help-28

[hot lily]

Oh wait

[viral ore]

a*sqrt(3) is opposite to 60 degrees, a is adjacent, and 2a is hyp

[hot lily]

I thought you were dividing by sqrt 3 lol

[viral ore]

which side is the hypotenuse here?

[torn jolt]

5 squared 3 is the hypotenuse?

[viral ore]

close

[torn jolt]

💀

[viral ore]

the hypotenuse is always the longest side, so it is across from the largest angle

[torn jolt]

OHHH

so its y here in this situation?

[viral ore]

it is probably confusing because it is rotated compared to the other ones

[viral ore]

yes

[torn jolt]

alr thank you so much man ive been stuck on this for like almost a good hour

[viral ore]

glad to help

if you have any more questions feel free to ping me

[torn jolt]

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[woeful anvil]

Could I get Guidance on this problem?

[woeful anvil]

(not sure where to begin)

[gritty rose]

Maybe write both lines as y=mx+b form

Hmmm but it's a line segment

[waxen raven]

Sorry I don't mean to interrupt but js wondering what level of maths is this?

[woeful anvil]

calc 2

[waxen raven]

Ty

[woeful anvil]

you also take a little bit of parametric in pre calc

but not much

[waxen raven]

I thknk I got it

[waxen raven]

Sub in t=0 into both equations, what do you get?

[waxen raven]

You get a point

Now sub in t=1

You get another point

[woeful anvil]

but isnt t suppose to equal x and y

[waxen raven]

Wdym?

[woeful anvil]

because its parametric

t is the parameter

of the equation

its defined by xy

[waxen raven]

Mhm, and the question is asking "t is between 0 and 1. Describe the line segment that joins the two points at these two t values"

[woeful anvil]

and also how do you show that it joins x of 1 and y of 1

if u do that u eliminate it

so arent i suppose to rearrange for t

[waxen raven]

Sorry hang on let me rephrase it

[woeful anvil]

i dont think you can physically do that with that logic because of the way t is defined

i need x and y to be isolated on one side

[waxen raven]

Not necessarily

Parametric equations are (for example) x=3t and y=4t
If you sub in t=3, you will have x=9 and y=12
So the point at t=3 is (9,12)

[waxen raven]

Same sort of logic here, what happens when you sub in t=0?

[woeful anvil]

u get x1

nad y1

[waxen raven]

You get x=x_1, and y=y_1, so you have your first point P1 (x1,y1)

Now what happens if you sub in t=1?

[woeful anvil]

u get everything

[waxen raven]

We subbed in t=0 because that's where rhe domain "starts"

[waxen raven]

Now let's sub in t=1 because that's where the domain of t "ends"

[waxen raven]

Huh?

[woeful anvil]

nothing gets removed

by making it 1

[waxen raven]

Something does get removed

Try again

[woeful anvil]

so just x2

remains

[waxen raven]

$x = x_1 + (x_2 - x_1)$

[woeful anvil]

yea

[glossy valve]

Moonful

[woeful anvil]

can i ask

[waxen raven]

Yesss

Same sort of thing with y2

[woeful anvil]

what is the answer key telling me with this?

how do they get that relation equation

where does it come from

[waxen raven]

The first part of the marking key is what we've done so far

[woeful anvil]

i see

i have difficulty comprehending

sorry

[waxen raven]

Well we know that at t=0 we have the point P1(x1,y1)

And at t=1 we have the point (x2,y2)

That's what we've shown so far

Now find the equation of the line connecting those two points

[woeful anvil]

how would i do this?

slope form?

idk

[waxen raven]

How do you find the equation of the line connecting for example (1,2) and (2,4)?

[woeful anvil]

i actually forgot idk i never worked with graphs in calc yet until now but improbly just making excuses

let me look up

trust

equation

i mean

Is this a formula

you just know

ive never seen it lol

not in calc at least

prob before calc

[waxen raven]

Do you know that third equation?

[woeful anvil]

yes

[waxen raven]

Hang on let me write it on paper

[woeful anvil]

okay

[waxen raven]

So from that third equation

m is the gradient

Do you know how to get thr gradient of the line passing through 2 points?

[woeful anvil]

i do not know what a gradient is

im sorry

[waxen raven]

Do you know $\frac{\Delta y}{\Delta x}$?

[glossy valve]

Moonful

[woeful anvil]

yes

[waxen raven]

That's what the gradient is

[woeful anvil]

i see

[waxen raven]

How do we get it given 2 points (x1,y1) and (x2,y2)?

[woeful anvil]

y2-y1/x2-x1

[waxen raven]

So that's m

If you sub that in for m

[waxen raven]

You get this right here

[woeful anvil]

i see

cool cool

[waxen raven]

Now in that equation

[waxen raven]

This y1 and x1 represent any coordinate on the line

So we're gonna do is sub in (x1,y1)

I did that so its correct out of technicality but the equation hasn't changed right now

[waxen raven]

So we still have this equation rn

Nwo what were gonna do is rearrange the equation to get

This right here

Yeah?

[woeful anvil]

okay yea

[waxen raven]

So the equation of the line we have that passes through (x1,y1) and (x2,y2) is this bottom equation here

What you have to show

[waxen raven]

Is that when you convert these parametric equations to cartesian

You will get the same equation

Do you know how to do that?

[woeful anvil]

cartesian is just in terms of t

[waxen raven]

Cartesian means in terms of x and y

Not t

[woeful anvil]

yea

that

[waxen raven]

So how do you do that?

[woeful anvil]

replace t in like a y equation

in terms of x

[waxen raven]

You could do that but that takes a while

[waxen raven]

Why not just rearrange for t in both equations and equate them?

[woeful anvil]

solve for t then make them equal to each other?

[waxen raven]

Yes

[woeful anvil]

$x = -2 + [3 - (-2)]t = -2 + 5t \quad \text{and} \quad y = 7 + (-1 - 7)t = 7 - 8t$

[glossy valve]

Light

[waxen raven]

Huh

What did you do?

[woeful anvil]

right?

uhh

[woeful anvil]

i plugged them back into here

then make them equal

to each other

?

[waxen raven]

What did you plug into there?

[woeful anvil]

the values given

from the porblem

on part b

[waxen raven]

That's part b

[woeful anvil]

oh

[waxen raven]

We are doing part a rn

[woeful anvil]

skull

[waxen raven]

Does this make sense to you so far?

[woeful anvil]

yes sorry iw as just writing that down

[waxen raven]

Since they both equal t

We can equate them

Then I rearranged the equation to get that third line

[waxen raven]

Now is that third line thr same equation as this?

[woeful anvil]

yes

[waxen raven]

So there you go

What we've done is that

1. Find the equation of the line that goes through P1(x1,y1) and P2(x2,y2)
2. Show that this equation is the same equation we get when we convert the parametric equations to Cartesian form

That's how you do part a

[waxen raven]

This is how you do part b, as you've done

Okay I've gotta get ready now

[woeful anvil]

ok thank u a lott

i apprecaite it

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[opaque locust]

could someone help me walk through this question

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[swift bridge]

I see but isn't the set Ext(S) = ${ x \in S^c : \exists \epsilon > 0 \text {such that} B(x, \epsilon) \subseteq S^c }$. Therfore shouldn't $(Ext(S))^c = {x \in E : x \in S \text{ OR } \forall \epsilon > 0, B(x, \epsilon) \nsubseteq S^c }$?

[glossy valve]

LXDL

[swift bridge]

#help-27 message continued from this convo

@granite torrent

[swift bridge]

i thought I understood your explanation, but then going back dont we need to include $x \in S$ in the negaation

[glossy valve]

LXDL

[swift bridge]

and then we do the same for $(S^o)^c$

[glossy valve]

LXDL

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[worn widget]

Could someone why the inequality turns out like this

And point out the main concept used here, I know I learned it a few months back, and I need to use it to solve a problem

But I can't remember why this was so! If you could refer to me the concept I'll go back and remind myself on it!

[onyx glen]

$x^2\leq a\iff -\sqrt{a}\leq x \leq \sqrt{a}$

[glossy valve]

ann.in.a.teacup

[worn widget]

Does this have something to do with parabolas?

[celest tapir]

It's easiest to see with an example

[glossy valve]

dyxn

[worn widget]

Ohhh right right

[celest tapir]

so you can interpret it as the distance of x from the origin is less than or equal to the square root of a

which means x can go both up until positive and negative square root of a

[worn widget]

Thank you!

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[woven grove]

Hi im having issues with finding the inverse of the cumulative distribution function for case 3. I get this for the third case where x goest between 0 and 1. Is this correct?

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@woven grove Has your question been resolved?

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@woven grove Has your question been resolved?

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@woven grove Has your question been resolved?

[gritty rose]

!original

[full forum]

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.

[woven grove]

The first image is a given CDF and second is my proposal for the inverse CDF for the third case in the 2nd image. My problem is finding the inverse CDF for this given CDF

[gritty rose]

says picture or screenshot

[woven grove]

Ok I don’t have access to making an image of the original problem at the moment. All info is given though! But i understand if i need to close the thread! 🙂

[gritty rose]

then the given info is malformed

the cdf is both 0 and 1 when x > 1

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@woven grove Has your question been resolved?

[woven grove]

Oh yeah ur right, Its supposed to only be 0 for x < -1 thats on me, sry!

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[flat python]

I think first step you need to put it as one over because you have a negative exponent

oh wait

like 1 divided by b root ((1/x)^a)

yes

yeah

when simplified did you get the right answer?

b is the index

if you raise it to the b i think you get (1/x)^(a*b)

wait no I was wrong

to get rid of the root you raise the numerator and the denominator to the power of b

[olive nymph]

its because of the minus sign, the minus kind of inverts

[flat python]

I feel like I'm making this way too complicated

i'm not sure you can raise everything to the power of b

oops

[olive nymph]

since you got (1/x)^-a/b first remove that - , so we have x^(a/b)

now this is equal to (x^a)*(x^(1/b))

[flat python]

ooohhh

[olive nymph]

x^a stays as is, and x^(1/b) goes to the root

be inverting 1/x

making it x/1=x

if you have - in the power, you just invert

okay forget everything

no

^ means power

so lets say you have 4/3 to the power of -2

4/3 to the power of -2, is the same as 3/4 to the power of +2

inverting the fraction

[flat python]

this might help

its my process

[olive nymph]

its numerator and denominator

[olive nymph]

i think this is complicated

if you have (x/y) to the power of -1 then that y/x to the power of 1, is that understandable?

[flat python]

does that make sense?

which part?

oh lol

yes it is

It works the same with multiplication

[honest hill]

$\frac1{\sqrt[b]{\left(\frac1{x}\right)^a}}=\frac1{\frac{\sqrt[b]{1^a}}{\sqrt[b]{x^a}}}=\frac{\sqrt[b]{x^a}}{\sqrt[b]{1^a}}=\sqrt[b]{x^a}$

[glossy valve]

Bonk

[flat python]

there we go

that's easier to read thx

I simplified the denominator of the bottom screen to 1

no matter what you do to 1 (with the exception of + and -) it will stay one

so you can completely get rid of everything around the 1 in the denominator

?

well if you divide by a fraction, it is the same thing as multiplying it by its inverse

lol

I'm so sorry I hope someone else can help I have to go

[honest hill]

$\sqrt[b]{1^a}=1$

[glossy valve]

Bonk

[honest hill]

1 to the power of anything is just 1

$1=1\cdot 1\cdot 1\cdot 1\cdot 1\cdot 1\cdot 1\cdot ... \cdot 1\cdot 1=1^n$

[glossy valve]

Bonk

[honest hill]

basically, $1^{\text{anything}}=1$

[glossy valve]

Bonk

[honest hill]

the anything can be a fraction

$\sqrt[c]{a^b}=\left(a^b\right)^{\frac1{c}}=a^{\frac{b}{c}}$

[glossy valve]

Bonk

[honest hill]

yup

yup

similar case for 0

$0^{\text{anything}}=0$

except for 0^0

thats an exception

[gritty rose]

did you mean anything^0

[glossy valve]

Bonk

[honest hill]

no 🙂

no, x^0=1

0^0 does not exist

im not going to go philosophical

go to #discussion for that

no comment

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[wooden aurora]

can someone explain to me like the notation of domain and range

[sharp lagoon]

Domain and range are sets.

[craggy tapir]

i've seen some people use $D_{f(x)}$ to refer to domain of $f(x)$

[sharp lagoon]

There are various ways to write sets.

[wooden aurora]

ive only seen of f(x)

[sharp lagoon]

You can write {1, 2, 3} if the set has exactly those numbers in it and nothing else.

[wooden aurora]

for smth like this

how would u write it

[wooden aurora]

wait so what is a set in a domain?

[umbral dome]

you can write it as the union of two intervals

[sharp lagoon]

You can write [1, 3] for all the real numbers between 1 and 3, including those numbers.

You can change the [ to a ( if you want to leave out the 1.

[wooden aurora]

oh so that means it includes all

[sharp lagoon]

You can change the ] to a ) if you want to leave out the 3.

So, like [1, 3) is the real numbers between 1 and 3, including 1, but not including 3.

You can also do this with infinity.

[wooden aurora]

yes

[sharp lagoon]

Like (-infinity symbol, 3] means 3 and lower.

Or (-infinity, infinity) means all real numbers.

Infinities have to have the rounded brackets.

[wooden aurora]

oh i see

what abt the {

[sharp lagoon]

Because infinity isn't one of the real numbers.

[wooden aurora]

is that just to define the thing

[sharp lagoon]

{} means you're listing things off.

[wooden aurora]

uh

[sharp lagoon]

{1, 2, 3} means 1, 2, and 3 are the only things included, not any other numbers.

[wooden aurora]

oh ye

i was doing function notation earlier and u just input the value into the thing right

[sharp lagoon]

With [ and (, it's interval notation, where you have an interval of real numbers and you're telling what's in it.

[wooden aurora]

and then the answer do u write it with {}

[sharp lagoon]

With the examples you have above, you have two intervals in the top example.

[wooden aurora]

this?

[sharp lagoon]

Like let's say the left filled in dot is 5, the next unfilled in dot is 6, and the next filled in dot is 7.

[wooden aurora]

oh cuz the white dot means its nothing

[sharp lagoon]

Right, it means that number's not there.

[wooden aurora]

yes

[sharp lagoon]

So, with 5, 6, and 7 for the dots, you have [5, 6) u [7, infinity).

The u means both of them put together.

It's not a lowercase u, though.

[wooden aurora]

what does U mean

is that and or or?

[umbral dome]

,, [5,6) \cup [7,\infty)

[glossy valve]

cloud

[sharp lagoon]

It's like ([5, 6) \cup [7, \infty)).

[glossy valve]

Chai T. Rex

[wooden aurora]

wai what does U mean

[sharp lagoon]

The u there means union.

[mystic tapir]

and

its union

[wooden aurora]

oh ok

[sharp lagoon]

It means those two sets put together.

[mystic tapir]

wait no

nevermind

its or

but thats a bad way to think of it

[umbral dome]

the union of two sets (denoted U) means the set consisting off everything which is in either set

[sharp lagoon]

So, you have all the reals between 5 and 6, including 5 and you have all the real numbers from 7 up, including 7.

[mystic tapir]

because its like any number thats in the first set OR the second set

but the better way to think is just "the range is this set and that set"

[umbral dome]

so for example {2,3,7} U {3, 5, 8} = {2, 3, 5, 7, 8}

[wooden aurora]

oh ye

wait how do i identify

whether i use or or and

U or the other

idk any symbols

[sharp lagoon]

You don't use or or and.

You use union [, (, {, }, ), or ].

[umbral dome]

$\cap$ is the intersection, but it's not really needed for writing intervals

[sharp lagoon]

And also U.

[glossy valve]

cloud

[wooden aurora]

oh okay

ohh i think i get it

[umbral dome]

the intersection is all the elements belonging to both sets, so ${3, 5, 7} \cap {2, 5, 8} = {5}$

[glossy valve]

cloud

[wooden aurora]

what abt like if we name them abc and use > or <

[umbral dome]

but for intervals you can just write a shorter interval

[sharp lagoon]

What do you mean by that?

Do you have an example?

[wooden aurora]

uhh not rlly it was on a board

but

hmm wait lemme try

[sharp lagoon]

You can say (-5 < x \le 6) or something, instead of ((-5, 6]).

[glossy valve]

Chai T. Rex

[wooden aurora]

would this make sense

[wooden aurora]

oh ye kinda like that

[sharp lagoon]

It's unclear what the vertical line is in the top part.

[wooden aurora]

hmmm

ok so my teacher made

an example from this

so this was one of the examples

but then he just made like the dots change and other stuff

[sharp lagoon]

OK, with that diagram, what's the interval notation for the left part?

[wooden aurora]

is that a

[sharp lagoon]

Yes, the part with a.

[wooden aurora]

interval notation?

my book

[sharp lagoon]

Yes, like [5, 6).

[wooden aurora]

oh its not using numbers

its only using a or b

[sharp lagoon]

Right, but a is a number.

[wooden aurora]

it says like u can write it as x < a or x > b

[sharp lagoon]

Variables are generally numbers.

[mystic tapir]

what

[sharp lagoon]

Right, you can write it like that.

[sharp lagoon]

You can also write it in interval notation, like ((-\infty, a)).

[glossy valve]

Chai T. Rex

[sharp lagoon]

That's the same as x < a.

[wooden aurora]

so my teacher used like these diagrams to like make smth like this but more complex ig

[wooden aurora]

inf ?

wait why is inf used as a negative

[mystic tapir]

if its all numbers less than a, then its all negative numbers too, all the way down to negative infinity

[sharp lagoon]

Negative infinity means less than any real number.

So, if you say it's between negative infinity and a, then it's less than a, and it can be anything less than a, not just some numbers less than a.

[wooden aurora]

how would equal to and sign look with that

_

smth like this

[wooden aurora]

how would u write like this

[sharp lagoon]

I'm not sure what you mean by 'that'.

[torpid perch]

(-∞, a) ∪ (b, ∞+)

[sharp lagoon]

How would I write what like that?

[wooden aurora]

what an example of using <_ sign which means like this

[mystic tapir]

x < a

[sharp lagoon]

That's any number lower than a, so x < a.

The negative infinity means you can go as low as you want, and the a means you can go up to a, but not reach it.

[wooden aurora]

oh

[sharp lagoon]

Like if a was 5, you could get 4.999.

[wooden aurora]

so negative inf > a wouldnt make sense

[sharp lagoon]

No, you wouldn't include that with less than or greater than.

It's left out of those.

[wooden aurora]

wym

left out

[sharp lagoon]

I mean you wouldn't write any less than or greater than or whatever with infinities.

You include infinities with interval notation only because you need a left side and a right side.

[wooden aurora]

oh

[sharp lagoon]

But you don't need both in an inequality.

[wooden aurora]

ohh so with infinities u use ,

[sharp lagoon]

You use a comma with all interval notation.

[wooden aurora]

yes

do u have vid

so i can pratcice

practice

[sharp lagoon]

Search for interval notation on this page: https://www.khanacademy.org/math/algebra/x2f8bb11595b61c86:functions.

[wooden aurora]

Oh alr Ty

[sharp lagoon]

No problem.

[wooden aurora]

ok ty ima check it

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[graceful python]

I need help on question 11 pls

[hot lily]

First, lets understand how to use interest

12% interest, means you add 12%, or 0.12

Adding 0.12P to your original P results in 0.12P + P = 1.12P where p is the original amount of money

@graceful python are you here?

[graceful python]

Yes

[hot lily]

Ok, so do you understand what ive written

[graceful python]

Kinda

[hot lily]

Ok, what do you need me to elaborate on

[graceful python]

The part where you add 0.12P down

[hot lily]

Ok, so we start with P = 8,000

[graceful python]

Yes

[hot lily]

Then we add 12% of that

Whats 12% of 8,000 (or whats 0.12 of 8,000)

(waiting for your response)

[graceful python]

Is it 960

[hot lily]

Now add that to the original value

[graceful python]

So 8960

[hot lily]

8,000 * ? = 8,960

[graceful python]

I'm confused

[hot lily]

So you added 8,000 + 0.12(8,000) = 8,960 right?

[graceful python]

Yeah

[hot lily]

If you factor that, you get1(8,000) + 0.12(8,000) = 1.12(8,000)

[graceful python]

Yeah

[graceful python]

Would didn't become 8960

[hot lily]

So you multiplied the original amount by 1.12

Do you see how that has a relation with 12%?

[graceful python]

Yesss

[hot lily]

So that 1.12 multiplier is applied once a year (for 12% annual interest rate)

So far so good?

[graceful python]

Yes

[hot lily]

For 11, the interest is applied for 3 years

So how many times is the 1.12 multiplier applied?

[graceful python]

Uhhhmm 3.36 if I thinking it right

[hot lily]

You need to multiply by 1.12 three times

$$8,000 \cdot 1.12 \cdot 1.12 \cdot 1.12 = 8,000 \cdot 1.12^3$$

[glossy valve]

King Leo [Ping For Help]

[graceful python]

Yess

[hot lily]

So what is that expression equal to

[graceful python]

11234.424

[hot lily]

Now round that to the nearest cent

[graceful python]

Wait were we doing the 11 a( i)

[hot lily]

Ya

[graceful python]

Okii

[slate violet]

e moment

[graceful python]

I got the answer we just got before but it wasn't right so am I missing a step

[hot lily]

You need to see how much interest was paid

You started with 8,000 and ended with 11,234.42

So how much did you gain

[graceful python]

3234.42

Omggg I knew I was missing a step

Wait

[hot lily]

Wait you already solved this before?

[graceful python]

Noo

I did the first part to get 11234.42

[hot lily]

Bro you could have told me 💀

I thought you didnt understand it at all

[graceful python]

Is there any step because the answer that it said was 3239.42

[hot lily]

Uh one sec

[graceful python]

Sorry cause I didn't understand the part U said on top cause all I did was use the formula towards it

[hot lily]

My mistake, this subtraction messed up

[graceful python]

Uhm I'm confused

So did we do wrong

[hot lily]

#help-28 message

You only need to redo that subtraction

[graceful python]

Which part ??

[hot lily]

How did you get 3234.42

[graceful python]

Ohhh

I get it now

[keen pulsar]

Hello

[hot lily]

Hi, do you have a math question

[keen pulsar]

No just in case, I thought this for general chat sorry pl

Can you recommend nr s channel which can help me with this

And also operations on exponents

[hot lily]

#❓how-to-get-help

[keen pulsar]

Ohk thanks

[keen pulsar]

No the problem is idk maths and it's concepts a channel like makit was great but it's for more complex matte

It's not that idk how to solve s problem but idk how to do maths

@hot lily sorry for the annoying distrubance I am causing in the first place cause am a newbie

Just need like a gud english vid that can explain thst stuff

Thank you

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[errant saffron]

is this correct? i’m seeing a bunch of different things

[hot lily]

✅

[errant saffron]

ty :>

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[hard siren]

I have a doubt in the line, Since U alpha is open, how have they assumed? How is it pre implied that they are open?

Lemme know if u need more context on how things are defined

[onyx glen]

can you highlight the exact line you're talking about?

oh you are wondering why they just say U_alpha is open

[hard siren]

yeah

[onyx glen]

that's easy: they are members of T, the topology

the topology is the collection of subsets of your space that are declared open

[hard siren]

but aren't we proving that this will be a topology?

[onyx glen]

yes we are. and we are showing that an arbitrary union of opens is again open

[hard siren]

so if i am not wrong we just simply are assuming that $U_{\alpha}$'s are open and their arbritaty unions is also coming out to be union, so this is "behaving" like a topology?

[glossy valve]

Gamin8ing

[onyx glen]

... yes?

[hard siren]

hmm, thanks a lot

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[silk kindle]

Anyone help

[onyx glen]

do you have any progress so far?

[silk kindle]

Yesss

[onyx glen]

show

[silk kindle]

Noo

[onyx glen]

also in general, when you post a problem here and have some progress already, then you should show your progress.

[onyx glen]

???

[silk kindle]

Not in this question

[onyx glen]

i was asking you about this question.

[silk kindle]

I know mothing aboutbthis concept

[onyx glen]

you know nothing about how decimal notation works?

units place, tens place, etc?

[silk kindle]

Ik that

[onyx glen]

(not even any further than tens place)

[onyx glen]

ok so

you have a two-digit number

its units digit is 3
its tens digit is unknown

[silk kindle]

Yes

[onyx glen]

when something is unknown but you want to find it, you should call it by a letter

[silk kindle]

Ok

[onyx glen]

so let the tens digit be x

the value of the number is?

[silk kindle]

Which

[onyx glen]

$\overline{x3}$

[glossy valve]

ann.in.a.teacup

[onyx glen]

this one

[silk kindle]

Ok

[onyx glen]

with x in tens and 3 in units

i want you to write it as an ordinary algebraic expression

do you understand what i am asking from you?

[silk kindle]

No

Can u explain it like this

[onyx glen]

we are on the third line of that exact solution

i was trying to get you to say it yourself

[silk kindle]

How did 10 come

[onyx glen]

the number = 10x+3

well

you know how place value works right

when we write a number like 72

[silk kindle]

Yes

[onyx glen]

the 7 means not just seven, but seven tens

[silk kindle]

Ah ok

Understood

[onyx glen]

so likewise, x tens and 3 units

[silk kindle]

Ok

[onyx glen]

i guess you already posted the given worked solution

can you tell me how far you get in there until you are stuck again?

[silk kindle]

I understood this auestion

[onyx glen]

ok

[silk kindle]

Solved it in my notebook

[onyx glen]

best to close this channel and open a new one for your next

[silk kindle]

I have another question

[onyx glen]

close this channel and open a new one

[silk kindle]

Ok

.close

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[grim chasm]

what exactly do you mean?

is the root alone? is the x^2 under the root?

The square root on the left side is wrong

when you take the square root, you elevate the whole thing to the power of 1/2

it would be $\sqrt{x^4 - x^2}$

[glossy valve]

Ryse

[grim chasm]

different from $\sqrt{x^4} - \sqrt{x^2}$

[glossy valve]

Ryse

[grim chasm]

you cannot do that

lets use an easier power to deal with

$(2+7)^2$

[glossy valve]

Ryse

[grim chasm]

$if you work out the thing inside the bracket you just get 9^2 and so 81$

ew

yeah

53 is not 81

for an easier visualization

are u listening

im not gonna help if you just dont read what im writing

ur not

well ask someone else for help

brother im tryina help you and you just ignore what im saying

not showing attitude

i wanted to make the power thing clear first

and you keep going down that path

alr sorry for earlier

but yes square rooting first is wrong because

• the square root is done wrong
• you also have to address cases in which it is positive or negative

i wouldn't use the square root approach first but if you wanted to use it

you can bring the x^2 to the other side

[grim chasm]

$x^4 = x^2$

[glossy valve]

Ryse

[grim chasm]

now taking the square root is much easier

but you have to remember to put the + and -

i would factorise everything

square rooting is also an alternative

i find factorising faster and easier

hmmm it's like rewriting the expression as factors (multiplications)

[mystic tapir]

move everything to the same side

[grim chasm]

you can first group the x^2

[mystic tapir]

so the other side is 0

[grim chasm]

almost not quite

[mystic tapir]

close

[grim chasm]

-1 inside

[mystic tapir]

not -x

[grim chasm]

and then since the result must be zero

one factor = 0 is enough to make everything zero

$x^2(x^2 -1) = 0$

[glossy valve]

Ryse

[grim chasm]

so $x^2 = 0$

[glossy valve]

Ryse

[grim chasm]

or $(x^2 -1) = 0$

[glossy valve]

Ryse

[grim chasm]

yeah

you can solve $(x^2 -1) = 0$ by factorising it further or just moving the 1 to the other side and + or -root

[glossy valve]

Ryse

[grim chasm]

$(x-1)(x+1) = 0$

[glossy valve]

Ryse

[grim chasm]

correct it cannot disappear

x^2 = 0

so x must be equal to 0

and it has double solution

[grim chasm]

here x can just be equal to 1 or -1

oh this

if we know that the entire thing is equal to zero

remember

any number multiplied by 0 is 0

so it is enough for a factor to be equal to 0

we can just consider the factors alone

which are x^2 and the brackets

if the x^2 alone is equal to zero, the whole equation is equal to zero

exactly

3 solutions

one of them is a repeated root

[grim chasm]

root also means solution btw

we could have also used the square root if you want to see how it could have been done

yes

a repeated solution would be more accurate

yes you can

b and c are 0

you can also just take the square root there

x^2 = 0

x = 0

it is a little bit weird tbh but you can imagine it as that both solutions are at that exact point

normally quadratic have 2 solutions

[grim chasm]

whether they are real, repeated or complex

you mean dividing by x^2?

you can, you have to check what happens if x^2 = 0

definitely possible

so before doing that you have to check x = 0

to check if it is a solution

$x^4 - x^2$

[glossy valve]

Ryse

[steel solar]

u=x^2

[grim chasm]

check x = 0

[grim chasm]

nah no need to do that

$x^4 - x^2$ if x = 0 it becomes 0 and that is true so 0 must be a repeated solution

[glossy valve]

Ryse

[grim chasm]

now we can divide by x^2

$x^2 - 1$

[glossy valve]

Ryse

[grim chasm]

yes

you can definitely do it that way too

a little bit unorthodox but correct nonetheless

yes

remember to bring the one to the other side first

$x^2 = 1$

[glossy valve]

Ryse

[grim chasm]

$\sqrt{x^2}= \sqrt{1}$

[glossy valve]

Ryse

[grim chasm]

${x}= {1}$ or $x = -1$

[glossy valve]

Ryse

[grim chasm]

we dont have any x^2 = x

oh wait im dumb

if you square root it yes you do

but it would be + or - x

you would have to deal with multiple equations by doing it this way

same base so yes

x^2 - x can be rewritten as
$x(x-1)$

no, its both

you would also need to set up inequalities

which is why i don't quite recommend using sqrt for this one

you can definitely do it but it takes way longer

it's a Union if you do it that way

no Intersections

yeah have you done inequalities yet?

like $x^2 - 2x > 1$

[glossy valve]

Ryse

[grim chasm]

not sure what you mean with this notation

i'd recommend revising that and factorising they are the most important things in Maths basically and they will always appear later on and you are expected to know how to do them

yes

yes you can do same denominator and just consider the top

ye

you can also just multiply by 2 both sides

(this equation is wrong and different from the beginning one btw)

yes

yeah normally you have to be careful but on the right you have 0 so everything cancels

you have to be very careful multiplying and dividing by a variable though

yeah because the equation you came up with is wrong

the answers should just be 0 repeated, +1, and -1

you actually did get it

remember x^2 = 1

x = + or - 1

(-1)^2 gives 1

1^2 gives 1

we dont have any imaginary numbers

they only appear when you have a negative inside a root

yes

do you agree that any number multiplied by itself gives a positive answer?

• times + gives +

. - times - gives +

yes

so when you take a square root

you always put a + or - in front

x^2 = 1 try plugging in 1 and then -1 inside the x

$(-1)^2 = 2$

[glossy valve]

Ryse

[grim chasm]

$(1)^2 = 2$

[glossy valve]

Ryse

[grim chasm]

replace x with 1 just to check and after having done that replace x with -1

sorry 1 is supposed to be there

$(1)^2 = 1$

[glossy valve]

Ryse

[grim chasm]

yes

$(-1)^2 = 1$

[glossy valve]

Ryse

[grim chasm]

well imma dip

gtg

-1 * -1 gives +1

when you take sqrt you want to make sure you include the negative number as well

valid for every even root

not odd because it doesnt change signs

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[merry swallow]

Suppose $(a_n)$ is non-negative. Use the Cauchy Criterion to prove that the infinite product ( \prod (1 + a_n) ) is convergent if and only if the series ( \sum a_n ) is convergent . Do not use the Monotone Convergence Theorem.

[glossy valve]

somethingwrong

[merry swallow]

Hi, i need help for this, I am not too sure what im suppose to do, if i just try to apply the cauchy criterion, i get

$\Longrightarrow$ If $\prod (1+a_n)$ is convergent, by the cauchy criterion, for any $\epsilon>0$ there exists $N\in \mathbb{N}$ such that $n\ge N$ implies $|\prod_{i={1}}^n 1+a_i-\prod_{i={1}}^N 1+a_i|<\epsilon$

[glossy valve]

somethingwrong

[merry swallow]

im not sure how this can relates to the series

[eternal stone]

Use log

[merry swallow]

hmm where would I use log

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[fast zinc]

what does it mean "where m, n ∈ Z"?? i get the "n ∈ Z its saying n is an integer but what does the "where m" part mean bc it doesnt say what m is?? is this a typo?? am i dumb??

[kindred grove]

m and n are integers

[onyx glen]

"m, n in Z" is shorthand for "m in Z and n in Z"

they just don't want to write the "in Z" part twice

[fast zinc]

so the comma is the same as &

its saying both m and n are ∈ Z

?

[kindred grove]

yes

[fast zinc]

okay thank you very much :>

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[covert flint]

Can anyone sketch the triangle described here, I'm confused on how it should look, ty 🙂

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@covert flint Has your question been resolved?

[hidden harbor]

I'm very sad reading this, since this is a problem that is usually solved in literally 1 line

[covert flint]

Lol some of the stuff I've seen that's 'hard' by the sat standard is crazy to me

But ty for the diagram

[hidden harbor]

that's all that is needed

if you wanna be detailed, you'd first say that ABC is congruent to ADE, and thus, the equation

[covert flint]

Also 'hard'

[torpid perch]

I feel smart knowing i can understand both, while being hard questions

which website is it?

even though it might be simple, basic proportionality theorem is really cool

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[solemn sand]

integral of cos^2x sin^2 x dx is it better to adjust 1/4 and 4 inside the antiderivative and make it whole squate for sin2x so 1/4intgeral of (1- cos4x)/2 or write cos^2 x as 1-sin ^2x open bracket and go from thetre

[solemn sand]

im getting 2 different answers both ways

[onyx glen]

can you show your two different answers?

they may not be as different as you think

[celest tapir]

if you add the + C they might look different but are still the same answer

If you want to make sure plot the two graphs on desmos

and if one is a raised version of the other then both solutions are valid

[solemn sand]

in one im getting x/8 - (sin4x)/4 +c and in other im getting x/2 - x/4 - x/8 - (sin4x)/32 + c

first one i did by adjusting 4 and taking whole square and second by identity

[solemn sand]

they are somewhat similar but not same

it becmes x/8

but sin term denominator is 8 times

[onyx glen]

ok gonna need to see your work for both then

to spot the mistakes

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[slim folio]

Hi, I was wondering can anyone tell me how to see the graph for P_theta vs theta?

[slim folio]

And how I combine to get the electron probabilistic cloud?

Nvm, I know for the P_ theta

[solemn sand]

this channels taken u mind if i dm it to you?

[cold agate]

open a new one cuh

[onyx glen]

yeah open a new one

dont dm me

i might not be available

[solemn sand]

i thought there was a limit to like 1 a day or smth

mb

[cold agate]

no it's as many as you want

[slim folio]

Hello??? Pls don’t chat here, I’m waiting others to guide me how to see the graph

[cold agate]

cant you use a graphing calculator

[slim folio]

Erm, no, hahaha I’m preparing for final, and this is the solution for the previous test

And I wonder how to combine the 3 graph and get the one for density cloud

[cold agate]

no calculators allowed?

dang

[slim folio]

Cannot use graphing calculator

Just simple calculator is allowed

[cold agate]

well then i can't help you w that
dk much about r-theta coordinates

[slim folio]

🥲

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[ashen silo]

Can someone check if this is correct?

[robust slate]

You’re differentiating 5-2y, not 5-2x

[ashen silo]

Shit

I’ll get back to it

Ty

.close

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[white karma]

why is $\frac 1n\sum_{k=1}^ne^{\frac kn}=\frac{e^{\frac 1n}(1-e)}{n(1-e^{\frac 1n})}$ instead of $\frac{(1-e)}{n(1-e^{\frac 1n})}$

[glossy valve]

pirateking0723

[buoyant plaza]

n is a constant

do the second term/first term

[white karma]

isnt in general like this : $\sum_{k=1}^n ar^k=\frac{a(1-r^n)}{1-r}$?

[buoyant plaza]

r^n not r^(n+1)

[glossy valve]

pirateking0723

[white karma]

ah yes

r^{n+1} if i start from k=0

[white karma]

and isnt a in my case equal to 1/n ?

[buoyant plaza]

no a is the first term

e^(1/n)

[white karma]

the first term is e^(1/n)/n

[buoyant plaza]

compute the whole series first

then just multiply by 1/n

ignore 1/n for now

[white karma]

ok so lets do this

let $r=e^{\frac 1n}$ then $\sum_{k=1}^ne^{\frac kn}=\sum_{k=1}^nr^k=r+r^2+\cdot\cdot\cdot+r^n$

[glossy valve]

pirateking0723

[white karma]

now let $S=r+r^2+\cdot\cdot\cdot+r^n$ then $S=r(1+r+r^2+\cdot\cdot\cdot+r^{n-1}+r^n-r^n)=r(S+1-r^n)=rS+r(1-r^n)\implies S=\frac {r(1-r^n)}{1-r}\implies\sum_{k=1}^nr^k=\frac {r(1-r^n)}{1-r}$

[glossy valve]

pirateking0723

[white karma]

ohh ok

i see

tysm for your help

.close

[full forum]

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[uneven rock]

hello i need help with this please

[honest hill]

factor

[full forum]

The purpose of this server is to help you learn, not to hand out answers. Do not ask someone to give you the answer directly.

[stiff musk]

!nosols

[full forum]

As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.

[stiff musk]

you're the one using a bot, bot

pls go away, you're not being helpful

[fringe silo]

who needs help ?

[stiff musk]

^ help has already been given, let's wait for the OP to respond to it

[honest hill]

are you a bot?

[uneven rock]

alr thanks

[honest hill]

sounds very human lol

[stiff musk]

sounds like exactly what a bot would say

[honest hill]

is this pingable? (to mods)

[void nova]

Everything started from this, which seems rude and out of context

Just that (I think)

[honest hill]

dont tell the AI how to improve bruh 💀

[full forum]

@uneven rock Has your question been resolved?

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[uneven rock]

yes thanks bro

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[uneven rock]

i lowkey am fucked, i have a math exam tommorow and i dont know shit, math is hard

[uneven rock]

i am probably gonna get my watch and use chat gpt

[nimble crane]

well... if there's only one day left, and you're feeling weak on everything, then it's going to be a challenge for sure

[honest hill]

!nogpt

[full forum]

Please do not trust ChatGPT or similar AI tools for mathematical tasks, as they often generate output which "sounds correct" but has numerous factual or logical errors. Use of these AI tools to answer other people's help questions is strictly against server rules (see #rules).

[stiff musk]

math would not be worthwhile if it wasn't hard 😁

[uneven rock]

yeah

[nimble crane]

it wouldn't be a bad idea to prepare in advance next time

[uneven rock]

and i cant find anything to explain shit to me

[uneven rock]

yeah i agree

AI isnt helping

[stiff musk]

try khan academy for specific topics

[uneven rock]

alr thanks bro

💌

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@uneven rock Has your question been resolved?

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[hard lichen]

[helping to teach another friend]
z³ = -8
z ∈ C

how do i approach this? i know the solution from wolfram alpha but is there an intuitive method for this?

[torn jolt]

yes! z = a+bi

[kindred grove]

'intuitive'

[stuck loom]

?

What is the question?

[torpid perch]

solve for z

[stuck loom]

I am disdainful towards people who use the word 'intuitive'.

Aren't real numbers under the set of complex numbers?

[lusty socket]

what?

[hard lichen]

z^3 = a+(8-a)i i guess?

[stuck loom]

z = -2

[hard lichen]

wait im on the bus

[hard lichen]

...?

[hard lichen]

thats not even the correct answer but go on

[stuck loom]

z = -2 satisfies z^3 = -8, doesn't it?

[hard lichen]

it does but thats not all the solutions

[stuck loom]

Ok sorry, I just understood what z < C meant. That z can be a complex number.

[torn jolt]

!nosols

[full forum]

As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.

[kindred grove]

you've the polar/trigonometric/exponential form of complex numbers ? @hard lichen

[hard lichen]

not really.., thats why

im just tryna find like a conceptual grasp because my friend is from germany and my uni doesn't go into this stuff as much