help-26

[calm jackal]

So how do I handle line 1

[opal vault]

start back from line 1, and regroup the terms in 3^k

[calm jackal]

How

[opal vault]

like how you regroup 7x + 6x

[calm jackal]

It has a 2(3) dangling on the other one

[opal vault]

2 * 3^k

[calm jackal]

How do I add $3^k and 2(3)^k$ altogether

[thorny flame]

ICANTCOMEUPWITHANAME

[opal vault]

i just told you, $2(3)^k = 2\cdot 3^k$

[thorny flame]

rafilou2003

[calm jackal]

How does that help though?

[verbal crater]

imagine 3^k as a variable like t

now i ask you, whats 2t + t

[calm jackal]

3t

[verbal crater]

so 2(3)^k + 3^k would be?

[calm jackal]

2(4)^k

[verbal crater]

what

[opal vault]

Let's write $X = 3^k$

[thorny flame]

rafilou2003

[opal vault]

$2X + X = ?$

[thorny flame]

rafilou2003

[calm jackal]

$2(6)^k$

[thorny flame]

ICANTCOMEUPWITHANAME

[opal vault]

2+1 =

[calm jackal]

3 bro

[opal vault]

2X + X =

[calm jackal]

3x

[opal vault]

2(3^k) + 3^k =

[calm jackal]

$3(3^k)

[opal vault]

yes

[verbal crater]

there you go

[calm jackal]

I did not know you could do that

So how do I make this $3(3^k) - 1 = 3^{k + 1} - 1$ equal?

[thorny flame]

ICANTCOMEUPWITHANAME

[verbal crater]

whats 3 * 3^k

exponent rules

[calm jackal]

Oh.

Its like the same as 3 x 3 x 3 x 3

[opal vault]

uhhuh

And so if you multiply by ANOTHER 3, how does that change the exponent?

for example 3 x 3 x 3 x 3 is 3^4. so if you multiply by one more 3, you get 3 x 3 x 3 x 3 x 3 which is 3^5

how did the exponent change?

[calm jackal]

By + 1

[opal vault]

yep

So applying what you just said, 3 * 3^k = ?

[calm jackal]

$3^{k+1}$

[thorny flame]

ICANTCOMEUPWITHANAME

[opal vault]

you did it

[calm jackal]

I graduated

🤧

[opal vault]

top of the class mathematics discord

[calm jackal]

🫠

[opal vault]

Alright I'll be going, take care

[calm jackal]

Tganms

Thanks

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[drifting patrol]

How to prove what I can't drow 5 squares inside one square

[marble gate]

you'll need to elaborate a bit more

because you can easily draw 5 squares inside one square if the one square is 1m x 1m and the little squares are 1cm x 1cm

[drifting patrol]

Not equal ones

[marble gate]

...you cannot fit 5 squares with side length n inside 1 square with side length n?

[drifting patrol]

Lemy check problem again

[ivory night]

I think its that you can't perfectly pack 5 squares with side length n inside 1 square with side length m, where m>n

[drifting patrol]

Problem is to prove what you can't fit 5 not equal squares in one big square with area of 5

[hollow bone]

sznd the exact wording

even if its not in english

[steep ibex]

Have you tried a proof by contradiction? Assuming you CAN?

[drifting patrol]

What's that

[ivory night]

@drifting patrol

[steep ibex]

Let there be a square of A=n. Assume you can fit 5 equal squares inside it.

I'm not saying this is the way to go, as I am taking the same class. But if a teacher gave me this problem, it's one direction I would attempt.

But this way, you would prove that 5 equal squares do not fit.

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[cosmic plaza]

Hello guys. I am stuck on integrating this

[cosmic plaza]

I am pretty sure I have to integrate by part but how to do so in the case when we have a substitution I dont know

[twin island]

that x^4 in the last integral is u

so your integrand is u e^u and you can integrate by parts

[cosmic plaza]

Oh wait so integration by parts U is the same u substitute ?

[twin island]

I didn't understand you question

[cosmic plaza]

Basically the integration by parts letter “u” in uv-… is the same as the u we use in substitutions?

[twin island]

you mean the letter "u" in the integration by parts as written like in this image?

[hollow bone]

why is there still an x after the change of variable?

[twin island]

For integration by parts I'm used to this notation with functions f and g and there is no "u" (which is the arbitrary letter you decided to use in the substitution)

[cosmic plaza]

Yes

[cosmic plaza]

Ah explains it

Thank you

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[burnt mountain]

I have this equation but I dont know if its true

[neon iron]

use the definition of the absolute value function

[burnt mountain]

,rotate

[burnt mountain]

Something like this

Or the -x² in the bottom function should be x²

[neon iron]

To solve the functionf(x) = x^{2}+x x>0 -x^{2}-x{if } x<0in a shorter way, you can simplify it by combining the two cases into a single expression.

For x > 0:
\( f(x) = x^{2}+x \)

For x < 0:
\( f(x) = -x^{2}-x \)

By combining these two cases, we can write the simplified function as:
\( f(x) = x^{2}+x \) for all values of x.

So, the simplified form of the function is \( f(x) = x^{2}+x \) for all x.

[burnt mountain]

Thanks

After derivating I notice that the signs are the same

So my initial equation is true

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[tidal flicker]

Can anyone help with this please ?

[weary umbra]

try to differentiate

[topaz sinew]

@tidal flicker Has your question been resolved?

[tidal flicker]

what about this one ?

[smoky sparrow]

same, differentiate using the chain rule for 4a)

e is a constant so differentiates to 0

for b use implicit differentiation

e.g d/dx (y^3) = 3y^2 * d/dx (y) = 3y^2 dy/dx by chain rule

and then rearrange for dy/dx

[tidal flicker]

thanks brother ❤️

what about this one ?

[smoky sparrow]

logarithmic differentiation means to take ln of both sides

so d/dx ln(f(x)) = f'(x)/f(x) again by chain rule

and then simplify the RHS using laws of logarithms

for 5b), ln 3x = ln 3 + ln x

[tidal flicker]

You're smart af i'm impressed

[smoky sparrow]

it is solveable by algebra

yeah, again being in uni for maths really helps

finished 1st year so

[tidal flicker]

i hate maths, I hate it so much that even your explanation made me lost ( you explained it fine, the problem is from my side )

[smoky sparrow]

do you need maths in the future?

I know that not all Indians are engineers / want to be engineers hehe

[tidal flicker]

I'm not indian though

I'm Egyptian, But I study IT, I kind of don't need maths in the future i guess ?

I'm just into coding

I hate calculus so much

[smoky sparrow]

oh lol

[smoky sparrow]

yeah

IT has really good job prospects if you can work your way up

from being junior technician and so on

you won't need calculus much and the calculus (or linear algebra) you will need, you can just search up

[tidal flicker]

Ikr

thank you for comforting me with this

❤️ xD

You seem Nice and smart, how old are you man

Would love to be around intelligent people

[smoky sparrow]

I'm 19 yeah as you could tell from the thing I said about uni

[tidal flicker]

Also, could you just give me a direct answer, ? I'm not getting the hang of it that much ngl

[tidal flicker]

19 !?!?!#?!#?!

I'm 20 yooo

now i feel dumber

[smoky sparrow]

so $\frac{f'(x)}{f(x)} = \frac{d}{dx} (3 \ln x + \ln(3 - 2x) - x - \frac{1}{3} \ln(3x^3 + 7))$

so $f'(x) = f(x) \left(\frac{3}{x} + \frac{-2}{3 - 2x} - 1 - \frac{1}{3} \frac{9x^2}{3x^3 + 7} \right)$

[thorny flame]

south

south

[tidal flicker]

me and my colleague got even more confused, wait hang on

is this A or B

[smoky sparrow]

A

[tidal flicker]

okay for B what should I do ?

[cosmic stream]

oh

[smoky sparrow]

B is much easier

[cosmic stream]

thx bro u helped me

[smoky sparrow]

so 3 ln x - (ln 3 + ln x) = 0

2 ln x - ln 3 = 0

should be nice now

[cosmic stream]

i needed lik 10 min to undrstand but i got it at the end

[smoky sparrow]

oh no worries!

thanks for letting me know that I helped you too

[tidal flicker]

You did man

thank you so much, you're a clutch

[smoky sparrow]

💖

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[soft thicket]

How to cure depression

[soft thicket]

Frfr

[torpid matrix]

See a psychiatrist

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[cobalt vortex]

I have a right triangle ABC with the right angle at C. Then there are points A¹,B¹ and C¹ which are reflections of A,B,C. A¹ is reflected about C, B¹ about A and C¹ about B.

[cobalt vortex]

Then I have to prove that

This is the first time I encountered a problem this weird to me. No matter how I look at it, I cant think of any way in which that makes sense. Could someone help me with this?

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@cobalt vortex Has your question been resolved?

[cobalt vortex]

<@&286206848099549185>

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@cobalt vortex Has your question been resolved?

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@cobalt vortex Has your question been resolved?

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[neon iron]

Hey I need help real quick, just a minute

[neon iron]

How do I solve number 19 and 21

I'm a quick learner

How do i use the formulas

It will take a minute or 2

Come on dude, don't have me waiting 10 minutes

[reef fjord]

https://www.youtube.com/watch?v=eDb6iugi6Uk

https://www.youtube.com/watch?v=eYSASx8_nyg&t

[neon iron]

Thank you

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[neon iron]

can someone help me out with specific parts of this problem?

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[cinder oxide]

https://math.stackexchange.com/questions/4105096/find-formulas-for-the-entries-of-mn-where-n-is-a-positive-integer

[cinder oxide]

I understand how eigenvalues were found

but how do you find multiplicity?

and eigenvectors

also, does it matter if you use
9 0 0 7

or
7 0 0 9?

it would depend on which eigenvalue you find first, right?

[hollow drum]

Multiplicity is founded during the process of the eigenvalues

[hollow drum]

It doesn't matter

[cinder oxide]

OK

so multiplicity is just whatever you subtract?

in this case 1 lambda multiplied by I?

[torpid matrix]

its how many times its a root of the characteristic polynomial

for example $x^2 = 0$ has eigenvalue 0 with multiplicity 2

[torpid matrix]

$x^100 = 0$ has multiplicity 100

[torpid matrix]

$(x-2)^3$ has eigenbalue 2 with multiplicity 3

[cinder oxide]

but here shouldn't multiplicity by 2 since it is the highest power?

[hollow drum]

No

Multiplicity is how many times each root occurs

[hollow drum]

This tells you how many roots there would be

[cinder oxide]

you mean here for roots?

each occur only once

[hollow drum]

Yes

[cinder oxide]

alright

so it's the highest root?

if 7 was found as a root twice for example

multiplicity of 𝜆=7 would be 2

[hollow drum]

It does matter if you are doing a decomposition though, $A = PDP^{-1}$ where D is the eigenvalue matrix and P is the eigenvectors, so you have to construct the matrix so the columns lines up. Like in that image, [3; 0] is the eigenvalue which is the first column so it coorelates to the first column of the eigenvector matrix so [1; 1]

[thorny flame]

CaptainNova22

[hollow drum]

Not sure what you mean

[cinder oxide]

im not sure either

i thought i understood but maybe not, in terms of roots to show multiplicity

[hollow drum]

But you are overthinking the terms, lambda is just a variable like x. As stated, multiplicity is how many times the roots occurred. If you have the equation, $(x - 3)^3(x+2)^4(x - 1) = 0$ would you be able to tell me the roots and multiplicity of each?

[thorny flame]

CaptainNova22

[cinder oxide]

x = 3 (mult 3)
x = -2 (mult 2)
x = 1 (mult 1)?

i think i need to expand but not sure how to solve for power 3 polynomial finding roots, only power 2

[hollow drum]

Yes this is exactly it

[cinder oxide]

Oh OK

was just guessing

so how do I find "P"?

I think that's all I need now to solve

[hollow drum]

That's the eigenvector

[cinder oxide]

but P is 2x2

no?

the eigenvectors shown above are 2x1

[hollow drum]

It will be

[cinder oxide]

I will need to combine both eigenvectors to get P?

[hollow drum]

P is the matrix of eigenvectors

[cinder oxide]

by row, or by column?

[hollow drum]

Columns

[cinder oxide]

OK

[hollow drum]

Do you know how to find the eigenvectors?

[cinder oxide]

Does it start with this?

I gotta do RREF on that, yeah?

or wait, maybe it's RREF on A

11 -2 4 5

[hollow drum]

No

[cinder oxide]

Oh I replace lambda with the value

[hollow drum]

Yes

[cinder oxide]

I will try that now

[hollow drum]

Then you reduce it

[cinder oxide]

I got a different answer

[hollow drum]

Why did you say x1 = - t?

You have x1 - x2 = 0

[cinder oxide]

x_2 is a free variable

[hollow drum]

Yes I get that

[cinder oxide]

so I back substituted

oops

[hollow drum]

But you moved it to the other side, to get it equal to x1, correct?

[cinder oxide]

mb

this should be it

for lambda = 9

[hollow drum]

Yes

[cinder oxide]

OK, I think I got it now

annoying lol, lots of work involved

[hollow drum]

That's why in classes, you only go up to 3 x 3 matrices, because it's a tedious process

Anything bigger, you would use software

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@cinder oxide Has your question been resolved?

[cinder oxide]

OK, good to know

[cinder oxide]

so I solved for the 2x2 but maybe I didn't need to do this

7 0 6 9

this is what they did instead, trying to figure out the logic here

[cinder oxide]

am I getting close?

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[surreal gazelle]

this is as far as i got, im not sure how to finish it

[surreal gazelle]

<@&286206848099549185>

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@surreal gazelle Has your question been resolved?

[sonic dawn]

you're quite literally 1 step from the answer

you got the linearization so now perform the approximation

[surreal gazelle]

I DON"T KNOW HOW

[sonic dawn]

stare at the expression and think what's the x value you should sub in

[surreal gazelle]

im thinking 16.04 but that does not make sense as to why I would do 16.04 - 16 to get a tiny tiny number of .04. And that wouldn't even give me anything.

i dont know, can i have hint

[sonic dawn]

you have a +8 at the end hellllo

[surreal gazelle]

so?

its just an 8

[sonic dawn]

which means you get a tiny number + 8

that's the entire point of linearization

[surreal gazelle]

okay, let me rephrase my question

"im thinking 16.04 but that does not make sense as to why I would do 16.04 - 16 to get a tiny tiny number of .04. And that wouldn't even give me anything. "

why do we do this

[sonic dawn]

because the entire point of linearization (and other methods of close approximations) is to approximate a function at a point close to somewhere that you have a known value of

here you know f(16) exactly so we make the assumption that f(16+h) is approximately f(16) + a small term for small h

where linearization models the change linearly, then you get the small term as f'(16)*((16+h)-16) = f'(16) * h

(and higher degree approximations involved higher degree derivatives)

[surreal gazelle]

thank you

but how do i actualy calculate the error of approximatino now

plug it in, get a number

and then what

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[surreal gazelle]

.reopen

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✅

[surreal gazelle]

wait

how do i solve for approximation of error

[sonic dawn]

uh
it's just the error
so f(16.04) minus whatever the approximation is

[surreal gazelle]

minus f(16) ?

[neon iron]

yes

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[tulip monolith]

$x^2 + \frac{2^2}{3 \cdot 4} x^4 + \frac{2^2 \cdot 4^2}{3 \cdot 4 \cdot 5 \cdot 6} x^6 + \dots$

[thorny flame]

ColdTee

[tulip monolith]

What test can I use for testing the convergence of the series

[tough forge]

That infinite series (from 1 to infinity) can be written as x^(2n)/(2n)!*(1/2) . Have you tried using the ratio test on this to get a radius of convergence?

[tulip monolith]

You mean the nth term

[tough forge]

Sorry, not sure what you're asking by this. Basically, you would need to do the ratio test for convergence on this infinite series and then plug the values of x in for whatever is left |...| < 1 in order to get a radius for what values of x make the series converge.

[tulip monolith]

$a_n = \frac{x^{2n}}{(2n!)*\frac{1}{2}}$

[thorny flame]

ColdTee

[tulip monolith]

How did you get this one

[tough forge]

There isn't a formula for seeing the sequence's expression, you just have to notice a pattern, as far as I'm aware.

[tulip monolith]

Hmmm

Can you explain

A little how the nth term that you found relates to the series

Its the same as writing $a_n = \frac{2x^{2n}}{(2n!)}$ right?

[thorny flame]

ColdTee

[tough forge]

Yes, this is correct. By the way, sorry for the late response, I'm still trying to express that multiplication in the numerator {1, 4, 4*16, ...} which is missing in the sequence.

[tulip monolith]

That I was wondering too

[topaz sinew]

@tulip monolith Has your question been resolved?

[tulip monolith]

ah

need the nth term

why didnt i realize it earlier

$a_n = \frac{2^2 \cdot 4^2 \dots (2n)^2}x^{2n}{3 \cdot 4 \cdot 5 \cdot 6 \dots (2n + 1) (2n +2)}$

[thorny flame]

ColdTee

[tough forge]

Okay so I may have found the problem.

$a_n=\frac{2^{3n-3}}{(2n)!} \cdot x^{2n}$

[tulip monolith]

Hmmmm

How

Oh

[thorny flame]

RA4

[tulip monolith]

Smart

[tulip monolith]

What's wrong with this

[tough forge]

Have you begun learning about series' as a function of f(x)? Because you've defined your sequence without x in it.

$s_n=\sum_{1}^{n}\frac{2^{3n-3} * (2n)^2 }{(2n)!} * x^{2n}$

[thorny flame]

RA4

[tough forge]

this is what s_n would be above. It needs to also have x in it, because the series you're working with may or may not converge depending on what your value of x is. That's the main thing you're missing.

[topaz sinew]

@tulip monolith Has your question been resolved?

[tulip monolith]

Ah shit

@tough forge

$a_n = \frac{2^2 \cdot 4^2 \dots (2n)^2 x^{2n}}{3 \cdot 4 \cdot 5 \cdot 6 \dots (2n + 1) (2n +2)}$

Uh oh

Why does look so bad

[thorny flame]

ColdTee

[tough forge]

Another big issue with your definition of a_n is that you can't just simply include ...

That would make it look like you're multiplying by an infinite sum over an infinite sum for each value of n. Everything should be expressed as an explicit formula with only operations and the variables "x" and "n".

[tough forge]

The explicit formula will let you use the ratio test as a way to find the convergence of an infinite series at different values of x, going back to your original question.

https://www.youtube.com/watch?v=4L9dSZN5Nvg Here's a video on how this would be done. The video goes from sigma notation to the infinite sum while you are going to do the opposite, but the same rules apply.

[tulip monolith]

Thank you very much

I will look into it

Can I dm you if ever I want to discuss more about this

[tough forge]

Sure!!

[tulip monolith]

Thanks!

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[woven hatch]

Help

[woven hatch]

How does that simplify to 1/3

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[upper mountain]

I need help

[neon iron]

No you don't

Revise your theory

[dreamy drum]

[neon iron]

Fine

[neon iron]

Y = mx + b

Or

Y = mx + c

Is the general equation of a line

Where

m is the slope or the gradient or "dy/dx"

c/b is the y intercept, the value of y where x coordinate is 0

[ruby mural]

||I thought c was the y intercept...||

[neon iron]

It is...

[ruby mural]

You wrote c/b

[neon iron]

No it's

C OR b

[ruby mural]

Oh

[neon iron]

Idk what they call it

[ruby mural]

Ok

Mb

[neon iron]

No no

[upper mountain]

THIS IS EXACTLY WHY I AM CONFUSED

[neon iron]

It's a fair doubt

[neon iron]

Shut ur ass up

[ruby mural]

Ayo

[neon iron]

You didn't study shit did you?

Like no offense but

HOW do you not KNOW the y intercept...

I would understand if the question was posed in a tricky way

But like come on.. that is like asking what's the first letter in the word apple

And it sure isn't b

[ruby mural]

Yes that's right
The first letter is p

[neon iron]

[ruby mural]

xD

[neon iron]

That is correct

Ppple is indeed a ppple

[ruby mural]

Ok but here's another way to think about it
When does a point lie on the y axis?

[upper mountain]

I learned this today..

[ruby mural]

I mean what's special about it's coordinates?

[upper mountain]

Holy shit

[neon iron]

Okay fine

My bad

[upper mountain]

On my dick for something i learned today

[neon iron]

"Learned"

No you didn't

[upper mountain]

Attemped 2 learn

[neon iron]

See?

That's better

[ruby mural]

...

[neon iron]

About

[ruby mural]

The q was for the other guy

But alr

[neon iron]

Oh 😭

He isn't even in the chat

[neon iron]

@upper mountain

[upper mountain]

hi

[ruby mural]

Hey

[neon iron]

Answer.

[upper mountain]

I don't understand this

To me yall are speaking gibberish

[neon iron]

🤦‍♂️

Dude..

Like

I don't understand people man

[upper mountain]

Ay bro

Its alright

.close

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[upper mountain]

ngl

[dreamy drum]

😦

[upper mountain]

i did nothing 2 you btw

i just needed help

[neon iron]

I am not mad at you or shit dude

[upper mountain]

was never that serious

but thanks

[neon iron]

You should be, like

[topaz sinew]

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[dreamy drum]

:/

[neon iron]

Uhm

[ruby mural]

Close it

[dreamy drum]

@neon iron 👎

[neon iron]

Come on 😭

Okay my bad

@upper mountain I am sorry man

I didn't mean it in an offending manner

[upper mountain]

Its whatever

Thanks though

[neon iron]

Sorry

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[swift finch]

@topaz sinew

[swift finch]

If p is "I love Lucy" , then the second is an implication, but I don't get how to proceed from here

@near panther sorry to ping, but could you help me w/ this by any chance?

[snow nimbus]

https://tenor.com/view/hello-there-hi-there-greetings-gif-9442662

[snow nimbus]

What did you do so far?

[vale furnace]

Why do ppl always ping bot

[swift finch]

I get that one is p, the other is p->q

so to get p->q true or false, q will always be true

because if q is false, then T->F will be false, even if F->F is true

The question says they're both either true or both either false, so if I want them both to be true or false at the same time, then q(loves Vivian) will always have to be true, right?

I don't know how to go from here, or if this is even right 💀

[snow nimbus]

So you got if P is true then Q is true, too. That is always right. Seems like we get like P <=> Q

[swift finch]

but if p is false, then I don't get how p->q is also false

I'm kinda confused

Yeah that's where I'm confused

The question says P is false and P->Q is also false and I don't get that

Where do I go from here?
Is there something wrong w/ the question?

@snow nimbus could you please help me figure this out?

[snow nimbus]

Thinking about it for myself. Looks like indeterminable for me. I'll ask in helpers channel to get another opinion

[topaz sinew]

@swift finch Has your question been resolved?

[near panther]

U still need help or did you figure it out?

[swift finch]

I still need help

How do I deal with
p=F and p->q= F

I don't get where to go in this question

[vital relic]

<@&268886789983436800> troll

[swift finch]

I'm losing my mind please help

[near panther]

I’m pre sure the strat here is to make a truth table for the statements

And then you see in each case whether the implication holds that he loves Lucy by examining each case

I’m not really sure what exactly they are asking for though, “determine whether John really loves Lucy” depends on which statements are true and false so I think they want you to analyze each case but it’s not worded clearly

[swift finch]

https://math.stackexchange.com/questions/1700834/propositions-logic

I found this answer but I don't get it

[near panther]

Have you covered truth tables ?

[swift finch]

I have, yes

I don't see what kind of table works here though

Like
For ->q is false AND p is false

What do I even do

[near panther]

yea so we have

P true q true
P true q false
P false q true
P false q false

Those are the 4 possibilities

And now you can analyze each statement independently

[swift finch]

P true q true, so p->q is T
P true q false, so p->q is F
P false q true, so p->q is T
P false q false, so p->q is T

but the question says P false, P->Q false

So P is false but P is also true and Q is also false?

So do we disregrard when he's lying

That when he's lying he's not being consistent (P is T and P is F) so that must mean that P is T, P->Q is T, and he loves Lucy (and also loves Vivian?)

[near panther]

Yea not every case will make sense, if it doesn’t make sense then you can just call it false I think as it’s not consistent

The thing is that the question isn’t really like p -> q. You can think of p as A and then q is A -> B

[swift finch]

If I do that, then how would I proceed?

💀 I'm sorry my braincells aren't working, the test is in 10 hours and I've still got a ton to cover

[near panther]

Nah these questions are really confusing cuz they super easy to overthink when you don’t set it up right

1 sec lemme make sure what I’m saying is actually correct then I’ll explain

@swift finch so this part make sense?

[swift finch]

Oh alright got it 🛐

Thanks a lot

[near panther]

oh like u figured out the whole question?

[swift finch]

I got what you were saying

[swift finch]

Is this the right way to think about the solution though?

Like
1 is true and 2 is true can happen, but 1 is false and 2 is true can never happen, so 1 is true?

[near panther]

1 is true and 2 is false can happen though

that would be like saying "i dont love lucy but If i did love lucy I would love vivian too"

the statement still makes sense

[swift finch]

yeah 1 is true and 2 is false can happen, but both can't be false at the same time right?

and the question is saying he's either telling the truth or he's lying in both cases

[swift finch]

.

[near panther]

essentially this is how I would solve this and this way you dont really need to think about contradictions or anything

the idea is just use symbols for statements or else you kinda get caught up in all the details of it and it gets way more confusing

if you cant make a conclusion for B then then you can just say that hes not consistent so his statement just dont make sense

[swift finch]

Thanks a lot!
Sorry for the late response

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[raven sparrow]

When you take the limit at infinity, you can assume that x is positive, so it works on your second method.

The issue is that when you take the limit at -infinity, then x is negative, so you can't just divide by x because the square root it always positive.

So you have to "leave" the sign behind in a way, hence why it turns negative.

In a visual way, it is pretty clear that if it goes to some value, it must be negative because sqrt(x^2 + 4) > 0 and x < 0.

[topaz sinew]

@neon iron Has your question been resolved?

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[edgy mantle]

Given this grammar, give the highest chomsky type and the language generated by the grammar

1. I know that this is type 3 in the hierarchy
2. Way I see it, we have bbbbbbbbbbb... ∪
   and basically {a,b}* \ {a, b, words with even length and words that end with b}

[edgy mantle]

but it seems a tad bit random to me . . is there a pattern here im not seeing?

[strange whale]

well if you start with b, the word can't terminate

[edgy mantle]

is a word that can't terminate not a word?

[strange whale]

if we allowed infinite words, then maybe bbbbbbbbb.... would make sense

[edgy mantle]

so i suppose bbb.. is not included . . but im not sure how to express/formulate the changing a and b pattern mathematically

[strange whale]

well alright

first we start with an a right

that's non negotiable

S -> aX

let's forget about the bT

[edgy mantle]

alright, we can get aa, ab - but then it grows exponentially almost . . and it will never end on even length/b as ending

[strange whale]

then we have whatever char we want

then we can terminate with an a

or we add 2 other chars then an a

etc...

so something like a {a, b} ({a,b}^2)* a for a regex describing this language

[edgy mantle]

can you explain the term in a bit more detail? i have never seen {a,b}^2 for example

[strange whale]

if you've seen *, you surely have seen ^n

[edgy mantle]

ah! i see

[strange whale]

all length 2 strings made of a and b if you want

[edgy mantle]

i dont think i have seen that/used it

but i think thats a complete enough hint for now - i will try it out first

thanks a lot!

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[onyx wolf]

how can i improve my problem solving ability for log exercises
i had a small question on a quiz and it took me a while to figure out, which i only did after i ran out of time.
i'm sure doing practice exercises helps, but it would help for exercises which resemble it. Ideally a student should be able to apply that knowledge to different exercices, but i struggle with that; it's only once I'm told how an exercice is done that i can do the similar exercices. So how can i improve being able to figure out a method to do a question.

i hope this isn't too vague

[young tinsel]

I love this! Figuring things out yourself is paramount, especially in Calculus, so applying this approach beforehand is great. For Logaritihims before doing practice exercises when you learn a concept best way to apply it is to learn the methods and don't do practice problems till u ensure that u fully understand the methods. Explain it to a 5 year old they say.

From what I recall when doing Logs here were the rules.

Power rule
Product Rule
Quotient Rule
Base change rule
Reciprocal Rule

One way of remembering these rules and applying them quickly is simply using numbers. For instance I may not be able to right off the bat memorize the quotient rule, but the way I am able to retain it is I apply it to a scenario where I can derive the rule really quick. Example:

log_2(8/2) = log_2(8) - log_2(2)

2^x = 7

log_2(2^x) = log_2(7)
xlog_2(2) = log_2(7)

Product Rule:

log_2( 2 x 4) = log_2(2) + log_2(4)

Reciprocal Rule:

log_2(8) = 1/(log_8(2))

Also if you are solving for something within a log function then ensure that the solution is not an extraneous solution by simply plugging it in and ensuring it abides the rules of logarithims.

[onyx wolf]

yeah that's true, every step really just comes down to applying a rule
but sometimes i don't know which step to take

like there are plenty of different steps that you can do, i'm just not sure which ones help or not. So i sometimes just brute force different rules

is there a way to know which rules you should apply

[young tinsel]

Never do that.

[young tinsel]

Ofc.

You have any questions you want to show me? Solve them in front of me, I analyze your critical thinking and we go onwards from there.

[onyx wolf]

like for example i had 2^x = 5^(x+2)

i rly struggled with this one
what's your though process for it

[young tinsel]

That way I'll be able to deduce your core issue.

mk, so the ultimate goal is to find a solution of x that satisfies both sides of the equation

now

2^x = 5^(x+2)

What is the first thing that comes to mind upon seeing this relation?

[onyx wolf]

they have a different base so i can't simplify it like that

i told myself i can split up the right side as 5^x * 5^2, but for no particular reason, idk if it helps or not

maybe i can also move it to the left side so i have all x on one side

[young tinsel]

Alright, so lets try your approach

[onyx wolf]

2^x - 5^x * 5^2 = 0

[young tinsel]

and see where that gets us

[onyx wolf]

kinda stuck here

[young tinsel]

Alright ima show you.

This is all u r essentially saying rn.

Now we could try various approaches here.

But its not going to work cus we need abide the rules of logs

Any other ideas that come to your mind?

[onyx wolf]

not really lol

[young tinsel]

Alright no worries we have to use the base change rule and I'm gonna explain the intuition behind the rule

So lets say I have an equation like this

assume z and b are some arbitrary constants

Now what is our prime goal here don't overthink it.

[onyx wolf]

get x to one side?

[young tinsel]

Right, cause our goal is to ultimately find an x value that will satisfy both sides.

[onyx wolf]

or make the base the same

[young tinsel]

Now when we find this x value and plug it in to both sides

what is going to do when we plug it in.

[onyx wolf]

it's going to be equal if that's what you're asking

[young tinsel]

YES THAT IS A HUGE THING

TREMENDOUS.

[onyx wolf]

loolllll

[young tinsel]

Your going

to understand why in a sec.

So because we know both sides are equal to one another for a certain x value.

Here is what we can do.

We can set the base to any base we desire, but knowing that both of them are equal to one another for a certain x value we input do you agree with this relation?

[onyx wolf]

yeah ok

[young tinsel]

Because we know this relation is true we can cross out the logs from both sides and would ultimately obtain the original equation.

Taking the log from both sides doesn't change the equation in anyway.

and here is a proof as to why both sides would be equal.

[onyx wolf]

this is all being said without looking at restrictions right

[young tinsel]

Of course we have to abide the log restrictions.

The restrictions is that the inside function cannot be negative, but we have no way of knowing this beforehand, until we obtain the x value only then we can determine if it will abide the restriction.

[onyx wolf]

oh yeah that's true ok

[young tinsel]

Okay.

One second.

Is this true?

[onyx wolf]

Dont think so

[young tinsel]

Okay what makes you say that? Have you learnt abt the rule
b^log_b(x) = x

[onyx wolf]

oh I didn't realize that

Yeah ok I see it now

[young tinsel]

Okay but do you understand why that rule works the one I just mentioned. Its okay if not I'll quickly resolve it for you.

[onyx wolf]

It's like an axiom right

Isn't that just how log is defined

I view log as getting the exponent with a base and product

So it's intuitive

[young tinsel]

Here is why that rule simplifies down to x ultimately.

[onyx wolf]

ok im following

[young tinsel]

Good.

now.

What would 10^log_10(b^x+10) simplify to

Write it down and use your thought process.

[onyx wolf]

10^ log... = b^x + 10

from there

i have to think about it

[young tinsel]

Exactly! So what we just did here was a proof that z^x = b^(x+10)

Essentially I showed you why we can take the log of both sides and solve for x, because at the end of the day we cannot change the equation or relation.

Now don't worry that is just the proof

[onyx wolf]

yeah that's cool to know

[young tinsel]

its just to get you thinking intuitively about how to approach these problems.

So we know that if we take the log of both sides like here for a certain x value since both sides we be the same it doesn't matter what we choose to take the base as from both sides.

we could say

log_5(z^x) = log_5(b^(x+10)) if we wanted to and would still be able to solve it.

Now don't worry Bikaty I don't expect you to understand all of this right off the bat it took me a while as well.

[onyx wolf]

hahha alright, i'm very slow with it but i'm sure if reread it'll help

[young tinsel]

Now, lets solve it.

Lets solve for x.

We know that taking the log of both sides doesn't change the relation as shown by the proof above.

[onyx wolf]

yeah

[young tinsel]

2^x = 5^(x+2)

so lets attack this problem and try your original method one more time.

2^x = 5^x * 5^2

[onyx wolf]

2^x -25(5)^x = 0

[young tinsel]

okay

there is no apparent way of solving for it by just manipulating variables.

Would you agree?

[onyx wolf]

yeah

[young tinsel]

Mk, so what is another approach we could possibly try here.

[onyx wolf]

log both sides perhaps

[young tinsel]

So lets do that.

[onyx wolf]

log of 0 is impossible tho right

[young tinsel]

🙂

that is exactly what I wanted u to realize

It is impossible indeed good job.

So solving it this way we can deduce it will simply not work.

[onyx wolf]

ok ok i see

[young tinsel]

There is no way of solving for it in this format do you agree?

Knowing this approach won't work

what is another way we can try solving for this?

[onyx wolf]

log both sides befores making one side zero

[young tinsel]

Mhmmm, so the relation that was initially stated right?

[onyx wolf]

2^x = 5^(x+2)

yeah

[young tinsel]

Yep and we know from the proof I did above taking the log from both sides does not change this relation at all.

[onyx wolf]

log(2^x) = log(5^(x+2))

[young tinsel]

That is really important to realize.

Yep.

We can say that because we know at the end of the day we are going to find an x that will ensure both sides are true.

now use the simplification rules.

[onyx wolf]

log(2x) = log(5x+10))

oh wow yeah it's easy from there

[young tinsel]

uhhhh what?

hold on.

5^(x+2) does not equal 5x + 10

[onyx wolf]

oh shit i messed up

it goes outside the log

[young tinsel]

Yep cus of the power rule

a simple of way of rembering the power rule is

[onyx wolf]

x log(2) = (x+2) log(5)

[young tinsel]

log_2(8) = 3
We also can say

log_2(2^3) = 3log_2(2)

that is one way to think abt the rules at a rapid speed ensuring u use em correctly.

[young tinsel]

yep.

continue

For instance when I solve a problem I jus don't apply the rules. I think abt why the rule is true also which saves me from making silly errors.

like a quick scenario you know for instance product rule

[onyx wolf]

i have no idea what the proof behind that rule is

shouldn't rly matter tho right

[young tinsel]

There is no proof as such behind this rule its simply a property that exists in nature.

Anyways.

continue solving.

[onyx wolf]

x log(2) / log(5) = (x+2)

hm

[young tinsel]

okay, but how does that help us solve for x?

[onyx wolf]

trying to put x on one side :p

[young tinsel]

Okay keep going.

[onyx wolf]

oh wait i should do the other term

*factor

[young tinsel]

Well you should expand it simply.

[onyx wolf]

x log(2) / (x+2) =log(5)

[young tinsel]

Mk.

And then?

[onyx wolf]

not sure

[young tinsel]

Your complicating the algebra.

Go back to the part before you manipulated it.

and check again.

[onyx wolf]

ah i got it

for me that is hard to see tho

[young tinsel]

🙂

[onyx wolf]

ok wait so to resume

[young tinsel]

Mhmm?

[onyx wolf]

you said moving everything to one side = 0 does not help because you can't apply any log rules right

and you can't manipulate it or simplify it

[young tinsel]

Exactly.

Right.

In the case you cannot manipulate and simplify it and have a 0 on one side, then your only option is using logs on both sides, but we know that doesn't work as log(0) is not possible.

[onyx wolf]

hypothetically, if there was not zero on the other side, you'd resort to loging both sides?

[young tinsel]

Well it depends on the scenario obviously.

[onyx wolf]

right ok

[young tinsel]

If its in exponential form sure we would take the log of both sides if there is no other way of simplifying using our rules.

[onyx wolf]

oh ok i see

alright so then we put it back to its og form

and logged both sides

because as you said, when it's exponential you generally resort to logging both sides right

from there it's developing and simplyfing

or factorizing

whatever

[young tinsel]

Back.

Depends on the scenario as mentioned.

[onyx wolf]

true ok

[young tinsel]

But yes

generally resorting to that.

any other questions?

[onyx wolf]

yeah thanks for patient 😄

so

[young tinsel]

Don't be shy lol.

I love getting hit wit questions.

[young tinsel]

Aye no worries.

[onyx wolf]

let me get exercice papers brb

[young tinsel]

Alr tell u what.

I can stay for up to 20 more mins

but i then gotta dip.

[onyx wolf]

👍 tyy

prolly won't take that long but who knows

anyways

7^(2x) - 7^(X) = 90

i was stuck on this

until a kid did a method where considered 7^x as y

then it was very easy to see that it's a simple quadratic

idk what im trying to ask here

[young tinsel]

uhhhhhhh quadratic has a degree of 2 on x.

[onyx wolf]

i meant quadratic when you subsistute 7^x as y

[young tinsel]

We could do that.

But what is the simplest way

of solving this problem?

[onyx wolf]

i thought this was the simplest way

unless your logging both sides again?

yeah i'll try that

[young tinsel]

In this question

[onyx wolf]

so this

[young tinsel]

one second.

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[onyx wolf]

you were very helpful, thanks a lot

very good teacher

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[left torrent]

I think I might have the right answer but I only have 1 attempt left so I just wanted to make sure

[rigid ivy]

screenshot please. No weird file formats.

[left torrent]

sorry that was a photo from my phone, ill try to fix it

Heres the work

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[neon iron]

i need steps please

[neon iron]

im literally gonna cry

idk how to fucking do it

they only taught me what to do if its 1 on the top with no fraction exponent

[hollow drum]

It's the same process

Except you just have fractions

[neon iron]

i need steps please

does it end up being something like 6√x^5

is that part of it

[hollow drum]

If it wasn't fractions, what would you do, for example $\frac{x^3}{x^2}$

[thorny flame]

CaptainNova22

[neon iron]

oh idk actually

i would multiply them-

so its x^6

[hollow drum]

No

Do you know your exponent rules?

[neon iron]

like

power of a power property

[hollow drum]

That's one of the rules but not the right one to use

[neon iron]

quotient property

?

wait yes it is

omggg my mind

so for x^3 / x^2 i wouldve done

x^3 / x^2 = x ^ 3 - 2

so x^1

right

so is it the same idea for fractions

[hollow drum]

Yes, that's right

[neon iron]

so x ^ 5/6 - 1/6

[hollow drum]

Yes

[neon iron]

so the simplest form would be x^ 4/6

or 2/3

[hollow drum]

Yes that's it

[neon iron]

thank you

omg

this guy yelled at me trying to explain it

is a single rational exponent just like

a normal number

like x^2

[hollow drum]

A rational exponent is an exponent that is a fraction

[neon iron]

so like 1/6

[hollow drum]

Yes

[neon iron]

but what if it was like

2/3

that isnt single is it

[hollow drum]

Can you post one of the problems that you're doing?

[neon iron]

this is the question

[hollow drum]

Single rational exponent means it has one exponent that is a fraction

[neon iron]

which means

[hollow drum]

It means what it means, you have an expression that one has one exponent that is fraction

[neon iron]

so when it says which is equivalent it just means which have the same answer

the first one would be x ^3/4 right

[hollow drum]

Yes

[neon iron]

but

[hollow drum]

It just means which answer choices are the same

[neon iron]

for b how would i

what property

would i use

[hollow drum]

Do you have a list of properties?

[neon iron]

would i do it as

1 divided by 1 / x^-1

thats not right is it

[hollow drum]

You didn't answer my question, do you have a list of properties?

[neon iron]

not like all on one page

well

wait

someone sent me this

[hollow drum]

Are you applying the negative exponent rule? Is that what you are doing here?

[neon iron]

ya i think

yes

[hollow drum]

If you were to do that, what do you get?

[neon iron]

well

im stuck there

hol

[hollow drum]

This is also an application of the negative exponent property, $\frac{1}{a^{-n}} = a^n$

[thorny flame]

CaptainNova22

[neon iron]

OHH

So its just x-1

no

x^1

[hollow drum]

Yes

[neon iron]

im suopposed to add the exponents together right

so its 11/10

wait

ya

[hollow drum]

Yes

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[solar rock]

Hi would anyone be able to help me with this problem, I have no idea on where to even start on it

returns of 10% and that returns between years are uncorrelated and normally distributed.
 
In how many years (integer) will it be before it has positive total (arithmetic, not compounded) returns with probability at least 97.5%?

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[solar rock]

<@&286206848099549185>

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@solar rock Has your question been resolved?

[vernal vale]

fwiw you might have better luck on the stats server

https://discord.gg/VfhEx5fw

its a slower server but ppl here are spooked by stats

[solar rock]

💀

3-5 business days for a potential response

[sweet shard]

Yes free help is often slower than paid help

You're not really doing business if it's free help

[solar rock]

i didnt mean it that way

[solar rock]

but anyway forgot to say ty

[vernal vale]

id help if i understood it ❤️

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[proper field]

how to do this pls

ping if anyone replies thanks

[molten vine]

use the exponential form for complex numbers (for -18i and x+iy)

[proper field]

is there a more simpler way?

bc i dont think ive learned that yet

[molten vine]

$(x+iy)^2=-18i \ x^2-y^2+2ixy=-18i$

[thorny flame]

Adam Chebil

[molten vine]

use system of equations

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[marsh storm]

Heyo people

[marsh storm]

I have a question about geometry, disguised as a complex number problem

[shy swan]

go on

ill try to answer

[marsh storm]

|z-2|/|z-3| = 2
Represent a Circle then the radius of the circle is , z being a complex number

I can easily solve this by assuming z = x + iy

But

the problem says that the ratio of the distance from a point to the distance from another point is a constant how does this represent a circle

I mean is this a geometry property of a circle??

Can somebody answer this please ??

<@&286206848099549185> , I guess?

.close

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[polar shoal]

how do u do this

[polar shoal]

@odd pagoda

[past cradle]

Sub ax for u

[odd pagoda]

dont ping me

[polar shoal]

ok mb

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[polar shoal]

.close

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[brisk geyser]

where do i start

[north timber]

hold on pls

[north timber]

You're solving for $$M(T)=\begin{bmatrix}a&b\c&d\end{bmatrix}$$ where $$\begin{bmatrix}a&b\c&d\end{bmatrix}\begin{bmatrix}1&1\2&0\end{bmatrix}=\begin{bmatrix}3&5\-1&4\end{bmatrix}$$

This is a linear system

[thorny flame]

luke1337

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