help-27

[sonic smelt]

Where is floor(x) - floor(-x) = 2x coming from?

[wooden axle]

I don't know my friend wrote this

[sonic smelt]

Anyways, it's false

You probably wanted to write floor(x) - floor(-x) = 2floo(x) + 1

[wooden axle]

These are the answers

[sonic smelt]

Does it say anything about a and b?

[wooden axle]

Nope

It's a bilingual paper in English

[sonic smelt]

None of them are true

Oh, wait, let me check actually

[toxic flower]

if the limit is same in both then answer would be option 'b'

[sonic smelt]

Hm? No

,w integral of floor(x) from 0.9 to 1.1 + integral of floor(-x) from 1.1 to 0.9

[woven radish]

Results provided by WolframAlpha

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Requested by A Lonely Bean#7998

[sonic smelt]

And 0.9 - 1.1 is not 0.4

The answer seems to be 2(b - a)

[toxic flower]

i said if* the limit is same in both -_-

[sonic smelt]

a = b?

[toxic flower]

from a to b in both

[sonic smelt]

Ah

Yeah sure

[sonic smelt]

But none of the options are correct btw

Look

[toxic flower]

hm

@wooden axle which publication book?

[wooden axle]

I don't know. I got this question from my friend

Maybe it's an old paper of something. I'm preparing for a maths teacher exam

[toxic flower]

if its arihant or sanjeev then congrats it has so many typo

[wooden axle]

What value for a,b?

[sonic smelt]

You can see those

[wooden axle]

What if I choose cengage?

[toxic flower]

its good

[wooden axle]

I would like to talk to you when you are free. Because i need some jee pdfs and videos so that I can enrich my maths to higher level

I didn't know there were many indians too haha

[toxic flower]

okk sure

[wooden axle]

All options are incorrect?

[sonic smelt]

Yes

[toxic flower]

i think you can help me also in a problem 😓

#help-18

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,close

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[wooden axle]

.open

I guess the option C is correct

[wooden axle]

@toxic flower @sonic smelt

Let me send you a solution

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[hollow sluice]

how to integrate this mf

i think i'm meant to use some sort of trig substition

[silent bronze]

Sub x for a trig function and try and get it to simplify

[hollow sluice]

that doesn't really help me out here lol

[silent bronze]

If x=tan(u), what does tan^2(u)+1 equal?

[hollow sluice]

oh shit

a tan substitution

is it

sec^2x

[silent bronze]

Yup

[hollow sluice]

hm

so eventually you'd get

$\int \frac{1}{sec^5x}$ ?

[woven radish]

chromium

[hollow sluice]

ah but then

i need to take into account

dx

because taht would change after the substitution

hm

let me try mess with it

[silent bronze]

Yeah you need dx

[hollow sluice]

holy shit

this is massive

[hollow sluice]

am i on the right track here, so.
$\int \frac{1}{sec^5\theta} \cdot sec^2\theta d\theta$

[woven radish]

chromium

[hollow sluice]

$\int \frac{1}{sec^3\theta} d\theta$

[woven radish]

chromium

[hollow sluice]

$\int cos^3\theta d\theta$ ? ???

[woven radish]

chromium

[toxic flower]

hm

now easy sinx=t

[hollow sluice]

wha?

[toxic flower]

$1-\sin^2x=\cos^2x$

[woven radish]

Hack With Techno Boy

[hollow sluice]

ah ye ye

i'm fine with integrating that

was just making sure i was doing it right

the previous shit

ty anyway

[toxic flower]

👍

[hollow sluice]

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[midnight finch]

i dont know how to prove it, can anyone help ?

[devout snow]

@midnight finch Has your question been resolved?

[midnight finch]

<@&286206848099549185>

[fast cedar]

sorry i don’t know how to help but just wanted to comment on your handwriting, it’s incredibly neat!

[midnight finch]

okay XDDD it's fine 🤣 u so nice!

[analog trellis]

You need to differentiate

$z(\lambda x, \lambda y) = \lambda^n z(x, y)$

[woven radish]

stabulo

[analog trellis]

with respect to λ.

[midnight finch]

λ😲

[analog trellis]

Then set λ = 1.

[midnight finch]

λ is constant?

[analog trellis]

The equation holds for any real number λ according to the question. It will hold for λ = 1.

[midnight finch]

ohhhh

after differentiate it, put λ=1

[analog trellis]

Yes.

[midnight finch]

let me try

[analog trellis]

To aid you should you need it:

[midnight finch]

i got this

[midnight finch]

let me save it

thanks!!

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[stone apex]

why is this true

I see that we get e from the first multiplier but why do we neglect the one to the power of 10

This is a limit as n approaches infinity btw

[pseudo basin]

what does 1 + 1/n approach as n -> ∞

[stone apex]

1 : )

I had a long day, thanks

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[narrow vortex]

prove L=R

[supple knot]

One of the angles makes the log return 0

Figure out when log(x) is zero and see which angle gives that x in tangent

[narrow vortex]

oh, logtg45=0?

thx

.clode

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[fallen elk]

can i get some help

[supple knot]

can't you just plug in your options to a calculator and check

or look at the unit circle

,tex .unit circle

[fallen elk]

i did look at the unit circle

[woven radish]

riemann

[supple knot]

you also need to know that sine is an odd function

sin(-x) = -sin(x)

[fallen elk]

so what can i do with that

[supple knot]

check your options

[supple knot]

e.g. sin(-pi/2) = -sin(pi/2) = - 1

[fallen elk]

ok

?

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[cyan fossil]

how is this

[cyan fossil]

wyh is this

can someone explain

[wicked turtle]

what answer(s) did you choose?

weird interface btw, why are some of the statements inside the boxes but others not

[cyan fossil]

ohio

[cyan fossil]

why did i ask here if i knew the answer at first place

[wicked turtle]

but you're asking "how is this" implying that someone is claiming a particular answer and you want to know why

if not, please clarify your question

[cyan fossil]

how is this, indicates that i don't understand any gibberish from that screenshot

especially W= (gibberish)

how do i solve it

[wicked turtle]

ah i see

do you know what a subspace is?

[cyan fossil]

a

a

part of a vector space, i don't what a vector space exactly is but it is what it is

[wicked turtle]

you will probably need to review the definition of subspace at least, in order to answer this question

do you have a textbook or notes or videos or something?

[cyan fossil]

i have a book but won't bother setting 4 hours just to understand a small piece of info

[wicked turtle]

ok, unfortunately the help channels aren't really meant as a substitute for your book though

i'll get you started though

the lower right option: (0,0,0) is not in W

is that true or false?

[cyan fossil]

Idk

for real

is it true because 0 0 0 are in R?

[wicked turtle]

no look at the definition of W

[cyan fossil]

a subspace in R

[wicked turtle]

it's the set of all (x,y,z) satisfying: 3x - y = 0

does (0,0,0) satisfy that?

[cyan fossil]

wait | means satisfying? what does it mean4

[wicked turtle]

yea the symbol $W = {(x_1, x_2, x_3) \mid 3x_1 - x_2 = 0}$ means, in words: the set of all points (x1,x2,x3) which satisfy the equation 3x1 - x2 = 0

[woven radish]

Bungo

[wicked turtle]

so to answer the lower right question, does the point (0,0,0) satisfy that equation

i.e the point x1 = 0, x2 = 0, x3 = 0

[cyan fossil]

0 and 0

where is x^3, where did they get 0 0 0 in the answer too,

the book is complicated

i think i understand your point

[wicked turtle]

W is a subset of R^3

which consists of 3-dimensional points

that's why there are three coordinates

[cyan fossil]

no

[wicked turtle]

the equation though:

3x1 - x2 = 0

just means:

"three times the first coordinate minus the second coordinate equals zero"

the third coordinate can be anything

[cyan fossil]

this is ohio math

[wicked turtle]

what does that mean

[cyan fossil]

thanks i understand now

not fully

[wicked turtle]

definitely recommend you check out your textbook and see if there are some similar examples

[cyan fossil]

what is a text book

[wicked turtle]

or whatever book/notes/something readable you may have

[cyan fossil]

i don't take note lol

[wicked turtle]

ahh

alas i can't help with that 😀

[cyan fossil]

do you see anything comprisable or useful here?

[wicked turtle]

yea i would recommend reading this example carefully

it's not identical to yours but the concepts are similar

[cyan fossil]

ok thanks

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[valid gale]

How to find x^2-x = 33

[valid gale]

It cant divide by x in 2 terms

Because it becomes multiplied by x

Quadratic formula doesnt work because there's no constant number

[unreal eagle]

solve as a quadratic

get it into quadratic form first

ax + bx + c = 0

[valid gale]

It cant cancel x^2

Theres no constant number as c

[unreal eagle]

33?

[valid gale]

It is in different term

[unreal eagle]

wdym

just subtract 33 from both sides

x^2 - x - 33 = 0

[valid gale]

Is it ±33

Because x^2-x = 33

Is just x = 33

[jaunty mantle]

From here

We can substract 33 from both sides

x² - x - 33 = 33 - 33

x² - x - 33 = 0

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[raven cape]

hey guys

[raven cape]

can someone help me factor this quadratic
3x^2 -2x - 5

I've tried factoring by grouping and got 3x^2 + 3x -5x - 5

[sturdy cairn]

@raven cape Here is a strategy to factor quadratics in the form of ax^2 + bx + c
Step 1: Find ac and b, in this case it is -15 and -2
Step 2: Find two numbers that multiply to ac and add to b. In this case it will be -5 and 3
Step 3: Create a 4 term polynomial with two terms that have a degree of 1 where you can factor by grouping using the two numbers that multiply to ac and add to b, in this case it is 3x^2 + 3x - 5x + 5
Step 4: Factor by grouping: 3x(x + 1) - 5(x + 1) = (3x - 5)(x + 1)

[raven cape]

oh I see, thanks.

[sturdy cairn]

I'll show you a visual of how it works if that helps too

It is called the cross method or something like that

There you go

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[molten tulip]

i could be dumb as hecc, i mean i am, but could it be A?

because for A, the max would be like 2-4 or something i think?

wherease for B, it only gives 0 as the largest value i think?

from what im gathering

oh btw this is all for the bottom one, i haven't looked at the top one

ok so for the top one, they do have functions which tells me that theyd want you to mess with those maybe? but if we ignore those, it would still be A given the large value of y=14, compared to the small amount of info given in the table

at least, those are my takes on that

[devout snow]

@restive river Has your question been resolved?

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[frail lake]

do the dots just mean that those particular digits are repeating? (im just used to the bar above the digits)

[frail lake]

so like, the decimal for 28 would be 0.4242424242...

[rotund fulcrum]

yea

[lyric hornet]

Yes the dots above it mean recurring:\~\

So, $x=0.4242424242...$\
$10x=42.42424242...$\
$10-x=42.4242424242...-0.42424242424242...$\
$9x=42$\

From there, the rest is easy. That's just the way I've always done these problems (there are probably others)

[woven radish]

XxMrFancyu2xX

[frail lake]

awesome, thank you so much!

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[formal oxide]

I need some assistance.

[formal oxide]

Given ⊙K with secants NS and NR, which expression represents PS?

I don't remember the process or directions to solve this, and I don't think my teacher went over this with us. If possible, does anyone have any tips?

[grave pond]

Use that

@formal oxide

[formal oxide]

Aight, I'll try that.

[devout snow]

@formal oxide Has your question been resolved?

[formal oxide]

So I've got this so far.

Do I just see which of the four answers can be simplified into that equation at the bottom of the picture?

hold up, i think i've figured it out. I'm all good now 👍
(thanks El_7AG for that reference thing)

.close

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[broken mason]

what is number 1 asking and how would i go about it?

[broken mason]

I think i know how to solve 2 but can someone check my answer

<@&286206848099549185>

[wicked turtle]

focusing only on row 2, how did you go from the first to the second matrix?

[broken mason]

r3-r1 then r2+2r1

[wicked turtle]

if you did r2 + 2r1, wouldn't the first column of the new row 2 be 2, not 0?

[broken mason]

ur right my mistake

the only thing you can do to get it in REF then is to multiply by -1 right

regardkess that wouldnt change the shape of the answer

so our solution would still be c=-2

<@&286206848099549185>

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[lost gazelle]

the stick is 4cm how do i find AF

[lost gazelle]

part a told me to find FC and i found it

[finite briar]

U found..?

[lost gazelle]

i found fc

[lost gazelle]

what?

[restive river]

can i get some help?

[lost gazelle]

can you send in new channel please

[finite briar]

#❓how-to-get-help

[finite briar]

What was the length u found?

[lost gazelle]

2sqrt3-2

[finite briar]

Which stick is 4 cm?

[lost gazelle]

both

the stick fell

[restive river]

no one’s is helping

and i cant send any pics

[lost gazelle]

what do i do

[devout snow]

@lost gazelle Has your question been resolved?

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[tall geyser]

i am keep proving the given information using the given information

i cant make any progress towards the answer

[devout snow]

@tall geyser Has your question been resolved?

[tall geyser]

<@&286206848099549185>

;-;

ok i have given up

[fallen elk]

can i get help solving this

[devout snow]

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[jovial mauve]

(claiming this channel)

[jovial mauve]

Is there a similar generalization for nPr as well?

like what would nPr(n,0)+nPr(n,1)+...nPr(n,n) equal?

[arctic field]

,w sum k=0 to n of n!/(n-k)!

[woven radish]

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Requested by 夢雪#2250

[arctic field]

e

[jovial mauve]

yikes

Gamma functions

e

[arctic field]

its the nth partial sum of e

[jovial mauve]

Does this help ${}^nC_r=\frac{{}^nP_r}{r!}$

[jovial mauve]

I will try substituting

Ok

[arctic field]

substituting where?

[jovial mauve]

Here ^

$\frac{{}^nP_0}{0!}+\frac{{}^nP_1}{1!}+\dots+\frac{{}^nP_n}{n!}=2^n$

[jovial mauve]

[pseudo basin]

weh

any reason why you're interested in this

or just curiosity

[jovial mauve]

Uhh, I'm curious

[jovial mauve]

I guess this cannot be simplified further

.close

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[lapis lodge]

Hi. How far off am I from the right answer?

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[lapis lodge]

.close

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[pine fox]

Prove that every tree with a vertex of degree 4 has at least 4 leaves

[pine fox]

i'm struggling with this proof
i'm not sure where to start
should i try induction on the number of vertices?

[pseudo basin]

do you need that

can you use the fact that a tree on n vertices has exactly n-1 edges

[pseudo stag]

you can also take the vertex that has 4 children. what can you say about the n.o children of children + n.o. leaves?

[violet wind]

is that supposed to be an arithmetic +

because in that case I'm lost too now

[violet wind]

only true of finite trees

[pseudo stag]

no. its supposed to not decrease

you start with 4 children, end up with at least 4 leaves

[pine fox]

do i combine that with the degree sum formula?

or is|V| = |E| - 1 enough to get me started?

[pseudo basin]

degree sum formula will be necessary in some way.

assume that there are at most 3 leaves, then your degree sum will be at least 4 + 3*1 + 2*(n-4)

which ends up as greater than the 2(n-1) which it should be

[pseudo stag]

ill state my sol again but in a bit more clear way: look at the subtree with root at the vertex with 4 children. Each child will contribute at least 1 leave

[pine fox]

just one quick question how did you know the remaining n-4 vertices would have degree of 2?

[pseudo basin]

i didn't.

their degrees are at least 2 though

because they are not leaves

[pine fox]

Ahhh I see I see

am i trying to get a contradiction by assuming there are at most three leaves?

[pseudo basin]

that's the approach im suggesting yes

[pine fox]

ahh i see i see

So if i can show that |V| = |E| - 1 does hold anymore

then i can get the contradiction?

[frosty cradle]

wrong server, see #old-network

[devout snow]

@pine fox Has your question been resolved?

[frosty cradle]

<@&268886789983436800> spam

read #❓how-to-get-help

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[stray linden]

How do I find the coordinates of the normal vector if the coordinates of the tangent vector are given? Like I get you have to switch the x and y coordinates, but when do I rotate counter clockwise and when do I rotate clockwise (which coordinates becomes negative)? like for example to find the center of curvature in P(1,1) of y=x^2, you rotate counter clockwise. I don't get it...

[pulsar dock]

Maybe I am misunderstanding the question, but it doesn't matter if you just want a normal vector.

For the centre of curvature, you would need to make sure the angle formed is acute or it would point the wrong way

[devout snow]

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[restive river]

and also if possible, how does one solve this

[stone stump]

desmos to graph it

you can solve it with eg wolframalpha after you set up the integral

or well, solve the integral manually

[restive river]

thank you

wolfram alpha doesnt seem to work for questions im keying into it, not sure if im using it wrong

[stone stump]

it's not that good

give it the integral and it can solve that

[frosty cradle]

graphing it is helpful to see the area you're going to integrate over

[restive river]

oh i thought it works like chatgpt

[stone stump]

no

[restive river]

is it this small red part here

[frosty cradle]

no

[restive river]

this one

[frosty cradle]

from t=1/2 to t=2

[restive river]

how does one get the integral then

sorry i really forgot all my math lol

is it by looking at the shapes

[frosty cradle]

for this specific problem?

you can integrate x^3 from 1/2 to 1, then 1/x^2 from 1 to 2 and add

[restive river]

so basically look at the shapes and derive from there

thanks a lot dude

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[mystic musk]

is it allowed to do a f B --> A,

[frosty cradle]

not as it is defined

[mystic musk]

why not actually

[mystic musk]

btw what does this symbol mean? equivalent?

in statistics it's to say which distribution it has

and in vector calculus it was meaning similair

[frosty cradle]

because A and B are different sets. f is defined as taking elements from A and "returning" an element from B

[mystic musk]

oh but I meant can't u introduce a new letter called z : B --> A

[frosty cradle]

then it's a different function

[mystic musk]

but will it still be one to one?

[frosty cradle]

it depends on how it is defined

you would need to check the conditions for it to be one-to-one

[mystic musk]

and how about this

[mystic musk]

in my case I think the conditions are met since every element has a pair and it only goes to 1

one to one funciton implies bijective too right

[frosty cradle]

bijective is one-to-one and onto

so no, it does not imply that

[mystic musk]

ah one to one is that every value in the codomain is being seleceted right

and onto is that every value of the range is chosen only once

[mystic musk]

and is this symbol saying equivalent?

[frosty cradle]

Read this if you are unclear on notation: https://en.wikipedia.org/wiki/Bijection,_injection_and_surjection

[frosty cradle]

I don't know that symbol

you'll need to consult the source for whatever that symbol is

[mystic musk]

how was that symbol called in statistics?\

[frosty cradle]

I don't know. it means "is distributed as" usually. the symbol in Latex is \sim, I believe. Consult the source or wherever you got those notes

[mystic musk]

\sim

[frosty cradle]

$A \sim B$

[woven radish]

cwatson

[frosty cradle]

that symbol?

[mystic musk]

ye

how is that called in wrods

[frosty cradle]

as I said, I don't know

[mystic musk]

my notes said it means equivalent

but I forgot why

[frosty cradle]

consult your textbook

[mystic musk]

what does consult mean?

[frosty cradle]

look in your textbook for the definition/information

[devout snow]

@mystic musk Has your question been resolved?

[devout snow]

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[fresh cypress]

how would you integrate this

[formal haven]

is that integral calculator

[fresh cypress]

I wrote it there to have it in a visual form which is easier to see instead of writing it out, but yes

[formal haven]

well what does integral calculator say 😝

[fresh cypress]

I dont understand what they did so I was asking on here hahah

[fresh cypress]

I am unsure of how they did the rearrangement in this part

I understand the substitution

but not how they essentially replaced the 3 in the problem with the x in the ln() power

[long pasture]

this is kinda playing a trick

you can prove it if you want, but

a^(ln(b))=b^(ln(a))

to proof it, you'll just have to change the LHS to
e^(ln(a)*ln(b))

because e^(ln(a))=a

hope that helps!

[fresh cypress]

ahh yeah thats true!

I never thought to do that

so they put it to the power of e, and then ln()e

ln() it in the power

so they can cancel out

that is smart

thanks!

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[serene bramble]

how can I TRIANGLE y={1} over {x+TRIANGLE x}-{1} over {x}
TRIANGLE turn y={- TRIANGLE x} over {(x+TRIANGLE x)x}

[serene bramble]

how do you make the one on the left the same as the one on the right?

[quaint citrus]

Triangle is called delta

You wanna make a common denominator on the left hand side

Multiply the numerator and denominator of one fraction by the denominator of the other

$\frac 1a - \frac 1b = \frac {b}{ab} - \frac {a}{ab}$

[woven radish]

Stephen

[serene bramble]

thank you

I asked a pretty stupid question

[quaint citrus]

Lol nah u good

It’s easy to get tripped up when we see symbols we’re unfamiliar with

[serene bramble]

thank you very much

It was a big help

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[hidden dragon]

Is it right to write this, (I)^m = I for all m belongs to R?

[hidden dragon]

Because I is the "one" of matrix world and one raised to any real number power is one

[restive river]

yes, but you need to be careful about how you define powers

it's fine if m is an integer

[hidden dragon]

I said power to be real number

[hidden dragon]

Oh

So fractions can't be the powers

[restive river]

fractions can be powers if you define things correctly

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

[hidden dragon]

Ok thanks for the source

[restive river]

I doesn't have distinct eigenvalues, so I'm not completely sure whether it's well defined

[hidden dragon]

I see

[restive river]

but if a matrix has distinct eigenvalues, you can diagonalise https://math.stackexchange.com/questions/732511/fractional-power-of-matrix

[hidden dragon]

Thanks for your help. I'll look into those links. For now I will consider m to be integer in the equation !

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[untold vector]

Why I'm getting a wrong answer.

[untold vector]

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[untold vector]

.reopen

[devout snow]

✅

[untold vector]

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[blazing edge]

hello i would like some help in finding a limit of a sequence and an explination of how to do it generally: $a_{n}=\frac{n^2}{2^{-2}+3n^3}}

[blazing edge]

$lim a_{n}=\frac{n^3}{2^{-n}+3n^3}}$

[woven radish]

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

[polar chasm]

limit as n approaches infinity?

[blazing edge]

yes

[devout snow]

@blazing edge Has your question been resolved?

[blazing edge]

<@&286206848099549185>

i am aware that it converges toward 1/3, but i want to find an epsilon proof for this

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[limber stag]

,tex Which of the following statements (one or more) are correct for an antisymmetric relation R in the set A? \newline
$R^{-1}\cap R={}$

$R\cap \mathbb{I}_A=\mathbb{I}_A$

$R\cup \mathbb{I}_A=R$

$R\cap \mathbb{I}_A={}$

$R^{-1}\subseteq R$

$R^{-1}\cap R\subseteq \mathbb{I}_A$

$R\cdot R\subseteq R$

$I_A\subseteq R$

$R^{-1}\cap R=R$

$R\subseteq R^{-1}$

[limber stag]

how can i solve this analiticly?

[woven radish]

Eric B.

[devout snow]

@limber stag Has your question been resolved?

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@limber stag Has your question been resolved?

[devout snow]

@limber stag Has your question been resolved?

[devout snow]

@limber stag Has your question been resolved?

[devout snow]

@limber stag Has your question been resolved?

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[restive river]

I feel like I know what I’m supposed to do but I’m just not certain of it

[acoustic vault]

so what do you think youre supposed to do

[restive river]

I feel like it’s got something to do with that other thing you helped me on, but what I’m not understanding is what number to plug into the function or how to get that number from the interval

[acoustic vault]

what you did before was an approximation

this is asking for an exact area

[restive river]

Yeah I got nothing then 😔

[acoustic vault]

what you did before is called a riemann sum

you divided the interval into 4 rectangles and calculated the area

[restive river]

I’m guessing that only applies when finding approximations?

[acoustic vault]

yes you would be missing this green portion

but its the same idea

[restive river]

Ahh okay yeah that makes sense

[acoustic vault]

instead of using 4 rectangles

what happens if we use 6?

the green portion will get smaller

8 rectangles?

even smaller

the idea is that you divide the interval into

an infinite amount of rectangles

each with a tiny tiny tiny width

[restive river]

Okay I’m understanding better, but how would I do that?

[acoustic vault]

this is what an integral means

$\int_{-1}^{2} f(x) dx$

[woven radish]

Køter

[acoustic vault]

this is a sum of an infinite amount of rectangles with width=dx and height=f(x)

on the interval [-1,2]

[restive river]

So I’d plug in my function into the (x), and find the integral from there?

[acoustic vault]

your function into f(x)

so its $\int_{-1}^{2} x^3-1 dx$

[woven radish]

Køter

[acoustic vault]

do you know how to find definite integrals?

or just integrals in general?

[restive river]

….no😭 I’ve been in the hospital all week 💀💀

But it’s just the opposite of derivatives right?

[acoustic vault]

yes the indefinite is

and the definite is F(b)-F(a)

where b and a are the bounds

so first just find the antiderivative

[restive river]

I’ve got no idea what the anti of x^3 is but for -1 I’m guessing it’s -1x

[acoustic vault]

yes

the antiderivative of x^3 is the opposite of the power rule

you add one to the exponent

and divide by it

you can see how that is opposite

[restive river]

1/4 X^4?

[acoustic vault]

yup

so F(x)=x^4/4-1x+C is the antiderivative

and you need to do F(b)-F(a) to find the definite integral

[restive river]

So I’d plug in the bounds and subtract my answers from that?

[acoustic vault]

yes, and the +C cancels out

so dont even consider it

[restive river]

So then my answer would be .75?

[acoustic vault]

yes i think so

,w (1/4 * 2^4-2)-(1/4 * (-1)^4+1)

[woven radish]

Results provided by WolframAlpha

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

Requested by Køter#5504

[restive river]

YEAAHHH

thank you so much for your help again : )

[acoustic vault]

np

[restive river]

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[slate willow]

An ethical hacker, also referred to as a white hat hacker, is an information security expert who systematically attempts to penetrate a computer system, network, application or other computing resource on behalf of its owners. The purpose of ethical hacking is to evaluate the security of and identify vulnerabilities in systems, networks or system infrastructure. It includes finding and attempting to exploit any vulnerabilities to determine whether unauthorized access or other malicious activities are possible.
An ethical hacker is required to randomly guess the correct pin code that consists of the number 0 through 9 that must be entered in the correct order to access a company system.

1. What is the probability that the ethical hacker will guess the pin code correctly on the first try?
2. There are many variations of this guess. Assuming the primary variation allows the ethical hacker to guess correctly if the four-digit in the number are selected in any order as long as they are the same four digits as set by the IT security of the company.
   For example, if the ethical hacker picks four digits making the number 2376, then the he will guest it right if 2376, 3726, 6327, 7632, and so forth, are entered. Consider the following four different versions of his presumptions.
   (a) All four digits are unique (e.g. 1234)
   (b) Exactly one of the digits appears twice (e.g. 2334, 8185)
   (c) Two digits each appear twice (e.g. 1212, 8855)
   (d) One digit appears three times (e.g. 2226, 8188)
   Find the probability that the ethical hacker will successfully guess the accurate pin code in the first try for each of these four situations.
   Show the necessary steps and explanation for the four presumptions stated above.

someone help me answer this please, i keep getting confused...

[slate willow]

<@&286206848099549185>

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[graceful elm]

what rule is there to differentiate this?

[sonic smelt]

You can rewrite that as e^(xln(3)) and use chain rule

Assuming you know how to differentiate e^x

[graceful elm]

I do not

[sonic smelt]

By definition the derivative of e^x is itself

[sonic smelt]

Do you know chain rule though?

[graceful elm]

yes

[finite briar]

Can't he use power rule?

[sonic smelt]

Alright, so what do you get?

Power rule is for x^n

Well if you wanna differentiate xln(3) using power rule then sure that counts

[graceful elm]

?

[sonic smelt]

The derivative of xln(3) is just ln(3) though

So it's 3^x * ln(3)

[graceful elm]

why

[sonic smelt]

$\dv{}{x} x\ln{3} = \ln3 \cdot \dv{}{x} x = \ln3 \cdot 1$

[woven radish]

A Lonely Bean

[sonic smelt]

$\dv{}{x} 3^x = \dv{}{x} e^{x\ln3} = e^{x\ln3} \cdot \dv{}{x} x\ln3 = 3^x \cdot ln3$

[woven radish]

A Lonely Bean

[graceful elm]

Huh

oh

[sonic smelt]

Okay, I will go through the chain rule then

[graceful elm]

nvm

I get it

[sonic smelt]

Alright

[graceful elm]

🙏 📿

[sonic smelt]

Pretty irrelevant

[graceful elm]

????????????

Very relevant

[sonic smelt]

Unless you are differentiating xln(3) with product rule

Then sure, that counts

As an overkill

[graceful elm]

Which I was ?_?

[sonic smelt]

In order to?

[graceful elm]

xln(3) requires product rule

x*ln(3)

[sonic smelt]

No, just bring the constant factor to the front

d/dx (kf(x)) = k d/dx (f(x))

Product rule makes it look unnecessarily complicated

[graceful elm]

I aint remembering all that

Thank you

.solved

[sonic smelt]

[graceful elm]

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[gray agate]

anyone know why 11 isnt onto?

[gray agate]

cant u do the same process for 11

[pseudo basin]

you mean for 11a?

well, try it.

you'll get the same scratch work starting from g(n) = y and get that n = (y+5)/4

[gray agate]

yea its the same thing

[pseudo basin]

knowing that n must belong to Z, being an input to g

[gray agate]

u get the same thing

[pseudo basin]

can you see where this could go wrong

[gray agate]

Oh wait when u do n= (y+5)/4

this is where u plug in the numbers?

for where it says Z->Z and if it isnt a integer then its not onto

[pseudo basin]

kind of missed my point here...

i dont like the wording "this is where you plug in the numbers"

the thing that "goes wrong" here is that (y+5)/4 need not be an integer.

for example when y=42069

[gray agate]

y=1

is not integer

so its not onto

?

6/4

[gray agate]

not be an integer?

It says Z->Z

[pseudo basin]

"need not" = "doesn't need to"

slightly more flowery way of saying it

[gray agate]

ur confusing me now

[pseudo basin]

but yes, this is how to prove that g: Z -> Z given by g(n) = 4n-5 isn't onto.

[gray agate]

ok thanks

thats all i need to know 💀 my exam tom

[gray agate]

but just to clarify if we plug in y=1 we dont get an integer

so that proves its not onto correct?

i just wanna make sure lol

[pseudo basin]

'plug in'

kind of meh as far as wording goes

but yes

[gray agate]

wb when its something like this

(2y,-x)

[pseudo basin]

"it"

[gray agate]

?

[pseudo basin]

i don't know what you mean by "it"

[gray agate]

something like this

[pseudo basin]

do you expect me to know exactly what kind of generalization you expect me to make

is this another function whose onto-ness you want to investigate

[gray agate]

to check if it is onto

do we also use

y=u/2

to check as well

[pseudo basin]

what letter is this?

[gray agate]

u

[pseudo basin]

looked like it was halfway between u and v to me.

handwriting coul be better.

[gray agate]

-x=v

[gray agate]

im using a mouse on my laptop on snipping tool 💀

[pseudo basin]

this is a mouse-drawn u that can be told apart from v just fine

anyway

so you've found that G(x,y) = (u,v) is solved by (x,y) = (-v, u/2)

no issues arise here as the domain of G is R^2

therefore yes this function is onto

[gray agate]

how do we check for domain

y= u/2 and x = -v?

and plug in numbers

[pseudo basin]

my god, what is "check for domain" supposed to mean

[gray agate]

Like in this question

(y+5)/4

to see if it is an integer

if the domain was was Z^2 or something

how do we check for that in this question

[pseudo basin]

in this question the domain is R^2

[gray agate]

ik

[pseudo basin]

are you asking what would happen if the function were Z^2 -> R^2 instead?

[gray agate]

no just Z^2

instead of R^2

[pseudo basin]

"if the domain was Z^2"

and you said nothing of the codomain and expected me to replace that with Z^2 too

[gray agate]

Z->Z x Z->Z

[pseudo basin]

task failed successfully.

now you've written something that is almost certainly nonsense.

[gray agate]

bro

u knwo what i mean ur trying ot make me suffer

Instead of real numbers

its Integers only

[pseudo basin]

please don't call me bro.

also i am not trying to make you suffer. i am trying to make you articulate things properly.

[gray agate]

ik that but Im on a tight time schedule 💀 exam tom

[pseudo basin]

in my experience, people who can't articulate things properly get confused, lost or stuck way more often.

do you want to get confused, lost and stuck on your exam tomorrow?

if you do, then say so, and i will cope with your bad wording.

maybe even respond in kind.

[gray agate]

Well i dont think i will

because your just checking to see if the number matches the domain

I just dont know how to check it

[pseudo basin]

you dont know how to check whether (-v, u/2) belongs to Z^2 for every u, v ∈ Z

[gray agate]

yes ik how to check (-v,u/2)

u/2 is not an integer

but im talking about other functions that may not be so simple to see

[pseudo basin]

there is no panacea for this.

there is no one size fits all solution.

[gray agate]

like is it the same thing as we did before n = (y+1)/5 and check if n is a integer or something

same thing here? we plug into u/2 and see if integer or not

[pseudo basin]

jeez

yes sure we plug "it" in and see if "it" is an integer or not and then "it" is onto or not onto accordingly

and then it is solved

or maybe not, and then it's over

[devout snow]

@gray agate Has your question been resolved?

[devout snow]

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[kind elk]

How does the first expression simplify into the answer? The part I mainly don’t understand is the one in brackets: 8sin4xcos4x
I’m assuming it’s double angle identities but I don’t know how it got the answer.. pls help

[chrome girder]

yes
2sin(4x)cos(4x) = sin(8x)

[kind elk]

Is that a formula u just need to know?

[hushed wraith]

its the double angle formula

2sinxcosx = sin2x

when u have 4x instead of just x

[sturdy cairn]

sin(4x) also equals 2sin(2x)cos(2x) and you can expand it

Using the identities multiple times

[kind elk]

Oh so if it was sin(16x) would that be 2sin8xcos8x

[sturdy cairn]

Sin(16x) is 2sin(8x)cos(8x) because 16x is 8x multiplied by 2

[kind elk]

Ohhhhhh I see ty

[sturdy cairn]

However expanding it would take a lot more work because you also have to deal with the cosine

[kind elk]

Yeaaa but I think I get how it works, why does it become 4 in the answer then?

[sturdy cairn]

cos(2x) = cos^2(x) - sin^2(x)
cos(2x) = 2cos^2(x) - 1

Oh

[sturdy cairn]

Because think of it this way: If you wanted to find sin(24x) and you know sin(2x) = 2sin(x)cos(x) you basically think what is "x"? Which is 24/2 = 12. Then use the identity

[sturdy cairn]

Oh was the sin(2x) multiplied by 4 or something?

[kind elk]

Ok but y is it not 2 on the outside but 4?

OHHH WAIT I GET it now

[sturdy cairn]

When you use the double angle formula you multiply by 2. What was the original question?

[kind elk]

The first expression is the original question

It’s basically the last big from a product rule question

That’s the differentiated answer but now it’s been simplified

[sturdy cairn]

ln(8) times 8sin(4x)cos(4x) + (sin(4x)^2)/x?

Oh you differentiated it

If you are supposed to differentiate it use the chain rule

Because you have a function inside another function

[kind elk]

Sorrryyyyy it’s supposed to be:

ln(x) times 8sin(4x)cos(4x) + (sin(4x)^2)/x
Not 8, I wrote it wrong

[sturdy cairn]

The derivative of ln(x) is 1/x
The derivative of sin(x) is cos(x)
The derivative of cos(x) is -sin(x)

So then you will have to use the chain rule multiple times

Oh and the product rule too

[kind elk]

dy/dx = ln(x) times 8sin(4x)cos(4x) + (sin(4x)^2)/x

[sturdy cairn]

And the quotient rule

[kind elk]

That’s what it’s supposed to be, then u just have to simplify it

[sturdy cairn]

Oh so this is the derivative?

[kind elk]

Yes

I don’t understand the 4 in ‘4lnxsin8x’

[sturdy cairn]

Oh

[kind elk]

In the answer

Yea, sorry for the confusion 😭

[sturdy cairn]

How did you get sin(x) in the ln(x)?

I thought you just multiply?

[kind elk]

So it seems like in dy/dx = ln(x)(8sin(4x)cos(4x) + (sin(4x)^2)/x

The 8sin4xcos4x = 8(2sin4xcos4x) from the double angle identity

WAITTT A

SORRY I GET IT

[sturdy cairn]

And you also want to use the power identity

[kind elk]

Ok nvm I got it

[sturdy cairn]

Here is what I did one sec

[kind elk]

Dw I understand it now!!! But tyyy

[sturdy cairn]

🙂

Sorry I wrote down one identity incorrectly

[kind elk]

Ahh I see lol

Ty anyways!!!

[devout snow]

@kind elk Has your question been resolved?

[devout snow]

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[somber shore]

hi

[somber shore]

can I have a small hint please

btw

I should not use hop

[chrome girder]

tan(x) ~ x
cos(x) ~ 1-0.5x^2

[somber shore]

sorry I forgot to say

neither hop nor maclaruin

[chrome girder]

ree

[pseudo basin]

if you're willing to wait like 30 min until i get back home i could sketch up a one-hand-tied solution

[somber shore]

sure
thank you

[devout snow]

@somber shore Has your question been resolved?

[pseudo basin]

ok im back

did this algebra here

used the ideas of sin(t)/t and tan(t)/t extensively here

plus some fraction manipulation and conjugate shit

u(x) is a bunch of triggy shit that goes to 1

[somber shore]

lemme read it

ty

[somber shore]

I got it tysmmmm

.close

[devout snow]

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[devout snow]

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[signal crag]

,rotate

[signal crag]

I tried wearing the lengths of the vectors

Here is my working out for that

However there seems that there are more solutions and also one of my answers is incorrect

[lethal shard]

@signal crag I texting you in private

[signal crag]

why

cant you just write the stuff here

[lethal shard]

@signal crag sabes español

?

[signal crag]

no

[lethal shard]

Xddd

Alse speak Spanish in this server?

[pseudo basin]

don't private-message people at random.

[lethal shard]

Xd

[devout snow]

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[signal crag]

so i tried to equate the lengths of the vectors

here is my working out for that

however, the answers are saying that there are more solutions and also that one of my solutions is wrong

[near belfry]

I don’t understand what you’ve done wrong tbh

Can I see the answers please?

[long fog]

Sure

[near belfry]

@signal crag

Ty

[signal crag]

there are like two other cases

[long fog]

Yes

[signal crag]

how do those other cases work

[signal crag]

also why doesnt my negative answer work

[long fog]

Well depends on the equivalent sides

There are three sides and hence you have 3C2 = 3 cases

ku - v is the line joining ku and v

Which can be equal to one of the other sides aswell

[signal crag]

oh yeah

that makes sense

but what about the negative answer

why didnt that work

?

[long fog]

I think that's a mistake in solution

It should work

[signal crag]

cool

[signal crag]

also is this like a general thing

[long fog]

General in what way?

We are choosing two sides that are supposed to be equal out of 3

So yeah there are going to be 3 or 3C2 cases

[signal crag]

ah okay

that makes a lot of sense

thanks so much

.close

[devout snow]

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[long fog]

You're welcome

[devout snow]

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[frail igloo]

why it be like that

[frail igloo]

can you just look at this and know?

(the answer is False btw)

[hazy totem]

This may be stupid, but ℤ₅ is finite, so there are only finitely many vectors with coefficients in it, so at most finitely many solutions

[frail igloo]

no my friend

it is i who is stupid

:C

thank you

[hazy totem]

that's a trick question honestly

don't feel bad

[magic thicket]

It's a legitimate question

[frail igloo]

i mean if you think about it

it's so obvious now that you say it

[magic thicket]

Sometimes linalg questions can be about other fields

For example, is rank conserved when you change field ?

[hazy totem]

yeah but it's clearly meant to catch off-guard an unsuspecting linalg student that misses the field it's defined over

[magic thicket]

Teaches you how to read properly

Very valuable

[frail igloo]

wait... so different fields might be infinite?

have infginite solutions*

[hazy totem]

well yeah, ℝ

[magic thicket]

The number of times I've had to read something 3 times to actually understand the question

[frail igloo]

OF COURSE R IS A FIELD

lmao

okay great. and the complex field too

[hazy totem]

and ℚ

[frail igloo]

but what about other Z_n fields, none of those, right?

[hazy totem]

all of them are finite so yeah

[frail igloo]

ofc.