help-10

[obtuse pebble]

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[covert karma]

i need help with algebra

[covert karma]

the question is

3x-yx45=987xy

is anyone gonna help?

thank you

[nocturne minnow]

What does this even mean?

[covert karma]

my teacher gave us this question

i dont underatand it

[nocturne minnow]

Can you post the full question with the instructions?

[covert karma]

um ok

[obtuse pebble]

@covert karma Has your question been resolved?

[obtuse pebble]

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[little sonnet]

can someon help me justify why vector 4 is the only one that cannot be removed, thanks

.close

[obtuse pebble]

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[obtuse pebble]

@timid silo Has your question been resolved?

[timid silo]

<@&286206848099549185>

[obtuse pebble]

@timid silo Has your question been resolved?

[timid silo]

<@&286206848099549185>

[crystal cloud]

have u tried multiplying the size of it by 3

then finding where A is

[timid silo]

you’re looking for A

what does the center have to do with it

take the original position of A then just multiply it for the given scale factor

lol idk

so 1,-1 would turn into 3,-3 but i entered that and it said i was wrong

yeah I have no idea then

I also thought it’s 3,-3 idk I haven’t taken this lesson yet

im on my 3rd attempt

so these two are singled out

[obtuse pebble]

@timid silo Has your question been resolved?

[obtuse pebble]

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[obtuse pebble]

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[violet sentinel]

need some help with abstract

[violet sentinel]

I think I was able to show that the order of f(a) is less than or equal to the order of a, but I'm not sure how to show that |f(a)| divides |a|

[zenith raft]

hmm I think you pretty much have it already

[violet sentinel]

https://tenor.com/view/you-what-it-you-you-are-its-you-gif-18663985

[zenith raft]

lollll

[violet sentinel]

I might be missing a rule somewhere then

cause I don't see it lol

[zenith raft]

if G is a finite group and g is in G and g^n = e, then o(g) divides n

have you seen that?

[violet sentinel]

oh I guess I haven't

maybe it's in my notes and I skipped over it

one moment

[violet sentinel]

oh wait I think that's where I'm confused. I always thought that if g^n = e, then o(g) = n.

I guess n would divide n lol

[zenith raft]

o(g) would divide n you mean?

[violet sentinel]

no, that it equals it, right?
If we have an element g of order, let's say 3, then that means the set would consist of the following multiples of g: {g, g^2, g^3}, where g^3 = the identity

so o(g) = 3

is that wrong?

[zenith raft]

if g^3 = e, then g^6 =e but o(g) is not 6

[violet sentinel]

-.-

I swear I will get this someday lmao

yeah I see what you're saying

[zenith raft]

🥲

[violet sentinel]

as always thank you

.close

[obtuse pebble]

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[sinful kraken]

if f(x)=sin(x^3), what is f^(15)(0) (15th derivative)

how is the answer a) ?

[final thunder]

Taylor series expansion

[sinful kraken]

how long should it take to solve this problem (approx) ?

[final thunder]

I mean you look at the x^15th term in the expansion of sin(x^3) then times it by 15!.

[sinful kraken]

i have to derivate 15 times ?

[final thunder]

(Because x^15 ->15x^14->…->15!

No

What’s the maclaurin series of sinx

[sinful kraken]

idk

[final thunder]

Plug x^3 in there

[sinful kraken]

y u deleted it

[violet sentinel]

I need to work on it, I realized it had a flaw in logic

[sinful kraken]

oh ok

[violet sentinel]

This is the key here.
You know that the Taylor expansion of sin(x) = x - x^3/3! + x^5/5! - x^7/7! + ....
If we plug in x^3, we get:
sin(x^3) = (x^3) - (x^3)^3/3! + (x^3)^5/5! - (x^3)^7/7! + ...
sin(x^3) = x^3 - x^9/3! + x^15/5! - x^21/7! + ...

[sinful kraken]

i agree

[violet sentinel]

did you know that when you take the derivative of x^n n times, then you get n! as the answer

for example

f(x) = x^3

f'(x) = 3x^2
f''(x) = 6x
f'''(x) = 6

at the 3rd derivative of x^3, we get 6, no matter what x is

and 3! = 6

[sinful kraken]

interesting

[violet sentinel]

so what Pure was saying is that, if we have a power of x^15, and we take 15 derivatives

then we have 15! as the answer

in the expansion above, we see that x^15 is divided by 5!

[sinful kraken]

ooooh

but how am i supposed to know that its insane

[violet sentinel]

it's just noticing things about sin and cos

for example, even though we are plugging in x = 0, we still get an answer out

that's because the 15th derivative of sin(x) has a cosine involved 🙂

and cos(0) is 1

so we will get a non-zero answer

[sinful kraken]

however why are we allowed to replace x by x^3 in the series

[violet sentinel]

you can put whatever you want in there

it's because the series expansion is for sin(x). But you just replace the x in the expansion with whatever you're passing into sin

[sinful kraken]

and if we do so then why is the 5th derivative becoming the 15th derivative?

[violet sentinel]

From this

(x^3)^5/5

[sinful kraken]

yhea but its still the 5th derivative

[violet sentinel]

(x^3)^5 = x^(3*5) = x^15

well dont think of it as the "fifth derivative"

it's the 5th power in the series

or the 3rd term if you will lol

[sinful kraken]

then how do we link the derivatives to the series ?

[violet sentinel]

but it's just the fact that (x^a)^b = x^(a*b)

well since the series is a polynomial, it's easy to take the derivative of that

all the taylor series is is an approximation (a damned good one at that)

[sinful kraken]

ok i see

[violet sentinel]

it's the polynomial expression of sin(x) in this case

but you could pass in whatever you want

like sin(tan(x)) = tan(x) - tan^3(x)/3! + tan^5(x)/5! - ...

I don't reccomend doing that but you could if you wanted lmao

[sinful kraken]

ok i see

thank you so much for helping me

.close

[obtuse pebble]

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[timid silo]

This makes no sense to me idk where to even start

so far i tried
p(8) = e^8k = 61440
p(4) = e^4k = 1920

Then Divided:
e^4k = 32

K = 5/4ln2 or 0.86643

[violet sentinel]

yep that seems reasonable for (a)

so it would be 86.66%

did it say that was wrong?

[timid silo]

it was right but im not too sure how to approach the other questions

[violet sentinel]

ah okay

so essentially we found the growth rate k. So we know that, for example, that p(4) = 1920 = Ae^4(0.8664).
So can you solve for A with 1920 = Ae^4*0.8664?

[timid silo]

what do i do with the A

[violet sentinel]

the A is our initial population

it comes from the equation y = Ae^(xk)

where A = initial population, x = time in hours, k = growth rate

[timid silo]

if i take ln of both sides will that get rid of the e

[violet sentinel]

nah

just divide both sides by e^4*0.8664

e^4*0.8664 is an actual number

[timid silo]

ohhh

[violet sentinel]

it equals about 47.30384

[timid silo]

okay okay

didnt know i could do that

[violet sentinel]

yep! You really only need to take logs of both sides if you are simplifying your calculations or if you have a variable exponent

[timid silo]

47.3 didnt work

[violet sentinel]

no that' snot the answer, that's just an example

1920 = Ae^4*0.8664

[timid silo]

oh gotcha

[violet sentinel]

divide both sides by e^4*0.8664

[timid silo]

ok i got 60

[violet sentinel]

bingo!

that's the initial populatio

[timid silo]

then c should be
y(t) = 60e^0.8664t

[violet sentinel]

yep bingo!

[timid silo]

im on e now

its just 60e^0.8664t * 0.8664

right?

[violet sentinel]

nah look at the unit of measurement

(d) tells you total bacteria at time t = 5.5

what it's asking for is like the average rate of growth (not in percentage)

[timid silo]

how do you get that

[violet sentinel]

so if you had 7041 bacteria in 5.5 hours, how many bacteria per hour?

[timid silo]

just divide?

[violet sentinel]

yep!

[timid silo]

7043 / 5.5

i got 1280

[violet sentinel]

hmm well it depends on how you round. It should be 1280.545, so if you round up on 0.5, then try 1281 if 1280 doesn't work as a correct answer

[timid silo]

still no

[violet sentinel]

oh wait I"m dumb

you were right

it's the derivative lol

60e^0.8664t * 0.8664

then plug in 5.5

[timid silo]

where into t?

[violet sentinel]

yep

[timid silo]

60e^0.8664(5.5) * 0.8664

that?

[violet sentinel]

correct

[timid silo]

weird

i got

679.99 and plugged into 679/680

both dont work

[violet sentinel]

try parentheses

60e^(0.8664(5.5))* 0.8664

[timid silo]

oh yeah

lol

thanks

then finally

[violet sentinel]

no worries! order of operation is a bitch lol

[timid silo]

66,000 = 60e^0.8664t

[violet sentinel]

yep! just solve for t. This time you will use the natural log ( but divide first before you do that)

[timid silo]

got em all

thank you for your help

[violet sentinel]

dab

yep no problem!

[timid silo]

coudl you help me with this last problem as well?

its a 2 parter

[violet sentinel]

actually I gotta head out

I reccommend that you .close this thread and open a new one

so you get pushed to the top of the stack

[timid silo]

oh alright

well i really apreciate your help

thank you very much

have a great rest of your day/night

[violet sentinel]

yep of course! Best of luck to you 🙂

have a good one!

[timid silo]

.close

[obtuse pebble]

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[obtuse pebble]

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[swift parcel]

3 1   % of 225 is equal to?
  -
  8

what is this

and how to solve?

[nimble dew]

31% of 225 is 225% of 31

[swift parcel]

ok

[obtuse pebble]

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[swift parcel]

3 1   % of 225 is equal to?
  -
  8

[swift parcel]

how to solve this

the options are

7
70000
7000
700

[nimble dew]

literally none of those are true

[swift parcel]

oh uhm

what will the answer be

225% of 31?

[royal basin]

do you mean three-and-an-eighth of a percent?

[swift parcel]

i dont know

i just have this question

i am really confused about it

[royal basin]

you had that question typed exactly as you showed it here???

[swift parcel]

yea

[royal basin]

or was it printed

[swift parcel]

its exactly like that

[winter geyser]

can you show a picture of it?

[swift parcel]

wait

[nimble dew]

yeah its 3 and 1/8s % of 225 if the answer is one of those

[swift parcel]

what will the naswer be

[royal basin]

we don't give out answers

[swift parcel]

i mean

[royal basin]

do you know in general how percentages work?

[swift parcel]

i want to learn the process

[swift parcel]

yes

[royal basin]

okay then what is troubling you?

[swift parcel]

the - 8 thing

its weird

[nimble dew]

they are trying to say 3 and 1 8ths percent

[royal basin]

i thought we had established that this is a poorly written fraction

[swift parcel]

oh

[swift parcel]

ah

so is it like

3 x fraction(1, 8) % of 225

[nimble dew]

no its

(3 + 1/8) % of 225

[swift parcel]

oh thanks

tysm

[obtuse pebble]

@swift parcel Has your question been resolved?

[obtuse pebble]

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[obtuse pebble]

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[frigid harness]

I'm slightly confused on this question

[tranquil quiver]

you can start by finding the domain of each individual term

and then intersection of all the domains

[frigid harness]

i can't find the domain of the last term

[tranquil quiver]

what part are you struggling with

hint: something divided by zero is undefined

[frigid harness]

i think i get it now thanks

.close

[obtuse pebble]

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[rigid void]

I have problems with finding new limits

they want me to sketch the domain of integration

i know that y = x

but idk how far the limit goes?

[obtuse pebble]

@rigid void Has your question been resolved?

[obtuse pebble]

@rigid void Has your question been resolved?

[obtuse pebble]

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[gloomy valve]

S includes all numbers of the form -1, 3, 7, 11, ... is that finite?

[silver plover]

sry

mb

meant infinite

was probs half sleep lol

[silver plover]

what about intersection is that correct?

and what do i do for union

[gloomy valve]

note that you can represent any odd number 2m + 1 as either 4k - 1 or 4k + 1.
why? 4k - 1 is some odd number, the next odd number would be (4k - 1) + 2 = 4k +1, the next odd number is 4k + 3 = 4(k+1) - 1, etc.

[silver plover]

so is union all odd numbers?

[gloomy valve]

^

[silver plover]

huh?

[gloomy valve]

just an explanation

[silver plover]

oh wait

so it is all odd numbers?

for union

?

[obtuse pebble]

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[obtuse pebble]

@silver plover Has your question been resolved?

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[zenith moon]

what did i do wrong 😭

[glossy basalt]

reading

d(sin(x))/dx = cos(x)

first line first term

[zenith moon]

its sin(3xy)

and since theres a chain rule you have to apply the implicit differentiation

[glossy basalt]

yea

so, the first term will be cos(3xy)

not negative

[zenith moon]

oh

wait lemme try if that was right

[glossy basalt]

sure

[glossy basalt]

hmmmm, lemme continue reading

3rd line

the negative in the middle of LHS should be positive

[zenith moon]

lhs?

[glossy basalt]

left hand side

[glossy basalt]

andddddd lemme continue reading again

[zenith moon]

sorry 😭

[glossy basalt]

2nd line

you distribute the 3 inside the brackets

therefore secodn term should be
3y cos(3xy)

positive

[zenith moon]

OH MY ALGEBRA 😭

now its correct

thank you 😭

[glossy basalt]

yey

high five

[zenith moon]

🙋‍♂️

[glossy basalt]

btw, you can close this channel if you are done

[obtuse pebble]

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[zenith moon]

i forgot sry

[obtuse pebble]

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[dark fossil]

Does anyone have any non-ridiculous idea for showing that this is lebesgue integrable?

[kind hawk]

sin(x)/x is bounded

[dark fossil]

One of the things I'm considering is splitting it into two integrals over $(0, \frac{\pi}{2}), [\frac{\pi}{2},\infty)$,
which should work.
I'm finding showing that the parts of h are measurable kind of hard though.

[warm shale]

CoffeeMan

[kind hawk]

continuous therefore measurable

or can you not use that

[dark fossil]

I can only use it for R to R.

And here it wouldn't be continuous on R.
Out of curiosity, does it work in a more general setting?

[kind hawk]

does what work in a more general setting

[dark fossil]

The use of continuity implying measurability

[kind hawk]

yes

a function is continuous if the preimage of every open set is open

and as the borel measurable sets are generated by the open sets you get the claim pretty fast

[dark fossil]

I see. So if we have a subset of R, then, as long as all the open sets of the subset are in the sigma algebra we choose for the space, we have that continuous maps imply measurability?

Ah, and that is what the borel sigma algebra is.

I see

[obtuse pebble]

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[forest rivet]

Hello, I am trying to solve this equation but I am stuck.

[spice citrus]

notice that sqrt(2) is nothing but e^ln(sqrt(2)), which can be even futher simplified if you write sqrt(2) as 2 ^ (1/2)

you probably forgot a square root on the bottom right btw

[forest rivet]

thanks

what are the missing solutions? ln(sqrt{2})+ i5pi/4. and ln(sqrt(2))+ ipi/4?

[spice citrus]

where did you get those from?

[obtuse pebble]

@forest rivet Has your question been resolved?

[forest rivet]

I am not sure how to get to the other solutions.

[spice citrus]

okay, so if z is a solution then so would z + 2pi be

because e^2pi = 1

and e^(z+2pi) = e^z * e^2pi

[obtuse pebble]

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[cursive onyx]

I am having some trouble understanding how to do this proof

[wild swallow]

you need to show that cauchy sequences converge

so first when given a cauchy sequence x1, x2, ...

you need to find something it can converge to

say y in l^2

and then you need to demonstrate that x1, x2, ... actually converges to y

so basically you need to do two things

show that some y exists in l^2

the show that the limit of xn is y

[obtuse pebble]

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[timber fox]

so I posted this earlier and still have no real idea how to approach it with full understanding,

I know the domain is [0,inifinty) because any negative number to the even power is in the complex plane and not reals,

I can set it up as ;

lim h-> ( (x+h)^(5/2) - x^(5/2) ) / h

[timber fox]

A helpful dude earlier said to multiply by the conjugate which I did but I'm not sure where it gets me to

[drowsy girder]

I proved power rule i think few times now imma try and find my msg.

[wild swallow]

keep doing the algebra after multiplying by conjugate

you get like (x+h)^5 - x^5 on the numerator

[timber fox]

yep

not sure what to do with that

[wild swallow]

factor it

[timber fox]

not really sure how

tbh

wait

okok

[drowsy girder]

$$\lim{a \to 0} \frac{(x + a)^n - x^n}{a}$$
$$\lim{a \to 0} \frac{\displaystyle\sum{k=0} ^{\infty} \left( {n \choose k}x^{n - k}a^k \right) - x^n}{a}$$
$$\lim{a \to 0} \frac{x^n - x^n + {n \choose 1}x^{n - 1}a + {n \choose 2}x^{n - 2}a^2....}{a}$$
$$\lim{a \to 0}\frac{a\left({n \choose 1}x^{n-1} + {n \choose 2}x^{n-2}a + ...\right)}{a}$$
$$\lim{a \to 0} {n \choose 1}x^{n-1} + {n \choose 2}x^{n-2}a + ...$$
$$nx^{n-1}$$

[warm shale]

Pluton

[wild swallow]

a^5 - b^5 = (a-b)(a^4 + a^3 b + a^2 b^2 + a b^3 + b^4)

[drowsy girder]

Ok my latex is broken

[timber fox]

(x+h)^(5/2) = ((x+h)^1/2)^5

[wild swallow]

no

multiply by the conjugate

[timber fox]

oh right

[wild swallow]

which is (x+h)^5/2 + x^5/2

[wild swallow]

then use this factorisation on the numerator

[timber fox]

so since we get whole number power now we get x^n - y^n

ok nice

ill write it out and post it in here

2 seconds

[drowsy girder]

I found the right latex

$$\lim_{a \to 0} \frac{(x + a)^n - x^n}{a}$$
$$\lim_{a \to 0} \frac{\displaystyle\sum_{k=0} ^{\infty} \left( {n \choose k}x^{n - k}a^k \right) - x^n}{a}$$
$$\lim_{a \to 0} \frac{x^n - x^n + {n \choose 1}x^{n - 1}a + {n \choose 2}x^{n - 2}a^2....}{a}$$
$$\lim_{a \to 0}\frac{a\left({n \choose 1}x^{n-1} + {n \choose 2}x^{n-2}a + ...\right)}{a}$$
$$\lim_{a \to 0} {n \choose 1}x^{n-1} + {n \choose 2}x^{n-2}a + ...$$
$$nx^{n-1}$$

[warm shale]

Pluton

[wild swallow]

bruh this isn't the power rule with integers

[drowsy girder]

Its using general binomial theorem

[wild swallow]

and you can prove it without spamming binomial coefficients

the generalised binomial theorem is proven with taylor etc

by that point you've already calculated the derivatives of x^n

[drowsy girder]

Agree. Still a cool way ig

[timber fox]

wait why is this thing so long

im going insane

[wild swallow]

?

you really shouldn't have anything very long

[wild swallow]

there are only 5 terms here

in the second bracket

[timber fox]

its x^n - y^n = (x-y)(x^n-1 + x^n-2 y ....) isn't it?

or am I tripping

[wild swallow]

yes?

[timber fox]

ITS SO LONG :((

[wild swallow]

i literally wrote all the terms out

n is 5

there are 5 terms

do not expand any of the brackets containing (x+h)

[timber fox]

ok so what do I do with this factorization now

because im still slightly lost if im being honesty and don't full understand the steps and why we are taking them

[wild swallow]

well you should find that (x+h - x) is just h

and that cancels with the h in the denominator

[timber fox]

ok lemme see

[wild swallow]

at which point every other h can be replaced with 0 in the limit

and you have your answer

[timber fox]

wait when you multiply by the conjugate its top and bottom right O_O , so i'm not wrong in having the h((x+h)^(5/2) - x^(5/2))?

[wild swallow]

that should be a +

in the denominator

[timber fox]

yes typo

mb

ohh I see

OHHH

is it looking tasty?

[wild swallow]

looks fine

cancel the hs

sub in h=0

and you're done

[timber fox]

OMG

that was banging

ty

that is godlike

NOW BACK TO THE POWER RULE!

ok so just for clarification

like for understanding

is the reason we multiply by the conjugate and factor out, so we can try and achieve our h cancellation on top and bottom ? like what other motives are there for the things we do

the thing I can think of is

[wild swallow]

conjugate so we don't have fractional powers

[timber fox]

ultiplying by the conjugate here would give a whole number power then we can do factorization of it

ok nice

[wild swallow]

integer powers have nice factorisations

[timber fox]

perfect

does this work for all fractional powers?

[wild swallow]

yes

[timber fox]

ok thanks so much guys

[wild swallow]

you just get progressively more terms

and by fractional i mean rational

[timber fox]

yep

ty ty ty

have a good rest of yoru day

.close

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[timid silo]

this

is scary

[obtuse pebble]

@timid silo Has your question been resolved?

[timid silo]

<@&286206848099549185>

[timber fox]

@timid silo

is it just asking us when you expand all terms is it equal?

like check if LHS = RHS

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[warm palm]

Find an expression for y'' only in terms of x and y, knowing that x^6 + y^6 = 1.

[warm palm]

i dont quite get the question, can someone explain?

[slim leaf]

You have to find the double derivative

[warm palm]

ah, with implicit derivative

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[lusty oar]

help I’m stuck and idk if I even did the right thing to begin with…

[slim leaf]

@lusty oar you just have to check differentiability at x=-1, so lim x-->-1^+ f'(x) = lim x-->-1^+ 2x = -2, lim x-->-1^- f'(x) = lim x-->-1^- (-2) = -2

and so the function is differentiable at x=-1, and we know that it is already differentiable everywhere else

[obtuse pebble]

@lusty oar Has your question been resolved?

[lusty oar]

for this part: lim x-->-1^- f'(x) = lim x-->-1^- (-2) = -2

why did we put a negative in front

[obtuse pebble]

@lusty oar Has your question been resolved?

[slim leaf]

Limit from left side of -1

@lusty oar

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[timid silo]

If f(-4)=5 and f(5)=5 what is the equation of the line

[chrome mesa]

do you know how to find the equation of a line from two points

[timid silo]

yea

but I have gotten it wrong twice

[hexed agate]

Show work

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[molten lintel]

What will be the characteristic equation for this matrix ?

[molten lintel]

<@&286206848099549185>

[obtuse pebble]

Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.

[wooden cipher]

oh thats a cool nnew feature

[timid silo]

New?

[wooden cipher]

unfortunately i dont know enough about this stuff but it may help to group the x terms and the y terms

[final thunder]

I just did !15m then deleted my message

[wooden cipher]

oh lmao

[molten lintel]

You have to make a matrix from the quadratic form

How would pairing the terms help

[wooden cipher]

i have limited knowledge, but it looks similar to factoring an equation for a circle/ellipse

i dont know enough to help you through the problem but since i was here anyway i thought it might help to try to contribute

[molten lintel]

.close

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[sullen seal]

In trigonometry, how to I calculate the angle of a right triangle without a calculator

[restive ridge]

90°?

[timid silo]

which angle? and which side lenghts are given?

[obtuse pebble]

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[distant sphinx]

I was playing around with desmos and made a representation of a doodle I used to do all the time with graph paper and was wondering what the curve made by the lines are, I always assumed it made a circle but I see it isn't now what do I need to look up to find the relevent curve?

[obtuse pebble]

@distant sphinx Has your question been resolved?

[distant sphinx]

<@&286206848099549185>

[distant sphinx]

ok

[distant sphinx]

I've Figured it out it's a Quadratic Bezier Curve I'm pretty sure maybe not idk

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[scarlet ridge]

pls

[scarlet ridge]

i need help

[obtuse pebble]

@scarlet ridge Has your question been resolved?

[scarlet ridge]

<@&286206848099549185>

[scarlet ridge]

<@&286206848099549185> bruh

[upbeat gazelle]

Use the quotient rule for the derivative

[scarlet ridge]

how u do it

[upbeat gazelle]

$h'(x) = \frac{f'(x)(g(x)+3x) - (g'(x) +3)(f(x)+1)}{(g(x)+3x)^{2}}$

[warm shale]

black_couscous

[upbeat gazelle]

Plug $x=2$ and you've got yourself an answer

[warm shale]

black_couscous

[scarlet ridge]

@upbeat gazelle

thats what i do?

[upbeat gazelle]

Yes

[scarlet ridge]

can u tell me if my answer correct

after i do it

bruh idk what to do

😭

[upbeat gazelle]

Just plug $x=2$ and look for the appropriate values on the graph

[warm shale]

black_couscous

[scarlet ridge]

but what it mean for prime

and stuff

f prime of 2

what do i do for that

[upbeat gazelle]

Use the slope on the graph

[scarlet ridge]

idrk how to don that

with derivatives

[upbeat gazelle]

Just compute the slope of the function in the points you're interested in

[scarlet ridge]

idk how to do that

like would it be 1

and -2

?

[upbeat gazelle]

For which one?

https://www.physicsclassroom.com/class/1DKin/Lesson-3/Determining-the-Slope-on-a-p-t-Graph#:~:text=Pick two points on the,rise%2Frun or slope).

[scarlet ridge]

@upbeat gazelle would it be 1/100

[upbeat gazelle]

Yup

[scarlet ridge]

betty

ily

[upbeat gazelle]

Thx

[obtuse pebble]

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[obtuse pebble]

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[gusty halo]

can someone please explain what this means in english

[gusty halo]

i dont get it

[candid moth]

what part don't you get

[gusty halo]

i was learning about inverse matrices and then suddenly they show me this and its like im reading a different language

all of it

mostly the null thing

[candid moth]

this seems to be saying that if a matrix is invertible it has trivial null space

[gusty halo]

what is a null space?

[candid moth]

which intuitively should be true

the things the matrix sends to the 0 vector

[gusty halo]

can you give me an example?

[candid moth]

currently how are you thinking about matrices?

grids of numbers or linear transformations

[gusty halo]

the first one

[candid moth]

ah

you should be thinking of them as linear transformations

do you know what a linear transformation is?

[late stump]

3b1b ftw

[gusty halo]

its an online class so a lot of this stuff is hard to figure out on my own

[candid moth]

yeah I was about to recommend that

[gusty halo]

no

[candid moth]

look up "3blue1brown essence of linear algebra" on youtube

watch the whole playlist

it's like an hour and a half and it's an insanely good resource for learning linear algebra for the first time

like it's hard to overstate how good it is

[gusty halo]

i think i saw the calculus series

[late stump]

yess i can vouch!

its a great series

[gusty halo]

ill try the LA one thank you

[candid moth]

I am of the opinion that it should be in the syllabus of literally every intro linear algebra class

legitimately one of the best math expositions on all of youtube

or possibly the internet as a whole

[gusty halo]

yeah most of what i have learned in this class is basically nonsense to me

[candid moth]

yeah linear algebra is like that

if you think of things the right way it's super intuitive, but if you don't it's very much not

but anyways to give a brief answer to your original question here

a matrix essentially describes a linear transformation of your vector space

If you don't know what a vector space is, just think of R^n. If you do know what a vector space is, temporarily forget that and think of R^n

[gusty halo]

i know what a vector space is

[candid moth]

so a linear transformation is a mapping between vector spaces that is linear.

So T(av) = aT(v) and T(v + w) = T(v) + T(w)

it turns out that linear transformations are matrices are basically the same thing (once you've fixed a basis but you don't need to worry about that now)

do you know what a basis is?

[gusty halo]

uhh you mean like I_n?

[candid moth]

like the canonical basis for R^3 would be (1,0,0), (0,1,0), (0,0,1)

[gusty halo]

yeah

[candid moth]

really those are vectors not points but whatever

so basically any linear transformation is entirely described by what it does to the basis vectors

by the condition of linearity

as an example if you knew that (1,0,0) was sent to u, (0,1,0) was sent to v, and (0,0,1) was sent to w, then you know that (1,2,5) is sent to u+2v+5w

so if I wanted to describe a linear transformation all I would need to do would be to list out what it does to the basis vectors

[gusty halo]

ok so a null space is a vector that gets sent to the origin after this transformation?

[candid moth]

so suppose I was sending (1,0,0) to (0,5,7), (0,1,0) to (4,2,0), and (0,0,1) to (6,6,6) then I might list that out like
$$\begin{bmatrix} 0 & 4 & 6 \ 5 & 2 & 6 \ 7 & 0 & 6 \end{bmatrix}$$

[warm shale]

Gamma is an Algebraic Number

[candid moth]

oh wait that's just a matrix

The rule for multiplying two matrices just correspond to composing the corresponding linear transformations and the rule for multiplying a matrix by a vector just correspond to applying the transformation to that vector

then the null space is the set of all vectors that are sent to (0,0,0)

[gusty halo]

so when you multiply a matrix would that be like transforming the basis vectors twice?

[candid moth]

sorta yeah

you're transforming the space twice

applying one transformation then another

the linear map acts on the whole space, it is just uniquely defined by what it does to the (a) basis

[gusty halo]

ok i think i understand now

[candid moth]

now a matrix is invertible if its associated linear transformation has an inverse. But if there's an inverse then you can't be sending multiple things to the 0 vector since that wouldn't be injective. So any invertible matrix must have only one thing sent to the 0 vector. But you can check from the definition of linearity that the 0 vector is always sent to itself under any linear transformation so there you go.

[gusty halo]

ohhh

that makes a lot of sense

thanks

[candid moth]

it turns out that the 0 vector is special because it's always mapped to itself so the null space turns out to be very important.

In particular if something that's nonzero is mapped to the 0 vector then you must also have multiple nonzero vectors mapped to the same nonzero vector.

Since if T(v) = 0 then T(w+av) = T(w) + T(av) = T(w) + aT(V) = T(w) for any scalar a and vector w

[gusty halo]

thats pretty much what i was going to ask lol

[candid moth]

anyways I'm going to bed now but watch the 3blue1brown series. It goes into this in a lot more detail.

[gusty halo]

i will, thank you

[candid moth]

Glad I could help and congrats on starting to learn linear algebra. Once you get the hang of it it's easily one of the nicest areas of math.

[gusty halo]

very different from what im used to lol

[candid moth]

And it comes up everywhere too. Right up there with calculus in terms of applicability.

[gusty halo]

good night

and thanks again

.close

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[gritty berry]

f(x) =
x+2 3x-6

[obtuse pebble]

Please don't occupy multiple help channels.

[gritty berry]

how do i solve this?

[hybrid gull]

Close the old one before opening a new one

You can’t solve an expression

[gritty berry]

oh okay im sorry

i really need to learn how to math 💀

[hybrid gull]

All good - we all do, don’t we?

[high lily]

what's the exact wording of the question

did you make any adjustments/omit stuff

[obtuse pebble]

@gritty berry Has your question been resolved?

[tardy epoch]

.close

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[proven zephyr]

Khan Academy question

Since it's not one sided limit, you can't find the value right?

Therefore "limit doesn't exist"

OHH

i get it

5

[tardy epoch]

each individual functions one sided limit doesn't match, but their sum can

[proven zephyr]

Yeah

Thanks

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[brazen zealot]

why do we let P (n) denote the open sentence “Sn = 2^n”.

[stable rain]

well

its induction

u let Pn denote the statement u wanna prove

[brazen zealot]

why is it 2^n

im kind of lsot

[stable rain]

u wanan show

that left bit is

this

[brazen zealot]

oml im stupid

[stable rain]

they just arbitraryily let it be that

[brazen zealot]

thanks

.close

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[stable rain]

np

[obtuse pebble]

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[placid nova]

.close

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[high lily]

what have you tried

[spark vector]

X = sallly

3x

3x + 134 = 710

But I know that is not it

[high lily]

read what the statement says carefully

[spark vector]

I tried to

I don’t understand

[high lily]

John received 134 more than 3 times sally

3x + 134
is how much John received

J = 3x + 134

[spark vector]

Ok

Oh I see

Got it

Thanks

3/4 x 2/3

1/2 x 1/3

1/6x3/4

=1/8

[obtuse pebble]

@spark vector Has your question been resolved?

[spark vector]

@high lily

[obtuse pebble]

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[distant moth]

how is T^2 being written?

.close

[obtuse pebble]

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[sacred cove]

Hi can anyone explain to me what happens to the 1/t+30 S?

Is it a rule that it just disappears?

In the last line of working

[lethal sand]

it didnt disappear

the integral of 1/(t+30) wrt t is ln|t+30|, and e^ln|t+30| is just t+30 given that t+30 is positive

[sacred cove]

yeah i get that part

but when they find the integrating factor is t+30

and multiply it across

[lethal sand]

uh huh?

[sacred cove]

i just dont understand how they go from the second line to the fourth line

i understand the third line

do you know what happened to the Ds/Dt then?

[lethal sand]

after multiplying by the integrating factor you have found, you get ds/dt * (t+30) + s=70(t+30)

[sacred cove]

yeah i got that

[lethal sand]

and as you can see, the lhs is just the derivative of s(t+30) wrt t

where s is a function of t

[sacred cove]

how is the left hand side a derivative of that?

[lethal sand]

product rule

or if you are not forward with the method, you can do a painful ibp for the lhs when integrating both sides wrt t

[sacred cove]

wait sorry are we calculating the derivative d/ds or d/dt?

[lethal sand]

d/dt, because like i said, it is the derivative with respect to t

and s is a function of t, hence you would get the lhs as your derivative

[sacred cove]

wouldnt it be times by s instead of + s then?

[lethal sand]

not sure what you meant there

[sacred cove]

im having a look at this

[lethal sand]

of course if you tried plugging it into symbolab without giving any, any, context at all, it would treat s as a constant

like i have mentioned twice already, s is a function of t

hence the derivative of s(t+30) is ds/dt * (t+30) + s

[sacred cove]

right ok i will try to solve the product rule by hand and see how i go

thanks a lot for your help

[obtuse pebble]

@sacred cove Has your question been resolved?

[obtuse pebble]

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[sleek owl]

If we have three categorical features, where each feature can take four values, and we have five classes, how many probabilities do we need to calculate using Naive Bayes classification?

[stable rain]

feels cs

[sleek owl]

Huh?

[stable rain]

i think its 5*(3*4)

[sleek owl]

Okay I’ll try it now

Nope lol. Thanks anyways

@stable rain

[stable rain]

[sleek owl]

It’s 65 lol

[stable rain]

oh ye

oops

[obtuse pebble]

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[worn summit]

!help

[worn summit]

what did i do wrong here

wait hang on im missing an image

there

the algebra calculator says im wrong

[olive swan]

$0 = \frac{2x}{3}$

[warm shale]

JoDKrishna

[olive swan]

x is always zero

[worn summit]

what do you mean

[jaunty ibex]

Multiply 0 by three and divide it by two and it's still 0

[worn summit]

but why is this wrong

[red stag]

the second to last line

[jaunty ibex]

Cause 2/3x + 1/3 is not x

[worn summit]

waht

[jaunty ibex]

The line before the last line

It's wrong

[red stag]

what is even going in there

[worn summit]

but how

why

[red stag]

cause you are multiplying 2/3 by x, not adding 2/3 to x

2/3x + 1/3x would equal x

[worn summit]

oh ok

alright thanks

.close

[obtuse pebble]

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[worn summit]

.reopen

[obtuse pebble]

✅

[worn summit]

wait

then if i added 1/3x to both sides

then wouldnt i get 1/3x=x?

[red stag]

and?

you still get 0

that does nothing

[worn summit]

no but a third of something isnt a full thing of something

[red stag]

1/3 * 0 = 0

anything times 0 is 0

[worn summit]

no but if i added it

instead of multiplying

[red stag]

you just added

and got 1/3x = x

[worn summit]

i got 0.333 = x

[worn summit]

but if i did this

i would get 1/3 of x = x

right

[hexed agate]

I suggest doing it again

Start back from 0=2x/3

[red stag]

exactly

[hexed agate]

Now what is the next step?

[worn summit]

add 1/3x to both sides

so 1/3x=x

[red stag]

you multiply both sides by 3 to cancel the division in 2x/3

[hexed agate]

I mean you could do that but that doesn't help

[worn summit]

ok yeah i did that but im just pointing out that something doesnt make any sense

im wondering why that makes a nonsencicle answer

[red stag]

what makes that answer?

[worn summit]

despite the fact i did everything right

[red stag]

I dont understand whats wrong with this:

0 = 2x/3 | *3 (multiply both sides by 3 to cancel the division in 2x/3)
0*3 = 2x
0=2x | :2 (devide both sides by 2 to cancel multiplication by 2 in 2x)
0/2 = x
0 = x

[worn summit]

yeah that all makes sense to me

[red stag]

so then what doesnt?

[worn summit]

this

wait no

[worn summit]

this

because then 0+1/3x=x

and that doesnt make sense

so whats going on there

why doesnt it work

[red stag]

1/3x = x doesnt make sense?

is that what you are saying?

[worn summit]

yes

[red stag]

it does, if you put 0 as x

cause 1/3 * 0 = 0

[worn summit]

no but im adding

[red stag]

1/3x = x, when x = 0:
1/3*0 = 0
0 = 0

[worn summit]

what if you add

[red stag]

0 + 1/3 x = 1/3x

[worn summit]

ok look

2/3x=0

right

so

[red stag]

Ok I see this is going nowhere and I cant help you (and I have to go). Hopefully someone else will help you (or your teacher)

[worn summit]

what im asking is why doesnt adding 1/3x to both sides work

what rule are we violating

alright

[hexed agate]

You could add a banana 🍌 to both sides and still no rules are violated

2/3x + 🍌 = 0 + 🍌 is also correct

[worn summit]

ok

so what if the banana is 1/3x

why doesnt that work

[hexed agate]

What is your reasoning behind adding 1/3x?

[worn summit]

so i can get x

[hexed agate]

Yes that's the correct way of thinking, however by adding 1/3x you are again placing another x on the other side

So you have to isolate x again

[worn summit]

ok so 1/3x=x is true

[hexed agate]

Yes

[worn summit]

wha

[hexed agate]

Now I could solve 1/3x = x

[worn summit]

but that doesnt make sense

9 isnt 3

6 isnt 2

30 isnt 10

[hexed agate]

1/3 of 0 is 0 though

[worn summit]

but were adding here right

thats what we did

we added 1/3x to 2/3x

and we got x

and we added 1/3x to the other side

so a third of x is x

and that doesnt make sense

what am i doing wrong here

[hexed agate]

That makes sense if x=0

A third of 0 is 0

[worn summit]

oh wait

yeah

your right

ok thank you

.close

[obtuse pebble]

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[hushed wraith]

How do I determine the number of smoke detectors needed to cover an entire area of a house?

What I know is that we have to use the radius of detection of a smoke detector and use the area of a house, but I'm not sure if I'm right.

Is (total area of floor/area covered by a smoke detector) correct?

And afterwards, I have to place these smoke detectors in each room (excluding bathrooms) correctly, but I don't know how to think of an equation for it.

[obtuse pebble]

@hushed wraith Has your question been resolved?

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[lusty violet]

Hey there, can someone help me? I need to figure out how to answer this question.

[timid silo]

substract 2/5 and add 2/5

[lusty violet]

Can you explain to me why? I feel like it would do nothing as we are adding it back

[timid silo]

$\lim_{x \to infty} \frac{2x-2}{5x+6}-\frac{2}{5} + \frac{2}{5}$

ok so evaluate $\frac{2x-2}{5x+6}-\frac{2}{5}$

the xs cancel out nicely

[lusty violet]

the numerator has 2x-2 not 2x-5

[timid silo]

oh I made a typo

but my method still holds

[warm shale]

pure maths>>>physics

pure maths>>>physics

[lusty violet]

Ok, in order to subtract I need them to have the same denominator right?

[timid silo]

just add the fractions I mean

$\frac{(2x-2)5}{(5x+6)5}-\frac{2(5x+6)}{5(5x+6)}$

[warm shale]

pure maths>>>physics

[timid silo]

that's how u make the denominators equal if u didn't knew

[lusty violet]

Yea, cross multiply

[timid silo]

yes

so add up

two weeks ago I had this chapter, my method is to factor by the largest x and put negative powers in the parenthesis, then simplify

you can't calculate at first sight because you arrive at an indeterminable form of inf/inf

[lusty violet]

the xs in the numerator cancel each other out and alls that left is -22 in the numerator? denominator is 25x+30

[timid silo]

yes

so

now

$\lim_{x \to \infty} \frac{-22}{25x+30}+\frac{2}{5}$

[warm shale]

pure maths>>>physics

[timid silo]

after which it's obvious

[lusty violet]

{-50 + 50x}/ {125x + 150}

[obtuse pebble]

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[misty beacon]

helloooo

[obtuse pebble]

Please don't occupy multiple help channels.

[misty beacon]

helloo