help-49

[candid rampart]

Then use P3 for multiplication so

a {(1/a)[1/(1/a)]} = 1/(1/a)

And so this is a = 1/(1/a)

Does it make sense?

[tender sphinx]

let me look at it for a sec
= 1 [1/(1/a)] at this part is the one in front of the [ supposed to be there

[tender sphinx]

is the last line a =

[candid rampart]

Yeah

[tender sphinx]

so u mult the LHS of the top line by 1/(1/a)

[candid rampart]

Yes

Well, both sides

You have a(1/a) = 1 so multiply both sides times the inverse of 1/a

[tender sphinx]

ohh yeah, but why did you do that?

[candid rampart]

Why not?

[tender sphinx]

fair

[candid rampart]

So a(1/a) is a real number, do you agree

[tender sphinx]

yeah since it equals 1

[candid rampart]

So just call b = a(1/a). So when I multiply a(1/a) times the inverse of 1/a we are multiplying b times 1/a so we are multiplying two real numbers

And the reals are closed under multiplication so you can do that

[tender sphinx]

i like that way too, what does closed under mult mean?

[candid rampart]

That if you multiply two reals you still get a real

Same with addition. If you add up two reals you get a real

[tender sphinx]

oh ok, i think i saw that in the book too

well thx for the assitance, u a math major

[candid rampart]

nah

[tender sphinx]

goated
.close

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[amber dove]

So there's a rectangle with
perimeter of the base is 52 cm.
The length and the height difference is 8 cm.
And the widsth minus the height is -2cm, what is the volume of the block?

idk how to solve this

[small jasper]

Let the length be x, the width be y, and the height be z

Then, you can formulate 3 equations from the 3 pieces of given information

What are these equations?

[amber dove]

Uhmm

the volume is xyz then

x-z = 8
y-z = -2

x = z+8
y = z-2 ?

z(z+8)(z-2)

i dont get it

(z²+8z)(z-2)
z³-2z²+8z²-16z

[small jasper]

perimeter of the base is 52 cm.

don't forget this

[amber dove]

that makes x+y = 52/2 right

so x+y=26 ?

[small jasper]

so now you have $$x+y=26$$ $$x-z=8$$ $$y-z=-2$$

[grand pond]

Civil Service Pigeon

[small jasper]

and you can start solving

[small jasper]

what you did here is a good start

[amber dove]

so uhh
x = 26-y
x = 8+z
z = 2+y

i stilm dont know ehat to do in the next step though

[small jasper]

||You could've substitute these into x+y=26||

[amber dove]

Ohhh

z+8 + z-2 = 26
2z = 26-8+2
2z = 20
z = 10

y = -2+10
8?

x = 26 -8 which is 18

TYSMM

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[tight isle]

Can anyone walk me how to do this

Through

Derivative is 3x^2 + 3

Also here

I did every other part I just dunno how to find the inverse derivative

Here's an example if yall wanna see

<@&286206848099549185>

[old lintel]

Let me see

Which question are you needing help with?

You’ve got quite a few screenshots

[tight isle]

Only the second one

I figured out the first one

[old lintel]

This?

[tight isle]

This one

[tight isle]

Yep

The third screenshot is just an example for you

Of example answers

[old lintel]

Well you first set f(x)=-14

And then you can multiply both sides by 27

[tight isle]

Right

Ok

x = -3

Alr did that

[old lintel]

What

[tight isle]

And derivative is

1/27(5x^4 + 15x^2)

[old lintel]

Can you explain where did x=-3 came from..?

I didn’t catch that part

[tight isle]

Oh I set the original equation equal to 0

Wait I mean

Set it equal to -14

And solved

I used a calculator

[old lintel]

Ok

Right.

[tight isle]

Oh wait

Now all I do is plug -3 back in the derivative

Alr

And get -20

So it's 1/-20

[old lintel]

Ok

[tight isle]

Wait that's wrong nvm

Shit idk

[old lintel]

The inverse of this function is essentially the same function, but f(y)=x

[tight isle]

@old lintel

The answer is 1/20

[old lintel]

Ok

[tight isle]

Idk why tho

Can u explain why

I got -1/20

[tight isle]

Right

[old lintel]

I would say because this time you have an even number for your power

So the negatives cancelled out

That would be my guess

[tight isle]

OH

YOU'RE RIGHT*

Right

No I just did bad algebra 💀

Soery

TYSM

[old lintel]

💀

[tight isle]

LMFAO

[old lintel]

Glad to help.

[tight isle]

Bro I'm using desmos

Forgot to put parenthesis

Around -2

Lmao

[old lintel]

Right!

[tight isle]

Imma leave this open cuz I need more help with inverse functions

[old lintel]

Ok

[midnight plank]

@tight isle Has your question been resolved?

[tight isle]

Yes

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[latent seal]

please help me understand how to determine if these are negative, positive, or zero given the information. these are the correct answers but i don't get it.

[midnight plank]

@latent seal Has your question been resolved?

[midnight plank]

@latent seal Has your question been resolved?

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@latent seal Has your question been resolved?

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@latent seal Has your question been resolved?

[last slate]

hi do you still need help with this?

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[orchid gazelle]

Why is 5 subtracted here ? I didn't understand in example 7

[orchid gazelle]

I didn't quiet understand what it means by leaving remainder 5 because i did the euclid division method without subtracting the 5 and didn't get any remainder as 5 anyway

Can someone please help me if u understand

[nimble copper]

,rcw

[midnight plank]

@orchid gazelle Has your question been resolved?

[wary thorn]

did you loud understand?

[orchid gazelle]

Loud understand?

[wary thorn]

well you didn't quiet understand

[orchid gazelle]

Lol yeah i looked at a similar problem i got it

[wary thorn]

:|

[orchid gazelle]

Lmfaoo i got what u were saying now

I was so confused

[wary thorn]

okay but as for your problem

think simpler terms

what's the largest number that divides 10 and 26 leaving a remainder of 2 in each case

well

let's try 8

because obviously 10-8=2

you want something that when you divide 10 by it, it has a remainder 2

the obvious choice is 8

and for 26, the obvious choice is 24

for the same reason

[orchid gazelle]

Owhh i see i see

Thank u

[midnight plank]

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[fast elm]

im stuck in functions proof

[fast elm]

Let F:A->B, where A!=emptyset then if F has cancelation property from left, then it has left invertible, how to prove it?

[runic hamlet]

well thats false

[fast elm]

that's actually what my lecturer wrote on the board

wait im sorry so much

i updated the statement and wrote a proof if you could check

[midnight plank]

@fast elm Has your question been resolved?

[runic hamlet]

how about you actually show the proof

[midnight plank]

No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for everyone if you just ask your question! See https://dontasktoask.com/

[fast elm]

second formalling it

[fast elm]

i got stuck on it :((((((

okkk

@runic hamlet i think i got it, ill send it

User
Let f:A->B, where A not emptyset then if f has cancelation property from left, then it has left invertible: is my proof correct?
Let f:A->B, f has left cancellation proprety, we must show f has left invertible g s.t g∘f = Ida.

first at all lets prove that f is injective: let x1,x2 s.t f(x1)=f(x2), thus x1 = ida(x1) and x2=ida(x2), thus f(ida(x1))=f(ida(x2)), now because of assumption that f has canceltion property from left, idA(x1) = idA(x2), idA is injective function thus x1=x2.

now f is injective, lets prove f^-1 is a function, let x,y1,y2 s.t (x,y1) belongs to f^-1 and (x,y2) belongs to f^-1, let show y1=y2.
(y1,x) belongs to f and also (y2,x) belongs to f, f is injective so y1=y2
so f^-1 is a function so f^-1 ∘ f = IdA -> f has left invertible
QED

@runic hamlet

[runic hamlet]

why are you inserting ida in the first part

not necessary

I wouldnt call the left inverse f^-1. just call it g

[midnight plank]

@fast elm Has your question been resolved?

[fast elm]

oh

Yes g ok no problem

but is the structure fine?

or i could proof it easier

i think i overkill this topic

[runic hamlet]

its fine. what you havent shown is that g is actually a function

right now you only have a partial function

you only have pairs (y,x) if (x,y) was a pair in f

what if y never appears in f?

[midnight plank]

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[fast elm]

.reopen

[midnight plank]

✅

[fast elm]

i did prove, firstly shown that f is injective, then shown f^'1 is a function?

[runic hamlet]

you have shown its a partial function. you havent shown that you can have a total function g

[fast elm]

im not sure what that means

[midnight plank]

@fast elm Has your question been resolved?

[midnight plank]

@fast elm Has your question been resolved?

[runic hamlet]

currently g(y) is only defined when there is an x such that f(x)=y

(and then you say that g(y)=x)

but what happens with g(y) if there is no such x

g is supposed to be a function, so its supposed to do something with those y aswell

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[worn blade]

what does this notation represent and what argument should I study to be able to calculate its derivative and a related(?) delta integral?

[timid slate]

the symbol means floor of x^2

round down

[worn blade]

ok thanks. what type of math is it related to?

[timid slate]

wdym

[worn blade]

it's part of a big exam called mathematical methods so I have zero clue what to study

[timid slate]

¯_(ツ)_/¯

[worn blade]

the exercise says to take that and calculate a derivative for x>=0

and the result is supposedly a sum

I'm so confused with those things because I wasn't able to follow the class and I'm studying it all by myself

[midnight plank]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[honest kelp]

read such an example, and rest you should analyse by yourself:

[worn blade]

translation: define f(x)=floor(x^2) where floor (y) is the greatest integer <=0. i) Calculate f'(x) for x>=0 ii)Calculate integral

[midnight plank]

@worn blade Has your question been resolved?

[worn blade]

<@&286206848099549185>
can you guys pls tell me what micro-field/argument that stuff is so I know where to start?

.closed

[grand silo]

close*

[worn blade]

sorry

.close

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[sacred cypress]

why/how did the left side get factored the way it did?

[torn compass]

they took tan(x) common

[last slate]

xy-x = x(y-1)

[torn compass]

its same as
2x^2 - x = 0
x(2x-1) = 0
taking x common here

[sacred cypress]

ohhhhh that makes sense

ty

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[unborn lance]

how do you deal with data like this on a ti 84 calculator?

[unborn lance]

i dont know how my teacher got to this point on written notes

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[final wedge]

I'm confused about number 11

[final wedge]

the context in this question is that the vowels are always together

So I assume the formula goes like this? 7!/3!

[cedar coral]

when it says "the vowels", does it mean all 3 vowels together?

If so:
HEXAGON
EAO
HXGN
I've divided the word in two sets. The set of vowels, and the set of consonants

now:
You know that all three vowels can be in any order. How many ways can you order three vowels? -I'm calling this a
You know that the four consonants can be in any order. How many ways can you order four consonants? - i'm calling this b
The block of vowels has to be either in between any two consonants (3 positions) or at the start or end (another two positions.
So you will have a*b*5 ways

[final wedge]

I guess so

[cedar coral]

can you get a and b?

[final wedge]

I got 720

Btw where did you get 5?

Im confused about the number 5

I wonder where you got that

[midnight plank]

@final wedge Has your question been resolved?

[old lintel]

I would say 5 is different ways you place the vowels

You have 4 none vowels, so there’s 5 ways to place the chain of vowels.

[midnight plank]

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[iron monolith]

$\int \sin^{-1}(x) \dd{x}$

[grand pond]

_Ω_

[iron monolith]

Im having some trouble with this equation

I found the answer but cant seem to find out how they got the answer

[raven abyss]

Whats the answer

[iron monolith]

$\sqrt{1-x^2} +xsin^{-1}(x)$

[grand pond]

_Ω_

[raven abyss]

👍ok

[iron monolith]

i get the xsin^-1 part

[worn halo]

Integration by parts?

[atomic magnet]

Integration by parts formula

Yeah

[iron monolith]

$xsin^{-1}(x) - \int \frac{1}{\sqrt{1-x^2}}*x \dd{x}$

[grand pond]

_Ω_

[light beacon]

substitution

[iron monolith]

so u = 1-x^2

[worn halo]

Yes

[iron monolith]

and du would be -1x dx

[worn halo]

-2x dx

[iron monolith]

yeah

but i mean shouldn't you also get the right answer by integrating by parts again?

[worn halo]

Yeah, but why would you?

IBP again is overcomplicating things

this integral is just u-sub + power rule

[iron monolith]

at first i didn't think of u sub and i tried to do it again but got a wrong answer. I think i just need some more training with u sub

[worn halo]

Yeah, practice is the best thing to get better at spotting u-sub

[iron monolith]

yeah i know, anyways thanks everyone

.close

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[timid tangle]

Hello

[timid tangle]

How can I visualize this?

That D_vector u f(vector a) is referring to directional derivative

From where is the angle coming from?

Where is the starting point of that angle?

[wary thorn]

recall that

the directional derivative in the direction of u

is the same as

the gradient dotted with u

right?

[timid tangle]

Yes

[wary thorn]

$$D_{\vec{u}}f(\vec{a})=\nabla f(\vec{a})\cdot \vec{u}$$

[grand pond]

Austin

[timid tangle]

That's some fast edit 😎

[wary thorn]

now if we are considering, where is this the largest or smallest

take the magnitude of both sides

$$||\nabla f(\vec{a})\cdot \vec{u}||$$

[grand pond]

Austin

[timid tangle]

wdym largest or smallest?

[wary thorn]

I thought you were asking about this

[timid tangle]

Oh um

I was asking about where exactly was theta

[wary thorn]

We'll get to that

[timid tangle]

Okay okay

[wary thorn]

theta comes in in the next step

do you know any formulas for $$||u\cdot v||$$

[grand pond]

Austin

[wary thorn]

because there's one key formula that they use

[timid tangle]

||a||||b||•cos(theta)?

[timid tangle]

how do I stop spoilers

[wary thorn]

put a \

before one of the |

[timid tangle]

Ahh

Ummm what next?

[wary thorn]

well earlier we described how the directional derivative in the direction of u was this

so now apply the formula above

[timid tangle]

Ohhh since they are both vectors

[wary thorn]

yes

[timid tangle]

Ohhhhh wait

Does that mean the angle is referring to the angle between nabla and u?

nabla I mean gradient

[wary thorn]

yes

[timid tangle]

Ohhh

I see I see

Tysm

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[twilit rampart]

i have to solve the following differential equation : $\frac{dH}{dt} = rH - \tau$

[grand pond]

LF

[twilit rampart]

i know its a separable equation but something confuses me

at some point after integration we arrive at $|rH - \tau| = Ae^{rt}$

[grand pond]

LF

[twilit rampart]

the solution from the manual (and wolfram alpha) seems to complety ignore the absolute value

what is the justification for that?

[honest kelp]

the constant A assimilates it

Sometimes, in books we write, A with tilda, or another symbol to show it

[twilit rampart]

but since H depends on t, couldnt the sign of rH - tau change at some point ?

making A not truly constant

[twilit rampart]

is that what the tilda means (that its only constant for some interval)

[honest kelp]

here, A is constant, i personally write big C

[twilit rampart]

right but depending on the sign of $rH(t) - \tau$ youd have $H(t) = \frac{\tau}{r} \pm Ae^{rt}$

[grand pond]

LF

[honest kelp]

if you ve got IVP, some initial condtion, that helps to make solution better, but in generalsolution we do nto worry ab constants

H is funciton of t

not a

[twilit rampart]

i guess my point is more that A is a function of t

[honest kelp]

it is not, i can show you my solution of yoru equation if you like

but you did well

[twilit rampart]

thats nice but im okay i think

its just the absolute value that worries me

[twilit rampart]

but what if rH-tau was negative?

[honest kelp]

A is a constant, that assimialtes all pluses minuses and eventually other constans,

yoru A = +-A

yoru A eats plus minus

informally saying

[twilit rampart]

yeah i see what you mean but my problem is that whether its plus or minus depends on t

[honest kelp]

maybe, you have more info about r and tau

then it can help you to select

[twilit rampart]

yeah i have initial conditions so it fixes everything

i was just wondering what to write if i didnt have this info

[honest kelp]

so assume, there is an interval of t, where H > tau/r

then you take " plus"

[twilit rampart]

yes okay thank you

well put

so im not looking for a global solution

just a local one

right?

[honest kelp]

ok

[twilit rampart]

alright thank you very much

[honest kelp]

yw

[twilit rampart]

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[simple ivy]

quick question, is 3/x^2 + 5x - 1 = 0 a quadratic equation?

[simple ivy]

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[prime hornet]

!status

[midnight plank]

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

[prime hornet]

2 for a,b,c and 1 for d

i've gotten to basically the same point for a,b and c

but i'm a bit confused on what to do about the absolute values

as for d, i havent really much of an idea

[olive matrix]

everyone ignores the absolute value on these which good job for remembering them

[prime hornet]

issue is that i'm not sure if i'm supposed to ignore them or not

[olive matrix]

but yeah without bounds you can just make the assumption that tanθ > 0

or whatever

[prime hornet]

sec and cos as well?

also, what about d?

[olive matrix]

yeah, you can write the assumption to the side if you're uneasy about it

[olive matrix]

it tells you the sub right? try that

oh it's not immediately clear how to do that huh

[prime hornet]

i briefly tried, but i cant see where i'm really going with u = tan(x/2) tbh

[prime hornet]

okay, will do!

[olive matrix]

what's problem 19.11? have you looked at that?

[prime hornet]

and now i have to find 18.10 ugh

[olive matrix]

hmm that doesn't look super relevant

,tex .half angle

[grand pond]

ヘイリー

[prime hornet]

we have not yet covered the hyperbolic trig functions yet either

[olive matrix]

apparently $\tan(\sfrac x2) = \f{\sin x}{1+\cos x}$?

[grand pond]

ヘイリー

[prime hornet]

ngl i've never seen this identity before, but i suppose it's convienient for this huh?

[olive matrix]

yeah prove it for yourself (it's not too far from the one I called up earlier) but yes v convenient

[prime hornet]

got it

damn i hate the fact that i dont know the half-angle identities well

completely forgot them lol

[olive matrix]

tbh I don't know them either

I only know the double angle ones for sin and cos, and cos is kind of a guess

[midnight plank]

@prime hornet Has your question been resolved?

[midnight plank]

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[weary crescent]

how would i start this?

[lilac sage]

arc length of curve formula

[weary crescent]

there's no dy or dx though

[lilac sage]

x = rcos(theta)
y = rsin(theta)

can use this to derive the arc length for polar formula

[weary crescent]

so is it just x=theta(cos(theta))

[midnight plank]

@weary crescent Has your question been resolved?

[midnight plank]

@weary crescent Has your question been resolved?

[weary crescent]

<@&286206848099549185>

[weary crescent]

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[pastel tree]

Find the deriviate to $\frac{1}{\sqrt{x}}$

[grand pond]

Merineth

[pastel tree]

I rewrote 1/sqrt(x) to

x^(-1/2)

and then did the normal proceedure of

-1/2x^(-3/2)

which i then rewrote too $\frac{-x^{-3/2}}{2}}}$

[grand pond]

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

[pastel tree]

... ^ that

and then i moved the x down to the denominator

and swapped - to +

$\frac{-1}{2x^{3/2}}$

[grand pond]

Merineth

[pastel tree]

Is it possible now to rewrite x^3/2?

[vague remnant]

x√x

Is it necessary

[pastel tree]

I don't think so but it'd be nice to know how

How did you know that xsqrt(x) = x^3/2

i know that x^1/2 = sqrt(x)

[vague remnant]

Well x is 1 and √x is 1/2 so 1+1/2 is 3/2

I mean powers

[pastel tree]

aaah lol

Think i'll still get full points for?

[pastel tree]

that?

[vague remnant]

If your teacher said that it is necessary

To rewrite

Ig

[pastel tree]

Alright i'll make sure to remember/learn it

tysm

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[autumn canopy]

Sometimes also written as $\sqrt{x^3}$, depends on the preferences of your class.

[grand pond]

Kepe

[autumn canopy]

My teacher would accept x^(3/2) without problems

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[pastel tree]

$e^{-x^{2}}$ Find the derivative

[grand pond]

Merineth

[pastel tree]

Got any advice how to handle this derivative?

make u = -x^2 ?

[severe cloud]

You‘ll need the chain rule

[pastel tree]

Yes, but where does the 2 go?

infront of the -x or e?

[severe cloud]

I‘m not sure I get what you mean. Do you mean after taking the derivative?

[pastel tree]

Well the chainrule states that i find the deriviate from the most outer into inner multiplied with each other, right?

and the most outer is 2, then -x and lastly e

[severe cloud]

!nosol

Dsmn that doesn‘t exist anymore

[worthy wing]

Looks like now it’s ok to give sol

[pastel tree]

I already knew the solution however i want to learn the process

[fathom egret]

i think giving the solution is useful since it shows you what to do

[pastel tree]

So ideally i want to use substitution method here and not chain rule?

[fathom egret]

you can try similar problems using the same idea

but seeing it once helps

[severe cloud]

If you sub you‘ll need chain rule in the end because u is a function of x

But you can sub if it helps you

[pastel tree]

Okay so first i rewrite it as $e^{u}$

wtf is wrong with the bot today

[sacred galleon]

guys

I have a question

[grand pond]

Merineth

[severe cloud]

? It‘s doing fine

[pastel tree]

looks messed up for me

[severe cloud]

Please get your own channel under the "available channels" category

[pastel tree]

!occupied

[midnight plank]

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

[severe cloud]

Oh? It looks fine for me at least

[sacred galleon]

How do I get my channel?

[pastel tree]

!occupied

[midnight plank]

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

[severe cloud]

Scroll a bit up and you‘ll find channels called "help-X"

Where X is a number

Those are available

[sacred galleon]

I don't have a channel for this thing

[pastel tree]

PLEASE

[pastel tree]

!occupied

[midnight plank]

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

[pastel tree]

read it

#❓how-to-get-help #❓how-to-get-help #❓how-to-get-help

[severe cloud]

There‘s no need to spam it multiple times. But yes, go to this channel

[pastel tree]

He isn't reading it?

[sacred galleon]

Okay, I'm sorry for the inconvenience

[severe cloud]

You can resume Merineth

[pastel tree]

Anyhow, i substitute u = -x^2

What i don't get is that e^u = e^u unless there is a coefficient infront of the x

$e^{-x^{2}} = e^{u}$ where u = -x^{2} and u' is -2x

[grand pond]

Merineth

$e^{-x^{2}} = e^{u}$ where u = -x^{2} and u' is -2x
```Compilation error:```! Missing $ inserted.
<inserted text> 
                $
l.49 $e^{-x^{2}} = e^{u}$ where u = -x^
                                       {2} and u' is -2x
I've inserted a begin-math/end-math symbol since I think
you left one out. Proceed, with fingers crossed.```

[severe cloud]

as per chain rule, $\dd{x}e^{u(x)} = e^{u(x)} \cdot u^{\prime}(x)$

[grand pond]

ℑμΤ𝛄𝛗θ

[pastel tree]

aaaah

that makes it a lot clearer

[severe cloud]

Wrong differential symbol but you get the idea

[pastel tree]

i use chain rule on u(x) first and then deriviate e^u'(x)

Okay that wasn't too hard !

[severe cloud]

So you got the same answer as the other guy esrlier?

[pastel tree]

yes!

I don't really like the way we write dy/dx = dy/du etc

as that isn't sosmething i've remembered or learned

but seeing it like you wrote it as e^u(x)

makes it a lot clearer

tysm!

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[chrome hill]

e^x is convergent for all values of x?

[midnight plank]

Please don't occupy multiple help channels.

[quiet flame]

Yeah

[chrome hill]

Thanks

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[pastel tree]

What is the derivative of $x^{x}$

[grand pond]

Merineth

What is the derivative of $x^{x}$

[last slate]

well you have two options really

implicit differentiation or using the chain rule via taking the logarithm and exponential of the function

which one do you want to use

[pastel tree]

chain rule i think

[last slate]

okay then

[pastel tree]

so you mean i use the rule for a^x first and then deriviate of x^a?

[last slate]

note that [
\m fx = e^{\m\ln{\m fx}}
]

[last slate]

so do that

[pastel tree]

I have no idea what to do

$y = x^x \implies ln(y) = ln(x^x) \implies ln(y) = xln(x)$

Is this fine?

xln(x) isn't to hard to deriviate, but what about ln(y)

that doesn't become 1/y, right?

[tribal temple]

If middle is ln(y) then yea

[pastel tree]

I remember something with "respect to x"?

[pastel tree]

yeah my bad

[grand pond]

Merineth

[limpid yoke]

just differentiate e^xlnx

[tribal temple]

Almost though(!) remember implicit differentiation

[pastel tree]

i don't :c

was so long ago

[last slate]

take the exponential of both sides then

and then its chain rule

[tribal temple]

Tl;dr if you differentiate a function of y with respect to x, multiply the result by dy/dx

[pastel tree]

How does it work?

multiply the result by dy/dx

i don't even remember what that means

[tribal temple]

Alright then implicit differentiation another time

[tribal temple]

See here

[pastel tree]

why would i do that?

[tribal temple]

That’s what you get when you do this

[limpid yoke]

because u say u want chain rule

[pastel tree]

I mainly want to know how to solve it

but if

ln(y) doesn't become

1/y

then

idk what to do

and i need help showing it

I don't understand or rembmer how dy/dx or w/e that is means

[tribal temple]

basically “the derivative of y with respect to x”, you may have seen it denoted as y’ maybe?

Like if you differentiated y = x^2 with respect to x, that turns into dy/dx = y’ = 2x

Anyways, if you took logs of both sides that’s an implicit differentiation problem

[pastel tree]

So i just multiply both sides with 1/y?

[last slate]

y*

[pastel tree]

implicit differentials is when i make it into an equation?

[last slate]

that's very badly worded

[pastel tree]

I don't speak math

[last slate]

implicit differentiation is basically like chain rule

if you remember the chain rule states that like

the derivative of the "outer function" multiplied by the "inner function" gives you the derivative of your entire function

[pastel tree]

Yepp

[last slate]

so like when you have an implicit y, like with ln(y)

you are dealing with like

,, \m f{\m yx} = \m {f'}{\m yx} \cd\m {y'}x

[last slate]

which is what happened here, just replace f with ln

[pastel tree]

Like i'm sorry but i can't learn this way

[wide olive]

Then learn both ways

[pastel tree]

$ln(y) = xln(x) \implies ln(y) = ln(x) + 1$

[grand pond]

Merineth

[pastel tree]

what happens with the LHS

1/y?

[last slate]

you apply the chain rule as is done here to get $\ds \f1y \cd \dv[y]x$

[pastel tree]

so i take 1/y * (ln(x) + 1)

[last slate]

no

[pastel tree]

Then waht do i multiply it with then?

[last slate]

,align
\f1y\cd \dv[y]x &=\m\ln x + 1 \
\dv[y]x &= {\c b{???}}(\m\ln x +1)

[last slate]

what is ??? do you think

[pastel tree]

1/y would be my guess

[wide olive]

substitute $\dv{y}{x} = t$ for any variable

[grand pond]

Bishop

[wide olive]

treat like a real equation

$\frac{1}{y} \cdot t = ln(x) + 1$

[grand pond]

Bishop

[pastel tree]

y

[last slate]

,tex but if you did that you would get
\env{align*}{
\f1y \cd \dv[y]x &= \m\ln x +1\
\c b{\f1y \cd} \f1y \cd \dv[y]x &= \c b{\f1y \cd}(\m\ln x +1) \
\f1{y^2}\cd \dv[y]x &= \f1y \cd(\m\ln x +1)
}

[wide olive]

yes

[pastel tree]

So when it says "with respective to x" and we are not dealing with x we just multiply with dy/dx and solve it like an equation?

[wide olive]

wha

[pastel tree]

and then substitute y with x^x

[wide olive]

arent you trying to find dy/dx?

if so then multiplying by y basically makes you "find" it

[pastel tree]

But isn't how i worded it exactly what we are doing?

$ln(y) = xln(x) \
\frac{1}{y} * \frac{dy}{dx} = ln(x) + 1 \
\frac{dy}{dx} = y(ln(x)+1) \
\frac{dy}{dx} = x^x(ln(x)+1) \
y' = x^x(ln(x)+1)$

[grand pond]

Merineth

[tribal temple]

If you mean you have some function of y, and you’re differentiating that wrt x, basically yea

So for example say a circle, x^2 + y^2 = 1, if you differentiated y^2 with respect to x, that’s 2y * dy/dx, so that would turn into 2x + 2y(dy/dx) = 0, and you solve for dy/dx

[tribal temple]

And as here we knew what y was explicitly, we can put it back yep

[pastel tree]

Yeah that makes sense

So basically finding the derivative of y and then multiplying it with dy/dx, and treat it as an equation

so for example if i had

$x^3 +y^3 = 6xy$

[grand pond]

Merineth

[pastel tree]

It would become

$3x^2 + 3y^2 * dy/dx =$

[grand pond]

Merineth

[pastel tree]

Not sure what 6xy would become however

can i treat it as 6x * y where f(x) = 6x and g(x) = y?

Alright thinking that way makes it a lot easier

Some people forget however how hard math is I can't keep up with math language ;-;

[tribal temple]

Awww

[pastel tree]

quick question

if i have

$\frac{x^{-1/2}}{2}$

[grand pond]

Merineth

[pastel tree]

And i move down the x in the denominator

does it become + or * between the 2 and sqrtx?

[limpid yoke]

u get
1/2*sqrtx

[pastel tree]

Ok!

[tribal temple]

[pastel tree]

1/(2*sqrt(x))

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[lunar ocean]

...there are?

you might not normally say it with decimals but temperatures like 50.2 F are physically possible

(in fact the conversion factor between C and F is not an integer, which would be kind of weird if temperatures were discrete)

[midnight plank]

@civic drift Has your question been resolved?

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[cerulean linden]

Hey guys I just want to know if my homework answers are good and having trouble with some questions:

for this:

his pub has the following selection of drinks:
• 5 different types of craft beer, B1, B2, B3, B4, B5,
• 4 different types of cider, C1, C2, C3, C4, and
• 3 different types of soda, S1, S2, S3

Over the course of the evening you have 5 drinks. In each of the following scenarios,
count the number of possible drink combinations (the order of the drinks matters):
a) You have 5 drinks of any type and drinks may be repeated.

for a I put 12^5 which im pretty sure is right

[cerulean linden]

You have 5 drinks of any type, but do not have the same drink twice in a row.

for this i did 12 x 11 x 11 x 11 x 11

c) You have 5 drinks, but limit yourself to exactly 1 beer and exactly 1 cider.
for this i dont know? any1 can help me for this one

You have 5 drinks and at least 1 of the drinks is a soda.
i believe im right for this one: 3 × 12 × 12 × 12 × 12

[violet storm]

a, b seem right

[violet storm]

wouldn't this be 2^5? or does it imply that you HAVE to have both of them at least once? the wording is a bit ambiguous?

or does it mean that you are only limited in those categories but can have freely of the other categories? dunno, i feel like its very ambiguous

[cerulean linden]

i feel like it could be 1 * 1 * 12 * 12 * 12?

or no

or maybe 1 * 1 * 10 * 10 * 10?

[violet storm]

the way im interpreting (most likely) is that you are only allowed two types of drink (one beer and one cider) but it is very poorly worded and ambiguous, i could make arguments for like 3 different interpretations of this

[violet storm]

it doesn't say that the restrictions differ from drink to drink though, so i wouldn't interpret it that way

[midnight plank]

@cerulean linden Has your question been resolved?

[midnight plank]

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[upbeat plover]

im kinda confused

for this part

like where did they gt the 4 from

ok wait

never mind

😭

.close

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[midnight plank]

@chrome hill Has your question been resolved?

[chrome hill]

Should I have to use cauchy root theorem?

[fallow scarab]

Sure try it

[chrome hill]

I got xe

So for convergent it should be <1

Option C?

Should I have to check at boundary poimts?

@fallow scarab

Sry for unnecessary ping

[fallow scarab]

!show

[midnight plank]

Show your work, and if possible, explain where you are stuck.

[chrome hill]

@fallow scarab

I used ratio test

[fallow scarab]

Yea that looks right

You didn't finish the ratio test though

[midnight plank]

@chrome hill Has your question been resolved?

[chrome hill]

XE<1

X<1/E?

[fallow scarab]

No. Missing absolute value bars

[chrome hill]

.close

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[mellow tapir]

so I seperate the expression into 3 parts

8th root of x^9, y^8, and z^16

for x^9, it is x^8 * x^1 the x^8 comes out

[round palm]

yes

[mellow tapir]

so its x 8throotx

y^8 and 8th root cancel

and z^16 is gonna be z^2

so u combine to become

z^2 y x 8throotx

but where tf does the absolute sign come from

is it because it perfectly cancels out?

I got it right cuz my answer was the same as number 3

but idk why the absolute sign is needed

[round palm]

,, \sqrt{x^2} = \abs{x}

[grand pond]

kanna

[mellow tapir]

so yea?

[burnt flame]

well if it were $\sqrt[3]{y^3}$ it wouldn't have an absolute value

[grand pond]

🫎 ムースィー (Moosey) 🫎

[mellow tapir]

cuz its odd?

[burnt flame]

yes

[mellow tapir]

so if x, y, and z all were squared to 8

then it would be absolute value of all 3 variables?

[burnt flame]

mhm

[mellow tapir]

aight appreacite it

.close

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[fallow cradle]

Can u pls help me with this poly question

I don’t understand b

[wintry cave]

P(x) has only 1 real root (which is 1) ==> the equation x² + kx + 1 = 0 does not have any real roots

which means that the discriminant is strictly negative

[fallow cradle]

Wait how is it strictly negative?

[fallow cradle]

Oh yeah

[wintry cave]

cause P(x) has only 1 real root which is the root of the polynomial (x-1) so the other quadratic have no real roots

[fallow cradle]

So do I make am range for what values it could be

[wintry cave]

calculate the discriminant of the quadratic (in terms of k)

then solve the inequality: discriminant < 0

[fallow cradle]

Ok Tysm

Yeah it worked Ty

[wintry cave]

np

use .close to close the channel

[fallow cradle]

.close

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[vernal fox]

A basis of module M_R is a generating set E under the group operation of M.
Is there a name for a set S, such that M_R = R * S, i.e. without involving the group operation?

Motivating example:

• Ring R = Z/6Z.
• Group (A, +), where A = R^2, + is component-wise ring operation.
• Module M_R is a submodule of A_R, M_R has a basis E = {(3, 0), (2, 1)}.
• Module M_R = R * S, where S = {(0, 0), (1, 2), (2, 1), (5, 1)}.

[midnight plank]

@vernal fox Has your question been resolved?

[vernal fox]

Maybe <@&286206848099549185>?

[midnight plank]

@vernal fox Has your question been resolved?

[fresh sparrow]

try #advanced-algebra

[vernal fox]

Thanks! Sorry for choosing a wrong channel.

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[cosmic cobalt]

4.1 Exponential and Logarithmic Derivatives

[cosmic cobalt]

how can i solve this

i forgot what as limit goes to infinity mean

[olive matrix]

it means as x gets bigger and bigger

[cosmic cobalt]

so 13 is inf?

because e^x gets bigger faster

[green marlin]

Yes

[olive matrix]

yea

[cosmic cobalt]

thanks

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[weak cairn]

L'hospital chad

[sage helm]

the users of lhospitals rule in a situation where there exist other methods are of no substance

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[hallow mauve]

can someone help me with this question pls?

[hallow mauve]

idk what a reflection is

[shy anvil]

it should make a reflection when you draw it

[hallow mauve]

?

[shy anvil]

so X value has to be opposite than it was

so it should be negative

[hallow mauve]

WDYM

[shy anvil]

let me send you pics hold on

[slender walrus]

similar to mirror symmetry

[hallow mauve]

k

[shy anvil]

thats what i understood

[shy anvil]

yes, thank you for explanation

[hallow mauve]

oh so basically... does it mean that the reflected line is just opposite of the original line

[shy anvil]

yes, kind of

you can think about it like a mirror

[hallow mauve]

alright, how do you do the question i sent in the beginning

[shy anvil]

just add negative to the x part

so its -2x+4

which is also 4-2x

[shy anvil]

btw, may i know what grade you are on ?

[hallow mauve]

9

[shy anvil]

yes, thats exactly how to do it

[shy anvil]

ahh ok

so . . .

we are done right ?

[hallow mauve]

ye, tysm

.close

[shy anvil]

okii

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[shy anvil]

i wanted to do that XD

good luck

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[last slate]

steps i took

wait let me try this again actually

.close

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[cinder path]

can someone fix it ?

[olive matrix]

where did 6S0 come from on line 3

[cinder path]

the triple of the same population no ?

[olive matrix]

wait i'm confused

[cinder path]

Its 3So or

6So

[olive matrix]

what

what's f(x)

f(x) is the population after x months

[cinder path]

yeah

[olive matrix]

S0 is the population after 0 months

[cinder path]

So is the initiale population

yeah

[olive matrix]

so if you want the population to triple

you want f(x) = 3S0 right?

[cinder path]

yeah

[olive matrix]

so um

start with that

and solve for x

[cinder path]

so x=3ln3/ln2

[olive matrix]

yes which is equal to $3 \log_23$

[grand pond]

ヘイリー

[olive matrix]

$\underbrace{3}_{\text{every 3 months}} \underset{\text{doubles}}{\log_2} \underset{\text{to triple}}{3}$

[cinder path]

its the same as this N?

[grand pond]

ヘイリー

[olive matrix]

yeah change-of-base

$\frac{\log a}{\log b} = \log_ba$

[grand pond]

ヘイリー

[cinder path]

ok thank you

[midnight plank]

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[cinder path]

i have a question

[cinder path]

That's the interest formula for N per year

But how can I put it in semi annually?

[tulip token]

can you type out the formula or explain what you mean by "N per year"?

[cinder path]

no wait

[tulip token]

the compound interest formula for t periods at an interest rate r is P(t) = P_0(1+r)^t

[cinder path]

yes

this formula

how i put in semi annually ?

t/2 ?

[tulip token]

I can't read the letters in the particular formula you were taught

[cinder path]

yeah go with your formula

[tulip token]

in the formula I gave, "t periods" means how many times it's compounded and "interest rate r" is the interest rate per period

so if it were compounded semiannually for 5 years with an annualized interest rate of 10%, r would be 5% and t would be 10

[cinder path]

for me t was the year

[tulip token]

if you define t as the number of years and N as the number of compounding periods within each year, and r as the annual interest rate, you can get an equivalent formula:
P(t) = P_0 (1+r/N)^(tN), but it shouldn't be hard to see this is exactly the same thing as I gave

in any case, you probably shouldn't be trying to memorize the formula

[midnight plank]

@cinder path Has your question been resolved?

[midnight plank]

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[worn ocean]

x=????

[midnight plank]

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[sharp matrix]

hey! i have a question about how to solve this:

6x^2+24x-126

i get to 6(x^2+7x), but according to the answer key i have to factor it to 6x(x+7). it’s late here so sorry if this is a dumb question, but how is 6x a factor of 7x?

[tulip token]

6x^2 + 24x -126 is not equal to 6(x^2 + 7x)

do you have the right problem?

[midnight plank]

@sharp matrix Has your question been resolved?

[sharp matrix]

it’s equal to 6(x^2 + 7x - 3x - 21) and i factored by grouping and i got 6(x^2 + 7x) for one side

[tulip token]

oh okay

so are you asking why 6(x^2 + 7x) = 6x(x+7)?

[sharp matrix]

yep basically

[tulip token]

do you agree that x(x+7) = (x^2+7x)?

[sharp matrix]

yeah

[tulip token]

so if we multiply both sides by 6

we get 6x(x+7) = 6(x^2 + 7x)

[sharp matrix]

oh i see

i get it now ty

have a good day

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[neat sandal]

why is it that in question 3 the defined functions are connected with the limits

but in question 4 they are separate and hole discontinuities

[junior flower]

the conditions $\lim_{x\to 0} f(x) = 4$ and $f(0) = 5$ force a discontinuity

[grand pond]

chmonkey #1 simp

[junior flower]

you could throw a discontinuity somewhere in the graph in 3 and still satisfy the other conditions but it's unnecessary

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@neat sandal Has your question been resolved?

[midnight plank]

@neat sandal Has your question been resolved?

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[slender scroll]

hi

[midnight plank]

@slender scroll Has your question been resolved?

[fallow scarab]

.close

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[little dagger]

how do I do this?

[polar crystal]

.open

[polar crystal]

does that say sinthetha costhetha and tantheta?

it's kinda blurry bro

[last slate]

\env{alignat*}{{2}
\m\sin\theta &= \f{\t{opp.}}{\t{hyp.}} & \q \m\csc\theta &= \f{\t{hyp.}}{\t{opp.}} \\
\m\cos\theta &= \f{\t{adj.}}{\t{hyp.}} & \q \m\sec\theta &= \f{\t{hyp.}}{\t{adj.}} \\
\m\tan\theta &= \f{\t{opp.}}{\t{adj.}} & \q \m\cot\theta &= \f{\t{adj.}}{\t{opp.}} 
}

[last slate]

you should use this

[polar crystal]

smart

[last slate]

do you know how to relate the sides of your triangles

[polar crystal]

the scsc sec and cot functions are just inverses btw

[little dagger]

i always mix the opp and adj

like how do i know which goes where