help-33

[simple gull]

all good

idk if it will help but the context is in a perfect economy

so 1

But even if its Ax=1x

i dont see how it works

oh well

nvm i get it its so dumb

.close

[marsh citrus]

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[marsh citrus]

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[vivid bolt]

I saw x²(x+3) - (x+3) simplified into (x+3)(x²-1), why is this, how was this simplification done?

[vivid bolt]

Ah

Question is solved, sorry for the simple one

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[normal atlas]

Is there any proof of the closure property of natural numbers or is it an axiom

[sinful thistle]

closure under?

[proud ice]

closure under...?

[marsh citrus]

@normal atlas Has your question been resolved?

[marsh citrus]

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[gloomy merlin]

Hey can I get help with a). I would love to show some sort of work, but I have been clueless over the past half an hour or so, and I would appreciate some guidance.

[proud ice]

There's some information missing in the problems. Are the random numbers chosen with or without replacement? Does the order of the numbers drawn matter?

I'll just assume that it's modeling after the New Jersey Powerball, where the numbers are without replacements, and invariant of order drawn

So you need to choose 6 out of 46, and then compare that to your event space.

You should have a stats formula that explains how to do that

[gloomy merlin]

Maybe I don't know what the event space is supposed to be here

[proud ice]

The event space is every way to draw 6 numbers without replacement and invariant of order

[gloomy merlin]

So, 6!?

[proud ice]

6 numbers from 1 to 46

[gloomy merlin]

that has got to be 46C6 as well? Am I mistaken?

Right, so let's restructure our thoughts for a bit. The amount of ways to get a winning ticket from the set of 46 numbers is to choose 6 (matching) numbers from 46

I believe we have just established that

[proud ice]

No. Choosing 6 from 46 describes all possible ticket selections

[gloomy merlin]

Okay I see that makes more sense

So we have yet to find the amount of ways to get the matching numbers

once we find that, we can determine the relative frequencies of the two, hence the probability

Well, the numbers are set I would assume. I mean

[proud ice]

wdym

[gloomy merlin]

there is only "one" way of having the correct number

[proud ice]

correct

[gloomy merlin]

so wouldn't it just be 1/46C6 = q

[proud ice]

Consider it as such: You must choose 6 numbers from a set of 6 numbers.

[gloomy merlin]

Yes, very intuitive. Thanks. Very unlikely to be winning then

So moving on to b

The question inherently doesnt make all that much sense right now. What does it mean to find the PMF of the number of winning tickets besides yours?

The probability that there will be X amount of winning tickets, excluding yours?

[proud ice]

I believe that is what it is asking for

[gloomy merlin]

So Id think it is simply the amount of permutations you can arrange the 6 numbers in

or such

[proud ice]

I do not believe so

let's scale the problem down

let's say I flip a coin, and ten people each guess whether it'll land H or T

what is the probability that x people guess correctly?

I believe that is what b is asking

[gloomy merlin]

binomial distribution? assuming their "guesses" tend to be equiprobable between tails and heads, so P(x) = 10Cx 0.5^x 0.5^(10-x)

[proud ice]

that's what I think, yea

No guarantees because I don't do stats normally, but that's my best advice

[gloomy merlin]

So i guess we already got the probability of getting a winning ticket; the question does say that the amount of winning tickets are k, or k-1 excluding ours

there are 2n tickets sold

or, 2(n-1) excluding yours again

[proud ice]

2n-1 tickets, not 2(n-1)

[gloomy merlin]

ah nice catch

P(Kn = k) = (2n-1)C(k-1) q^(k-1) (1-q)^(2n- k)

I'd think?

[proud ice]

I think its choose k, not k-1

because k is already the number of other winning tickets

[gloomy merlin]

Ah I am describing the k in the third line

but maybe that is irrelevent to this

Since that k is constant id assume? and this distribution is meant to be some variant behaviour

[proud ice]

I believe so.

[gloomy merlin]

Ok so we modify that to k

or k_n for clarity's sake

[proud ice]

More like k is being defined differently in those two scenarios

[proud ice]

However you like is fine

[gloomy merlin]

I concur; anywho, I will call it quits at part c if you don't mind. Maybe I will reopen a help channel for the rest tomorrow

[proud ice]

yeah no worries. but fyi c comes quickly from b

expected value is just weighted sum, so that should be easy

(but that actual calculation will suck)

[gloomy merlin]

So is it meant to be the prize that an individual winner is meant to get?

or expected to get

[proud ice]

It is the expected value of your winning ticket

it must take into account the number of other winners

[gloomy merlin]

you'll lose 1$ if you dont win, and you will win however much 0.5*n/k is

well

no that isnt right

i think thats correct now

k is the amount of people who win, n is the pot's money

[gloomy merlin]

would my description above be correct then?

[proud ice]

tbh I am not sure if the expected value needs to consider the ticket cost.

The problem has started on the assumption that you bought a winning ticket already

[gloomy merlin]

we can just assume it is net 0 then i guess

[proud ice]

I think the problem is simply asking, now that you have a winning ticket, what are you expected to get from it

[gloomy merlin]

0.5*n/k, is the amount

if i would not be mistaken

well i guess we multiply that by the probability that you win

So, E[W_n] = q * 0.5 *n/k

this does seem pretty iffy tho

i think all i said was pretty much nonsensical, sorry

[proud ice]

close. The number of tickets sold was 2n, not n

[gloomy merlin]

ah yes

fair point

so simply n/k

E[w_n] = qn/k?

[proud ice]

I think that is still incorrect

consider when k = 0 (no one else won)

[gloomy merlin]

oh k is supposed to mean that?

[proud ice]

yes

[gloomy merlin]

honestly the variables here are a bit confusing

[proud ice]

k is how many other people won

k=0 means you win it all

k=1 means you have to split it in half

[gloomy merlin]

n/(k+1)

[proud ice]

k=2 means you get a third

[proud ice]

consider k=1

[gloomy merlin]

hmm

fixed

ok hooray

[proud ice]

Then yes, this is the amount you win when k other people won.

Now the expected value is just the weighted sum of all possible k

[gloomy merlin]

is there not supposed to be a probabilistic component in the expected value?

[proud ice]

wdym?

[gloomy merlin]

This is the definition of expected value im familiar with

So what exactly is our p(x_i) component here

[proud ice]

you have both x_i and p(x_i)

[proud ice]

It's part b

p(k) is the probability that k other people won

[gloomy merlin]

Ah I think I get it

so x_0 p(x_0) is when only you won

x_1p(x_1) is when you and someone else won

so the sum would be

wait but what would be the range

so it is from 0 to ...?

there is no specified max winning tickets

maybe they expect me to use k

[proud ice]

how many other tickets were sold?

[gloomy merlin]

2n-1 tickets were sold

but this range should be describing the amount of winning tickets am i incorrect?

This is very hard to visualise I wont lie...

But I think I get what it should be I suppose

[proud ice]

you're considering every possible value k could be

but what is the probability that k could be greater than 2n-1?

[gloomy merlin]

impossible i think

you cant win more than there exists

$$sum_i=0^2n-1 _(2n-1)C_i-1 q^i-1 (1-q)^(2n-i) * n/k+1$$

[elfin berry]

Aero

[gloomy merlin]

oh no its not working

[proud ice]

Allow me

[gloomy merlin]

how do you use this i have never used this thing before 😭

[proud ice]

$\sum_{i=0}^{2n-1} {{2n-1} \choose {i-1}} q^{i-1} (1-q)^{2n-i} \cdot\frac n{k+1}$

[elfin berry]

SWR

[proud ice]

Is that what you meant?

[gloomy merlin]

yes! but i should've made that last k an i instead

also wow you're very good with using it

how long did it take you to learn it?

[proud ice]

a while

I joined here 2 years ago

and I had no experience with latex before that

I just look up stuff I don't know

[proud ice]

So, $\sum_{i=0}^{2n-1} {{2n-1} \choose {i-1}} q^{i-1} (1-q)^{2n-i} \cdot\frac n{i+1}$?

[elfin berry]

SWR

[gloomy merlin]

yes

[gloomy merlin]

I see. I wonder who developed it. Sounds like the person who made this server really liked it to make a whole... formatting bot/language?

[proud ice]

btw choose i-1 seems off

[proud ice]

latex has world-wide use. This server simply uses it as well

[gloomy merlin]

oh interesting

[gloomy merlin]

ah right because at i = 0 we have negative

but that's what we made earlier hmm

pretty lost

[proud ice]

we did? where?

[gloomy merlin]

oh wait right we corrected thst

sorry my mind is mixing stuff hp

So, $\sum_{i=0}^{2n-1} {{2n-1} \choose {i}} q^{i} (1-q)^{2n-i-1} \cdot\frac n{i+1}$?

[proud ice]

no worries

[elfin berry]

Aero

[gloomy merlin]

oh YAY it works for me too

[proud ice]

that looks right

also good job

[gloomy merlin]

thank you

anyways i got the idea thanks! these problems are always about being able to connect these words into their mathematical representation

super hard to connect that expectation formula for example

[proud ice]

yeah it's a skill that takes practice

and it's one you should definitely work on, as things get easier once you hone it

and know your fundamentals. They save you so much

[gloomy merlin]

Aye aye sir!

anyways I'll catch you around! Thanks for the session

.close

[marsh citrus]

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[proud ice]

np happy to help

[marsh citrus]

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[wise oxide]

Why is un+1 including the x^n?? And also the original un. I thought x^n is not part of the un. It should be un which is 1/n! and then times x^n?

[wary kite]

what

this is the ratio test

wdym un+1 contains un?

[wise oxide]

I meant I thought the x^n isn’t supposed to be part of the equation cus it’s power series

[wary kite]

they’re literally just defining the sequence u_n as x^n/n!

hence u_n+1 = x^(n+1)/(n+1)!

[wary kite]

what?

it contains n

[marsh peak]

u_n is supposed to denote the nth term in the sum, that includes the x^n

[wise oxide]

Like this it’s not part of the equation

[wary kite]

bro what equation

a_n?

[marsh peak]

You are confusing the terms of the sum with the coefficients of the powers of x

[iron cipher]

Wait what math are you in

[wary kite]

prealgebra

[iron cipher]

I meant dellungi sorry

[marsh peak]

This is a topic of calc 2

[iron cipher]

Ok

[wise oxide]

Ok so I just count x^n as part of the original when I do ratio test

?

[marsh peak]

Yes

[marsh citrus]

@wise oxide Has your question been resolved?

[marsh citrus]

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[split vigil]

linear programming?

[marsh citrus]

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[heavy phoenix]

pls, can someone confirm my answers here

[heavy phoenix]

idk what an annulus is, seems to be a space between 2 circles

[marsh citrus]

@heavy phoenix Has your question been resolved?

[main idol]

yes. https://www.cuemath.com/geometry/annulus/

[marsh citrus]

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[heavy phoenix]

oh, thanks

[marsh citrus]

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[stable veldt]

I’m stuck I’ve tried but can’t seem to figure it out

[stable veldt]

<@&286206848099549185>

Bro

<@&286206848099549185>

<@&286206848099549185>

[acoustic bramble]

Have you calculated the average mass of one orange?

[stable veldt]

No

Would I do 1.126/8 to find the average mass of one orange

[acoustic bramble]

Yes.

[stable veldt]

0.140

[acoustic bramble]

So, if that is the mass of one orange, what would the mass be for n oranges?

[stable veldt]

OHHH

I get it now

[keen marten]

yes

good job

[acoustic bramble]

👍

[stable veldt]

Thanks guys

[keen marten]

glad we could help bro

😉

[stable veldt]

Lol

Need help with this

[marsh citrus]

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[marsh citrus]

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[twilit plaza]

Hi

I think you already solved the a, which is
total mass of the eggs in the carton - mass of the carton

then divide the mass by 12, since you have twelve eggs

which is 59.25g

for b part, you can find the u1 and the common difference and then substitute into the general formula for the arithmetic sequence.

[marsh citrus]

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[cosmic laurel]

A ruler dropped from the vertex of the rectangle to the diagonal divides the diagonal in the ratio of 1:3. From the point of intersection of the diagonals, the allowed length of the long side is 3.2 cm. Find the length of the diagonal

[cosmic laurel]

<@&286206848099549185>

[sand fable]

similar triangles

[marsh citrus]

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[gloomy merlin]

Background exercise (i already solved this): An oil drilling company ventures into various locations, and its success or failure is independent from one location to another. Suppose the probability of a success at any specific location is 0.25. (a) What is the probability that the driller drills at 10 locations and has 1 success? (b) The driller will go bankrupt if it drills 10 times before the first success occurs. What are the driller’s prospects for bankruptcy?

Actual exercise:
Consider the information in Review Exercise 5.90. The drilling company feels that it will “hit it big” if the second success occurs on or before the sixth attempt. What is the probability that the driller will hit it big?

[gloomy merlin]

so my issue with this is how i solved it vs how my book solved it

I considered P(2 <= X <= 6) where X is a negative binomial random variable, but my book considers P(X = 6) only

I feel like this is an incorrect solution to the question- since it also says "or before the sixth attempt"

[main pumice]

Did u get the same answer?

[gloomy merlin]

my answer involves sum of (x-1)*0.25^2*0.75^(x-2) from x = 2 to x = 6

which is 0.46 approximately

[main pumice]

Ok

[gloomy merlin]

so am i right 😭

[proud arch]

Hello @gloomy merlin

I believe you might be wrong here

[proud arch]

How did you get this?

What need to be understood is that the possibility of second success relies on the first one

If the first success is on 5th try

To hit big the second success must be on 6th

[gloomy merlin]

because i thought the question asked us to consider not the second success happening on the 6th success, but also the 5th, 4th, 3rd, or 2nd- which should be equal to the sum of all cases i had thought-

[proud arch]

It depends right?

If 1st success happened at 4th try

Then second success can happen at 5 or 6th try only

[gloomy merlin]

yess exactly

so whats wrong with my solution?

[proud arch]

Ok let's try and put values in your solution

Maybe then we'll see it

For x = 2

It is simply
(1/4)^2

What does this mean

[gloomy merlin]

yess it just means that there were two successes from the first two tries

seems fine

[proud arch]

Who said they have to be consecutive

[gloomy merlin]

nobody but there is only 1 way to arrange it in 2 trials no?

[proud arch]

It could 1 and 3

Or 1 and 4

?

[gloomy merlin]

yess but i think u r confused by what my formula is saying. It is representing P(X = 2) + P(X=3) + P(X=4) + P(X=5) + P(X=6). P(X=2) means that the second success happens at the second trial, P(X=3) means that the second success happens at the third trial and so on. I dont think im neglecting any of the cases with this?-

If it was P(X=3) for example it would be 2 * 0.25*2 0.75 and the 2 term is account for all the possible combinations which be NSS or SNS

[proud arch]

Ok

But in the answer from book

I think they are saying that only 2 successes

On or before 6th try

Correct?

[gloomy merlin]

yess

[proud arch]

But by your formula

You don't account for those

Like in x=2

[gloomy merlin]

but the issue is that their solution is assuming it happens at the 6th trial only

[proud arch]

First 2 are successes

What about the rest?

[gloomy merlin]

they dont exist

because negative binomial stops after second success

[proud arch]

Only 2 successes

Before 6th try

From your solution their maybe more than 2

Just because you didn't account for them

Doesn't mean they don't exist

They do exist

[gloomy merlin]

i diddd

its the sum

[proud arch]

And need to be marked as failures

[gloomy merlin]

😭

[proud arch]

Give me a minute

Let me reanalyze everything

[gloomy merlin]

X=2: SS
X=3: NSS, SNS
X=4: NNSS, NSNS, SNNS
X=5: NNNSS, NNSNS, NSNNS, SNNNS

X=6 is the same too ofc-

[main pumice]

SSSSSS would be valid, no? Because the second success does occur before 6. It doesn’t seem to specify only 2

[proud arch]

I think it should be more like
X=2: SSNNNN
X=3: NSSNNN, SNSNNN
X=4: NNSSNN, NSNSNN, SNNSNN
X=5: NNNSSN, NNSNSN, NSNNSN, SNNNSN

[gloomy merlin]

The problem cares about when there are two successes tho- not that many. X represents the random variable of having the 2nd success happen on the xth trial

[main pumice]

It only asks for the probability of hitting it big before attempt six

SSSSSS is valid no?

[gloomy merlin]

Even in their solution it would not be i think

bcuz k = 2 represents the amount of successes

there is only 2

[main pumice]

Oh mb the question seems like it’s asking for 2+ to me

[gloomy merlin]

why should the N's after the second S matter tho? again i think the random variable X is "The random variable of having the 2nd success happen on the xth trial" So if X = 3 you dont look at numbers bigger than 3

[proud arch]

You just said they matter

Of they didn't SSSSSS

Would exist

And you just said it shouldn't

I think this is the answer

@main pumice just said that

[gloomy merlin]

but then it just wouldnt be a negative binomial distributioin

[proud arch]

We use such when both events can vary in number

As in atleast 2 successes

If that were given

We would use it

But it isn't

Question implies EXACTLY 2 successes

[gloomy merlin]

Yes

my solution doesnt say there is more than 2??

[proud arch]

.-.

[gloomy merlin]

see here

there is only 2 S's per trial

[proud arch]

Your solution in x = 2

Accounts for only the first 2 tries

Even though the sample space is of 6 tries

It's as if I were to say to flip a coin 6 times

[main pumice]

U have to exclude the possibilities with more than 2 successes I think

[gloomy merlin]

Yes because it literally says 6 tries or before

this is the case for when there was only 2 tries and they won

[proud arch]

And find turns in which exactly 2 head turn out

You just neglected the other possibilities

So in our answer

More then 2 heads will also show up

That's exactly what's happening here

Just 2 heads

Just because you got it in the first 2 trns

Doesn't mean you can neglect the other 4

They have to tails

[gloomy merlin]

No because the question is actually asking "whats the probability of getting two heads in 6 coin flips OR LESS". If you get it immediately on ur second flip. you are done. there is nothing more to do

[proud arch]

And NEED to be accounted for

[main pumice]

So then SSSSSS is valid then, because you only need to get 2 successes and what happened after doesn’t matter

[gloomy merlin]

i never did

[proud arch]

Exactly 💯

We don't want that
we only want 2

[gloomy merlin]

well sure, but we are just recording a video and we stop it whenever they get the second success

if they continue they might a billion success

[proud arch]

No they don't

Where is that mentioned

Where is it written that once they hit big

They stop?

[gloomy merlin]

yes!! thats literally the defintiion of negative binomial

thats what they used in their solution

X is a negative binomial variable

you dont record after the final success happens

SSSSSS Is meaningless to us

because it says the 6th success happens at the 6th trial

but why dow e care?

we care about the 2nd success ONLY

[main pumice]

They have (0.75)^4 meaning 4 N

So they are considering all results with two S and four N

As the probability of N and S is 0.75 and 0.25 respectively

[gloomy merlin]

x = 6 and k = 2 which means there were six trials and the second success happened at the 6th point

x = 5 and k =2 which mean there were five trials and the second success happened at the 5th point

if it was x = 5 we dont care about the 6th trial anymore

because we only want to satisify the k

which is 2

[main pumice]

The solution accounts for all 6, being 2 successes and 4 failures

[proud arch]

Ok let's assume this is correct

If for x= 2
Nothing matters after that
Than that means
After SS anything can happen
Such as
SSNNNN
Or
SSNSSNN
And so on
Wouldn't all these favorable outcomes to hit big

But you said only 2 success

Where have you subtracted the overcounted cases

[gloomy merlin]

yes because once the company hit SS they hit it big

thats what they want

[proud arch]

So SSNNSS

Is fine?

[gloomy merlin]

NNSS dont affect the probability at all

because SS already had the effect of hitting it big

there can only be 2 successes

[proud arch]

But it's a viable route correct?

[gloomy merlin]

only

[proud arch]

You want to exclude any cases where their are more then 2 S

But at the same time you say their amount doesn't change the number of favorable outcomes at all

And therefore the probability

How is that possible?

[gloomy merlin]

genuinely, i dont want to be argumentative, but your argument to me sounds like "Ok, they hit two successes, so they achieved their goal of hitting it big, but for some reason, they continued recording even after their goal was satisified. Even though their probability calculations was aiming to check for when the 2nd success happens and not focusing on what happens after"

[proud arch]

It's a contradiction

[proud arch]

What goal

They said they thought

[main pumice]

Problem asks to consider interval of 6

[proud arch]

That they might hit big

It was a part of the plan

How do you know they stopped after that

You don't own that company

Maybe they tried 1 million times

Our sample space is just the first 6 tries

And 2 of those tries being the correct ones

[gloomy merlin]

the question literally says "What is the probability that the driller will hit it big?" thats the goal

[proud arch]

So as to HIT BIG

[gloomy merlin]

i dont know and i dont care

because this probabilty question

focuses on when they hit it big

[proud arch]

Interval also know as Sample Space

[gloomy merlin]

the question doesnt care

[proud arch]

The question cares about the first 6

You care only about the interval you get the success

Their calculation was to find the second success

In the first Six

We don't care after that

[gloomy merlin]

you conveniently are always omitting the "or before" they specified in their question

[proud arch]

Before that yes it has to be taken into account

I don't think worth the energy anymore

It's been 1 hour

[gloomy merlin]

Me neither

I will reopen because i remain unconvinced

have a nice day though!

.close

[marsh citrus]

Channel closed

Closed by @gloomy merlin

Use .reopen if this was a mistake.

[proud arch]

I believe it's best you to talk to your teacher

And DM me on who's correct

I will take this question into account and see where I get

[marsh citrus]

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[sly sinew]

I have this long ass equation that i have to differentiate, but i cant even differentiate the first part of the equation. Idk what im doing wrong

[sly sinew]

I will translate whatever needed if needed

[smoky spire]

what is the question asking

[sly sinew]

To differentiate very top thing on left picture

[smoky spire]

u need chain rule and quotient rule

[sly sinew]

ah that sucks

i use the quotient rule for x/x right?

[smoky spire]

chain rule for -sqrt x^2 + 1 and quotient rule for the entire thing

[sly sinew]

So i shouldnt split into x/x

[smoky spire]

no

[sly sinew]

How am i supposed to make my f(x) into the blue

[smoky spire]

u just derive it

[sly sinew]

With what rule

[smoky spire]

-sqrt x = -x^1/2

[sly sinew]

Should i skip the power rule?

[smoky spire]

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

[elfin berry]

pppoopoo

[smoky spire]

ye

[sly sinew]

Why does x become 0 instead of 1? If x^0 is 1

[smoky spire]

wdym

[sly sinew]

Like if i want to differentiate x, i do 1*x^1-1, which is x^0, which is 1

[smoky spire]

yes

[sly sinew]

My math program says its 0

[smoky spire]

where did u get 0 from

u will get 0 when u derive a constant

[sly sinew]

Nvm it changed

[sly sinew]

I think ive done something wrong

On the left side, is the x from 2x supposed to cancel out with x?

And -(- becomes +?

and the 2 cancels out with 2

[smoky spire]

which one r u talking about

what x from 2x

[sly sinew]

Like this

Inside the paranthese

[smoky spire]

the 2 cancels out

[sly sinew]

So just like ive written it?

[smoky spire]

doesnt look right

[sly sinew]

cap

[smoky spire]

u should get something like $\frac{x*\frac{x}{\sqrt{x^2+1}}-\sqrt{x^2+1}}{x^2}$

[elfin berry]

pppoopoo

[marsh citrus]

@sly sinew Has your question been resolved?

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[brazen cloud]

im confused, how do i find x,y and z?
the italic text is the translation

[marsh citrus]

@brazen cloud Has your question been resolved?

[sleek lake]

no idea

so it went +18% then +50%

maybe it goes +82% ?

there's like not enough context

[brazen cloud]

wait

does this help?

is table 2 related?

[marsh citrus]

@brazen cloud Has your question been resolved?

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[tropic meadow]

3. how do i do this faster instead of expanding using binomial
4. how do i do this faster again, not summing all of 1 to 19 and then subtracting the evens
5. im just not sure how to do this, i simplifed it down but it didnt help

[tropic meadow]

i was typing it

[night mica]

I hope so

[tropic meadow]

whoops 5 is the wrong one

should be this one

[tropic meadow]

?

[marsh citrus]

@tropic meadow Has your question been resolved?

[marsh citrus]

@tropic meadow Has your question been resolved?

[tropic meadow]

.close

[marsh citrus]

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[calm hamlet]

For those who are familiar with boolean algebra
Prove that F(x,y,z)=Sum(1,3,7)=Product(0,2,4,5,6)
My teacher asked for a 100% algebraic proof, no karnaugh map or something like that
Just algebra

[sinful thistle]

"boolean algebra"?

never even heard of that before

,w boolean algebra

[elfin berry]

Wolfram Alpha doesn't understand your query!
Perhaps try rephrasing your question?
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Requested by justarenegade

[sinful thistle]

and neither has wolfie apparently

[still temple]

https://en.wikipedia.org/wiki/Boolean_algebra

[calm hamlet]

Boole is the mathematician who invented it

[sinful thistle]

so it's literally just mathematical logic

why did they have to give it such a weird name lmao

[calm hamlet]

We need that in computer science apparently

[still temple]

huh

bro

[sinful thistle]

yeah ik

[still temple]

you invent something

[sinful thistle]

all logical operations are made of that

no no

[calm hamlet]

No fcin idea why

[still temple]

you wouldnt like to name it something good ?

[sinful thistle]

i get why it's called boolean algebra

but it's not typically called that--we usually call it mathematical logic

[slate yarrow]

its just first order logic in terms of algebra kek

[sinful thistle]

ik 😭

[calm hamlet]

Anyone wanna help here?

I don't need people mocking the 'ame

[sinful thistle]

never seen the "sum" and "product" things though

we're not lol

[calm hamlet]

Sum is or

Product is and

[sinful thistle]

we're just not familiar with these notations

ah, guessed

[calm hamlet]

The notation is for sum of minterms and product of maxters

Maxterms*

[quick dust]

Boolean algebra is a normal term dw

[slate yarrow]

why would your prof do this...pure evil

[calm hamlet]

cuz she's an idiot who likes wasting people's time

Anyway
Here is the message again so that people won't have to scroll up

For those who are familiar with boolean algebra
Prove that F(x,y,z)=Sum(1,3,7)=Product(0,2,4,5,6)
My teacher asked for a 100% algebraic proof, no karnaugh map or something like that
Just algebra

[slate yarrow]

alright so Sum(1,3,5)=X'Y'Z+X'YZ+XYZ
=X'Y'Z+YZ(X'+X)
=X'Y'Z+YZ
=Z(X'Y'+Y)
=Z(X'+Y) iirc

now we try the same for product(0,2,4,5,6)

[marsh citrus]

@calm hamlet Has your question been resolved?

[marsh citrus]

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[sinful pier]

Can someone help me figure out what I did wrong?

[sinful thistle]

(2x)^2 != 2x^2

[sinful pier]

?

[sinful thistle]

$(2x)^2 \neq 2x^2$

[elfin berry]

ren

[sinful pier]

How-

It’s a monomial-

OH SHIT WAIT

ITS 4

[sinful thistle]

the moment of realization

that's the only thing wrong it seems

[sinful pier]

Oopies

[sinful thistle]

happens to the best of us

[sinful pier]

Should I keep x^2 negative?

[sinful thistle]

??

wdym?

[sinful pier]

You have to move 4x^2 to the other side

And you can’t subtract 4 from 3 without it being negative

[sinful thistle]

sure i guess

you can move the other stuff to the LHS tho

which would be easier in my opinion

[sinful pier]

Cus if not it turned back into x^2 + 9x -8

[sinful pier]

LHS?

[sinful thistle]

left-hand-side

[sinful pier]

Ohhhh

[night mica]

you was helping two people simultaneously

[sinful thistle]

so

i do that all the time

[night mica]

you can do much better!

[sinful thistle]

sometimes 5+ at once

[sinful thistle]

i'm lazy :/

[marsh citrus]

Please read the channel description before posting, and stay on topic.

[sinful pier]

No that also makes it x^2 + 9x -8

That isn’t factor able..

[sinful thistle]

no it doesn't?

it's 4x^2 remember

[sinful pier]

Yeah

4-3=1

[sinful thistle]

but
the signs of 9x - 8 will change??

[sinful pier]

They were -9 and pos. 8

I think

My computer died

OH SHIT NVM

[sinful thistle]

all g

[marsh citrus]

@sinful pier Has your question been resolved?

[sinful pier]

Ok I think I messed up

I’m not sure if I can use log base 10

The example problem only shows log base e

But isn’t natural log..just base 10

<@&286206848099549185>

[strong portal]

u mean u cant use log base e ( ln )?

[sinful pier]

I’m not sure how

[strong portal]

ok

until ln(5x-8) = 5 is right

[sinful pier]

Oh also

[strong portal]

then exponentiate both sides using e

[sinful pier]

Ignore that

[sinful pier]

Huh-

[strong portal]

e as the base

[sinful pier]

Oh

So e^5 = 5x -8

[strong portal]

right

then slove it to x

[sinful pier]

Ok

So…. e^5 + 8 /5?

But it said exact value..

[strong portal]

yes

thats the exact value of x

if u want a number

[sinful pier]

Let me try that first

Ok that was it!

Thank you

[strong portal]

your welcome

[sinful pier]

.close

[marsh citrus]

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[brittle fog]

How do I find the 2 other roots?

[wary inlet]

z^3 - 8 = 0
factor and you can find the other 2

(use difference of cubes)

[brittle fog]

I know I have to factor

But I don't understand how

[wary inlet]

do you know the difference of cubes formula?

[brittle fog]

I don't think so

[wary inlet]

a^3 - b^3 = (a - b)(a^2 + ab + b^2)

[brittle fog]

This feels wrong

[wary inlet]

can you use this to rewrite z^3 - 8 = 0

[brittle fog]

Okay so you get the base sides and the quadratic?

[wary inlet]

yes

[brittle fog]

Alright

[wary inlet]

exactly

[brittle fog]

Is this right?

[wary inlet]

looks good

so we have

wait

it’s 2z not 4z

(z - 2)(z^2 + 2z + 4) = 0

it’s (a^2 + ab + b^2) not (a^2 + 2ab + b^2)

just to be aware

[brittle fog]

But

It would be 2z2

Which is 4z

Since b =2

[wary inlet]

it’s ab not 2ab

(a - b)(a^2 + ab + b^2)

[brittle fog]

Oh

Wait why

[wary inlet]

it’s just the factoring

if you wanna expand out and check you can

[brittle fog]

Alright

Brb

[wary inlet]

okay

[brittle fog]

Like this?

I can shorten it a little

Hold up

[wary inlet]

(z - 2)(z^2 + 2z + 4)
z^3 + 2z^2 + 4z - 2z^2 - 4z - 8
2z^2 and - 2z^2 cancel out
4z and - 4z cancel out
z^3 - 8

[brittle fog]

How do I get rid of -2²z²?

[wary inlet]

it should be (z^2 + 2z + 4)

[brittle fog]

This time

[brittle fog]

Where am I going wrong?

I just get this

[brittle fog]

Omg Idk how I thought you wrote - there ;-;

Sorry

[wary inlet]

all good

so we have (z - 2)(z^2 + 2z + 4) = 0

now can you use this to find the other 2 roots

[brittle fog]

That makes so much more sense

[wary inlet]

nice nice

[brittle fog]

I don't get how

[wary inlet]

if we have 2 numbers a and b, let's say that ab = 0. This means that either a = 0, b = 0, or a = b = 0

we can set each product to 0 and solve for z

z - 2 = 0
z^2 + 2z + 4 = 0

[brittle fog]

What

I don't understand where the second part came from

[wary inlet]

(z - 2)(z^2 + 2z + 4) = 0
either z - 2 = 0
or z^2 + 2z + 4 = 0

[brittle fog]

Are those two different roots?

[wary inlet]

z - 2 will give one root

z^2 + 2z + 4 will give 2 roots

[brittle fog]

What do I do with this to find them

[wary inlet]

solve for z

i would use the quadratic formula

as they are imaginary

[brittle fog]

Oh okay ofc

I don't get what I can have done wrong

Everything seems correct

Doesn't it?

[patent sandal]

under the square root, it should be 2^2 - 4.1.4

not 1^2

cus b=2

[brittle fog]

Aaah

I'm so inattentive today

This

This is correct

Right?

Or do I cancel the 2 infront of the √

I do

[patent sandal]

yea the 2 would be cancelled

when you divide by w

2

[brittle fog]

Yeah

Alright thank you both

.Close

[patent sandal]

[brittle fog]

Close.

How do I do it?

.close

[marsh citrus]

Channel closed

Closed by @brittle fog

Use .reopen if this was a mistake.

[marsh citrus]

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• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
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[mellow frigate]

How do I do this?🥲

[red nimbus]

You need to first understand what things the derivative tells us about the primitive function

[rich bough]

use first and second derivative test

[mellow frigate]

What are those?

[mellow frigate]

How so? I don’t have the function

[rich bough]

you have the graph

[mellow frigate]

How do I do it visually?

[rich bough]

use critical points

f'(x) = 0

[mellow frigate]

That’s where the peaks are right?

[red nimbus]

For example f'(x') = 0 and f'(x) goes + to - before and after x=x' then you would have a maximum at x=x'

[mellow frigate]

Where the curve changes direction

How does that tell me the derivative?

The left one for example the critical points are
X=-5
X=-3

Right?

[red nimbus]

yes

[mellow frigate]

How do I know which parabola it is based on that?

[red nimbus]

now notice at x=-6 the derivative stays non-negative meaning no change of sign therfore no local max/min at x=-6

[mellow frigate]

The one that touches the x axis at those points?

[mellow frigate]

How is that not a local minimum

Oh -6

Oh wait yea it’s -6

[red nimbus]

and we cant tell if the other is exactly x=-3

[red nimbus]

because the derivative keeps being non-negative which implies the function is still increasing

[mellow frigate]

But why is that not a local minimum

[red nimbus]

so it's a saddle point situation

[mellow frigate]

Even if it changes direction?

But it’s decreasing and then increasing

[red nimbus]

yea and?

[mellow frigate]

So it’s not a critical point?

[red nimbus]

that only tells us about the curvature

it's a f(x) = x^3 situation

where f'(x) = 3x^2

f'(0) = 0 but x^3 doesnt have a local min

it's a saddle point

[red nimbus]

it is

a critical value is one such that f'(x) = 0

[mellow frigate]

Never heard of saddle points

[red nimbus]

,w plot x^3

[elfin berry]

Results provided by WolframAlpha

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

Requested by adoniszgks

[mellow frigate]

But that’s where the curve changes direction no?

[red nimbus]

that sounds unbelievably untrue

[mellow frigate]

Or not necessarily

[red nimbus]

yes

from concave down to concave up

[mellow frigate]

If it changes direction it has to go from increasing to decreasing or decreasing to increasing no?

[mellow frigate]

Ohh that thing

So it’s inflection points not critical points

Honestly I’m beyond confused

[red nimbus]

both together is called a saddle point

[red nimbus]

this is an inflection point with f'(0) = 0

also called saddle point because f'(0) = 0

[mellow frigate]

Ohh lol

I see, and what’s next?

I only managed to find inflection points visually once and it’s because I had a point I could move on the graph

[red nimbus]

collect information and deduce which of the primitive functions suits the information you collected the best

[mellow frigate]

Otherwise I have no idea if I don’t have the function

[red nimbus]

What about the monotony?

[mellow frigate]

What’s that

[red nimbus]

When is f increasing/decreasing based on whether is f' > 0 or f' < 0

[mellow frigate]

Wut

How does the derivative tells you that

[red nimbus]

Ok, do you know what a derivative basically is?

Where it does come from

[mellow frigate]

The slope of the tangent line

That’s all I know

[red nimbus]

well that's kinda vague

so you haven't studied this stuff before thoroughly

[mellow frigate]

Slope means the angle I think

[mellow frigate]

Need to thank my current professor for that

[red nimbus]

sure, blame other

people

[mellow frigate]

I could’ve self studied but I wouldn’t care that much

Just need the grade

[red nimbus]

well now that you said it it makes sense to me

[mellow frigate]

What does

Anyway, what are the steps to solve it I still don’t understand

You gather the critical points

X=6
X≈-3

And then what

Which one does it match?

[marsh citrus]

@mellow frigate Has your question been resolved?

[marsh citrus]

@mellow frigate Has your question been resolved?

[mellow frigate]

<@&286206848099549185>

[still temple]

hii i can help with what oyu need 🙂

[mellow frigate]

thank you!

[still temple]

What are you stuck with 🙂

[mellow frigate]

im not sure how to do this

what are the steps

[still temple]

so are you matching the top one with one of the 3 down the bottom?

[mellow frigate]

yes there are more but these are the negative ones

[still temple]

alr

[mellow frigate]

cuz its a down facing parabola

[mellow frigate]

these are all of them

[still temple]

may i ask if your in high school or university?

[mellow frigate]

college, its calc1

[still temple]

ive seen one of these questions before in my grade 7 class but i forgot im so sorry 😦

i wish i could help but im stuck trinna do other stuff im so sorry

i hope you do find someone who can help you

srry once again

wish you luck 🙂

[mellow frigate]

its okay!! thank you anyway!

[mellow frigate]

<@&286206848099549185>

[astral kite]

hi

[mellow frigate]

hai!

[astral kite]

can you reply to thep roblem

*problem

[mellow frigate]

here

[mellow frigate]

and this is the overall problem

i just seperated it cuz these are the three options for that curve i believe

[astral kite]

ok

so just that problem?

[mellow frigate]

matching the curves to their derivatives

since i dont know the steps to do that

i have tons more of homework but i wanna start here for now lol

[astral kite]

ok

so i don't know if you know this

but the maximum and minimum of the curves are equal to 0's

for the derivative

so

[mellow frigate]

the critical points?

[astral kite]

which points look like a relative maximum or minimum

[astral kite]

yes

[mellow frigate]

X=6
X≈-3
?

[astral kite]

yes

[astral kite]

6 or -6

[mellow frigate]

whoops -6

[astral kite]

yeah

so now we know that -6 and -3 are 0's

[mellow frigate]

Ye

[astral kite]

so which graph matches this

[mellow frigate]

The derivative is 0 there

[mellow frigate]

How do we tell?

[astral kite]

we know that the original graph is cubic

[mellow frigate]

Ohh because of the x intercept points?

[astral kite]

yeah

like tha

[mellow frigate]

Ohh!!!

[astral kite]

yes

[astral kite]

ye thats right

[mellow frigate]

And this pair as well right?

Wait no

Oh yea

[astral kite]

yes

thats right

[mellow frigate]

How is this a pair?

There is no local maximum or minimum

[astral kite]

yeah

thats right too

[mellow frigate]

Why so?

How do you solve this one?

[astral kite]

look at the graph y=x^3

when you make a deriviave

we get y=x^2

now that graph is a translation of y=x^3

[mellow frigate]

Not 3x^2?

[astral kite]

oops

3 didin't type

but you get the poin tright

[mellow frigate]

I think so

I thought it has to be a critical point

[astral kite]

this is the one special case where there is no critical point

[mellow frigate]

Ohh I see

[astral kite]

yeah

[mellow frigate]

These are correct?

[astral kite]

yes

[mellow frigate]

Let’s goo

Now I have these

Is this correct?

X=-0.5

I took two points and used slope formula

(6,0) and (0,-3)

[astral kite]

ok

so there are two parabolas

[mellow frigate]

Wdym

[astral kite]

in the question

[mellow frigate]

Where?

Isn’t it a linear function

[astral kite]

yeah

so we find the line

[mellow frigate]

So the derivative is just a constant no?

[astral kite]

yeah

so we find the eq of the line

and then use an integral

[mellow frigate]

Woah

I’m not at integrals yet haha

I just have to sketch the derivative

Isn’t it just x=0.5

The thing I drew in blew

[astral kite]

sorry have to do somethi ngrq

[mellow frigate]

No worries!

I’m here

[astral kite]

ok

so the best way I think to compare this

[mellow frigate]

Wait what I drew is wrong?

[astral kite]

no

ot

[mellow frigate]

So what are you trying to explain I’m confused

[astral kite]

I'm trying to think of a way to do this

without integrals

[mellow frigate]

The slope formula?

[astral kite]

yeah

once you find the line

[mellow frigate]

That’s what I did

[astral kite]

you would integrate

[mellow frigate]

Why

That’s the answer

It asks to sketch the derivative

[astral kite]

because integerals are the opposite of dervitaves

[mellow frigate]

The derivative is x=-5

[astral kite]

ye

[mellow frigate]

So why do we need integrals

[astral kite]

we use integrals to work backward

[mellow frigate]

Don’t we just draw the line -5

But why do we need it

I’m trying to go forward not backward

[astral kite]

oh

yeah

[mellow frigate]

Haha

So the line in blue is -0.5

The derivative

[astral kite]

of which graph

sorry i'm so confused

[mellow frigate]

This graph

Haha

The red graph

Blue is derivative of red

[astral kite]

yeah

[mellow frigate]

But now I have this and no idea how to do it