up to neq but im sure you got that

well leq geq

[lilac delta]

$f'(x)>0,;x\geq2$

[flat frigate]

lILi

[marsh walrus]

sure yea

theres a floating edge case but it doesnt really matter

[lilac delta]

Ok so to recap, If I can show that x-lnx is positive on the given interval, then x must be larger than lnx because, if it were not, the result would be negative

[marsh walrus]

it just holds the concern that you cant compare two series that are identical

or the test isnt valid

[lilac delta]

and I can prove that it is by showing it is positive at the start of the interval, and that the derivative is positive for the rest of it

[marsh walrus]

yea

then argue ...

[lilac delta]

Alright yeah that makes sense

yeah

[marsh walrus]

[lilac delta]

Thank you

[marsh walrus]

np

[lilac delta]

.close

[safe radish]

Channel closed

Closed by @lilac delta

Use .reopen if this was a mistake.

[safe radish]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[noble dune]

i forget how do you solve for invariant points do you make the equation = +/- 1?

[safe radish]

@noble dune Has your question been resolved?

Channel closed

Closed by @noble dune

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[lean otter]

x+y+z = 63

[elfin mortar]

i think i knoew the answer

[lean otter]

x= 0.4(z+y)

[lean otter]

i dont want the answer

i want the method

[elfin mortar]

oh ok

[lean otter]

is this correct?

[elfin mortar]

mhm

oh sorry

you said not answer

[lean otter]

where did u get 45

[devout peak]

yeah and then

[lean otter]

substitute?

[devout peak]

ok

you get?

[lean otter]

0.4z+ 0.4y +y +z = 63

i still have 2 unknowns?

WAIT

[devout peak]

i think you see it right?

[lean otter]

1.4z + 1.4y =63

divide by 1.4

z+y = 45

0.4 x 45

18

damnn

[devout peak]

bingo

[lean otter]

thanks man

[fathom adder]

Gj wp

[devout peak]

nicely done

[lean otter]

.close

[safe radish]

Channel closed

Closed by @reef edge

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[storm coyote]

Currently learning triangle inequality. Stuck on this question (don’t know what inequalities to use) please help!

[rigid inlet]

,rccw

[safe radish]

@storm coyote Has your question been resolved?

[storm coyote]

<@&286206848099549185>

[storm coyote]

.close

[safe radish]

Channel closed

Closed by @storm coyote

Use .reopen if this was a mistake.

[safe radish]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[edgy mica]

this is a question on my math 142 practice test and I've been really struggling with inverses since it's an online class and would like to receive some extra help. i'm having trouble wrapping my head around converting a trigonometric function into an algebraic one

[safe radish]

@edgy mica Has your question been resolved?

[safe radish]

Channel closed

Closed due to timeout

Use .reopen if this was a mistake.

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[sinful iris]

Deduce the following formula from a^2 = b^2 + c^2

I honestly have no idea

[rigid inlet]

it'll help if you start by drawing a triangle with sides a, b, c and an angle B