help-19

[toxic monolith]

$\int_{}^{}f\left[ g\left( x \right) \right]g'\left( x \right)\text{ }dx=\int_{}^{}f\left( t \right)\text{ }dt\text{, }\\\text{ where }t=g\left( x \right)\text{, and }dt=g'\left( x \right)\text{ }dx$

[clever fjord]

Joanna Angel

[hoary jungle]

idk if i understand it but i can apply it xd

[toxic monolith]

that formual describes the theorem about substitution in integrating, sayign ti infromally

[hoary jungle]

yeah its like a reverse chain rule

[toxic monolith]

yes but it is harder to notice

but very alike

so in your integral iii)

you must decide what is

g(x)

and

what is g'(x)

if you identifgy it well, that is great

sometimes it needs slight manipulation too

before aplyign such formuala

[hoary jungle]

yeah i felt like i can manipulate it somehow by seeing that 1/x was lnx which would give me e^lnx

[toxic monolith]

1/x is a derivative of lnx

[hoary jungle]

yeah and whats under the integral is the derivative no?

[toxic monolith]

last integral iii) ?

then no any lnx theer

lnx is absoluteley useless

you have to look at yoru integrand

insuch a WAY:

first decude waht is inenr function, emans g(x)

inner = = inside

and then

calcualte its derivative

i shwo you diff interaglnow look :

$\int_{}^{}x^{2}e^{x^{3}}dx$

[clever fjord]

Joanna Angel

[toxic monolith]

yo can seehere,

multipliocation

of two things

exponential function

and quadratic function

addtioanlly

expoenthial function is not simple but it is compsoed

and you can see, that inside this expoent

you can find cubic function

do you see it ?

in oher words, cubic function is inside, inenr function

[hoary jungle]

yeah i see it

[toxic monolith]

so it can be our g(x)

so probably we can assume

t = x^3

but before

we do it

we must calcualte its derivative

derivative of x^3 is

3x^2

right ?

so it stands in your expresiosn

differnce is the constant only

3

but rememebrt

we never eve rare afraid of cosntants

since

every and each cosntat can be taken out of integral

hence

[hoary jungle]

yeah i know that

[toxic monolith]

we can see that g(x) = x^3

and

g'(x)dx = 3x^2 dx

[hoary jungle]

i know how to do it

[toxic monolith]

and now coming to your iii) integral

the "inner" function loks liek 1/x

so we can assume

that g(x) = 1/x

but now lets get its derivative

its derivative is equal to:

g'(x) = -1/x^2

and now, we can see some maybe problem

since

before exponent we can see

1/x^3

and it is diffeertn than our derivative

so wwe can be confused but now

becasue

1/x^3 = 1/x * 1/x^2

so we do such sligth manipualtion

$\int_{}^{}\frac{1}{x^{3}}e^{\frac{1}{x}}dx=\int_{}^{}\frac{1}{x}e^{\frac{1}{x}}\frac{1}{x^{2}}dx$

[clever fjord]

Joanna Angel

[toxic monolith]

$f\left[ g\left( x \right) \right]=\frac{1}{x}e^{\frac{1}{x}}$

[clever fjord]

Joanna Angel

[toxic monolith]

and

$g(x)=\frac{1}{x}$

[clever fjord]

Joanna Angel

[toxic monolith]

because:

$g'(x)=-\frac{1}{x^{2}}\Leftrightarrow -g'(x) = \frac{1}{x^{2}}$

[clever fjord]

Joanna Angel

[toxic monolith]

I suppose that you should do it now ,

[hoary jungle]

yeah

[toxic monolith]

after this substitution you should reach an integral, that will need integraiotn by parts s well

but first step is substituion

[hoary jungle]

i also need integration by parts?

[toxic monolith]

not at once, but after substituion

[hoary jungle]

yeah

[toxic monolith]

you shud get the integral that wil require

integraiotn by parts

i can see it

[hoary jungle]

okay thanks for the help

.close

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[pallid wyvern]

Hi I need help with getting all the solutions for x in [0,360degrees) for the equation

2 sin(2x)-tan(2x)=0

[pallid wyvern]

2x = y

2 sin y = tan y

2 sin y = sin y / cos y

2 sin y cos y - sin y = 0

sin y (2cos y - 1) = 0

sin y = 0
y = 0, 180

x = 0, 90

2 cos y - 1 = 0

cos y = 1/2

y = 60, 300

x = 30, 150

0, 30, 90, and 150 are probably not ALL the solutions within [0,360)

this was a quiz question recently and in the quiz i used double angle indentities and did not get all the solutions

can someone help?

oH WAIT 330 is also a solution

oh 180 as well

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[modern palm]

yo

[runic swift]

!status

[odd edge]

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[modern palm]

bruh

I used the cosine law

lets put it like that

and I didnt get what I wanted

[runic swift]

you're trying to find the distance between those two buildings?

[modern palm]

yea

I keep getting the wrong answer when I plug it in

[runic swift]

well whats your answer

[open onyx]

you sure your calculator is in the right mode?

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[slow fern]

can anyone help

[slow fern]

?

.close

[night heron]

Look at your denominator

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[clever fjord]

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

[gentle canopy]

nvm i got it

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[austere matrix]

I'm working on this problem and I honestly have no clue how to progress this to a point where it does not equal 0 over 0

[austere matrix]

I'm using L'Hopitals

I got (sin(x^2) - 2x)/(-2sinx)

which doesn't help

[long tinsel]

Your differentiation is incorrect

Can we have a look at your work

[austere matrix]

that wouldn't be very helpful considering this line is the only thing i wrote

[long tinsel]

Alright

$\dv{x} \int_{a(x)}^{b(x)} f(t) \dd{t} = f(b(x))b'(x) - f(a(x))a'(x)$

[clever fjord]

jan Nejon

[austere matrix]

so 2sin(x^2)

[long tinsel]

In the denominator you haven't applied the chain rule [correctly]

[long tinsel]

2xsin(x^2)

[austere matrix]

ah

yes

[austere matrix]

how so

[long tinsel]

Differentiating cos^2(x) gives 2cosx(-sinx)

[austere matrix]

oh

yes

sorry this is a review packet and we've been doing integrals for months

I don't know how I've forgotten everything

[long tinsel]

Don't apologize lmao

[austere matrix]

ok

so from there

what do I do to progress it

@long tinsel

<@&286206848099549185>

[trim agate]

(solving problems of simultaneous linear equations with two variables) im stuck in 3d=e+7 3e=d-8

[austere matrix]

#❓how-to-get-help

.close

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[pine drift]

Is the dependable variable x or y?

[mint mirage]

How I remember it is DRY MIX
D - Dependent
R - Responding
Y - Y axis
M - Manipulated
I - Independent
X - X axis

[pine drift]

Tysm!

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[hallow pelican]

how should i approach 16?

also 18

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[idle cipher]

tryna do this green's theorem exercise

[idle cipher]

but could someone give me a hint as to how i can find the bounds for the x domain?

i get in terms of why the region is bounded from x^2 to x

(il fix)

but for the the x domain itself i can't make heads of tails

im trying to understand the process intuitely because we wont be given graphing calcs or be expected to know how to draw some of hte more complex functions

or vectors rather

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[sharp karma]

Please just when I put a number like 2 is it then -2^2 or (-2)^2

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[flint heart]

-2^2

[sharp karma]

.close

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[simple pond]

Is there any numerical set in which √x=-1 has a solution?

[south plume]

no

[simple pond]

I think there's no solution even for complex numbers

[south plume]

sqrt is always positive

[odd magnet]

alright, sorry

[south plume]

yes

[south plume]

there are solutions for $x^{\frac{1}{2}} = -1$ tho

[clever fjord]

artemetra

[simple pond]

I heard that the sqrt only gives positive numbers because that's a function or something like that

[south plume]

oh nvm

[simple pond]

Sad but true

Thank you so much

[south plume]

kinda

formally speaking it only gives the principal root

[simple pond]

So in the complex numbers

The sqrt would be also defined only for values in which the complex and the real part are superior to 0?

[south plume]

not understanding

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[bronze cradle]

Channel closed so I'm opening a new one
https://media.discordapp.net/attachments/903481105888452609/1180131010889908305/image.png?ex=657c4da6&is=6569d8a6&hm=824ab9cb2e91e84886866bbee749f6946d2bceb57d64ca8dc0049b4662d8816d&

[bronze cradle]

All i need to prove is the reciprocal

yo @mystic saffron if you got any idea it would be appreciated

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[bronze cradle]

.reopen

[odd edge]

✅

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[bronze cradle]

.reopen

[odd edge]

✅

[prisma cairn]

ok so what does |z-a| represent?

if z and a are two points on argand plane

[odd edge]

@bronze cradle Has your question been resolved?

[bronze cradle]

Not sure, distance?

[prisma cairn]

yes

distance between z and a

so what would |z+a| represent?

[bronze cradle]

Distance between z and -a

[prisma cairn]

so you are adding the distances of a point z from 2 points a and -a

[toxic rose]

(u can substitute z=x+iy as well, I think)

[prisma cairn]

yes you can do that too

[bronze cradle]

To prove the reciprocal?

[prisma cairn]

yea do that, that would be easier

[bronze cradle]

Oke, then what's next?

[bronze cradle]

I could give an example of a number that fits the criteria and it should prove since it's an existence connector

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[lethal moon]

I thought for the limit divergence test

If it wasn't 0 it diverges

so why does it converge since it goes to 4.

[dreamy yacht]

You are mixing up series and sequences I guess

[lethal moon]

ohh i misread i thought it said series haha

for sequences is that the only test for it?

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[solid moth]

I proved that the vectors are linearly dependent but vector c doesn't belong to the line and I don't get why 😭

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@solid moth Has your question been resolved?

[solid moth]

.cloe

.close

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[median cedar]

Bekah has four brass house number digits: $2$, $3$, $5$ and $7$, and only has one of each number. How many distinct numbers can she form using one or more of the digits?

[clever fjord]

龖

[median cedar]

i have no idea what to do

wait i think i have. ideaa

so first single digits the possibilities are just : 2,3,5,7 : 4 numbers

double digits : 4x3 = 12 numbers

triple digits: 4x3x2 = 24 numbers

4 digits : 4x3x2 = 24 numbers

4+12+24+24=64 ?

can someone verify if my solution is correct

[heady marten]

That's right

[mystic saffron]

Correct

[median cedar]

ok thanks

.close

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[median cedar]

In writing the integers from $10$ through $99$ inclusive, how many times is the digit $6$ written?

[clever fjord]

龖

[median cedar]

i got 16,26,36,46,56,76,86,96 and 60,61,62,63,64,65,66,67,68,69

so i got 18 but its incorrect

[nimble blaze]

66 has two sixes

[median cedar]

oh when they say written do they mean like the number 66 would count twice

[static totem]

they do

[median cedar]

so its 19?

[static totem]

yes

if we start from 0 and write 00, 01, 02.. 99, we would get 200 digits total, and no digit would be special, so each would appear 20 times

so 19 makes sense mathematically, since we skip the 06

[median cedar]

yeah it is thanks

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[median cedar]

What is the sum of all the positive two-digit integers where one of the digits is three times the other?

[median cedar]

i got 10-19 = 13

20-29 = 26

30-39= 39,31

40-49 = 0

50-59 = 0

60-69 = 62

70-79 = 0

89-89 = 0

90-99 = 93

so i got 264

is that correct

[static totem]

it looks right

[median cedar]

Let $\Delta ABC$ be an equilateral triangle. How many squares in the same plane as $\Delta ABC$ share two vertices with the triangle?

[clever fjord]

龖

[median cedar]

no idea how to do this

isnt it just 6

like each two vertices of the triangle can connect 2 squares

so its 6?

but thats wrong

[ember oak]

consider a diagonal of a square

[median cedar]

ohh

so its 9?

@ember oak

[ember oak]

I haven't counted myself, but you seem to get the idea

9 sounds right

[median cedar]

yeah its correct

thanks

[ember oak]

👍

neat problem

I thought 6 too until you said the answer was wrong

[median cedar]

yeah aops problems are all like htis

.close

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[median cedar]

How many numbers can you get by multiplying two or more distinct members of the set ${1,2,3,5,11}$ together?

[clever fjord]

龖

[median cedar]

multiplying 2 numbers:10 possible values

multiplying 3 numbers:4 possible values

multiplying 4 numbers:1 possible value

but for example if i get 30 as a value in 4 numbers and also as a value in 3 numbers does it count twice?

did i do it correct

like if you multiply 5 digits together you get 330, but multiplying 4 digits together also gives you a value of 330 so does that mean theres no possible values of 5 numbers?

[supple mural]

Basically

Everything is coprime except 1, which doesn’t change the value

So I think you can just take 1 out of the set and every subset of the new set will give a distinct number

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[latent plaza]

someone help me please

[odd edge]

What step are you on?
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3. I got an answer but I was told that it's wrong.
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5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

@latent plaza Has your question been resolved?

[toxic monolith]

To find the length of the sides in a dilated image, multiply the length of the pre-image sides by the scale factor

[latent plaza]

wouldn’t i be multiplying them values by 2 though

if the dilation is halfing them

because we dk the pre image only the image

[mystic saffron]

what is bro yapping about

[toxic monolith]

|A'B'| = 20

[latent plaza]

how😭😭

[toxic monolith]

FORMUALA

[latent plaza]

yes but

what are multiplying…..

[toxic monolith]

adn base on it, you have to find AB

[latent plaza]

aren’t we reversing the dilation though

[toxic monolith]

i said, A'B' = 20

not AB

right ?

[latent plaza]

oh

yes

yes yes

[toxic monolith]

🙂

[latent plaza]

so it’s 40

?

[mystic saffron]

No it's 5

[latent plaza]

if A’B’ is 20 would that mean AB is 40

[toxic monolith]

right

becasue

40* (1/2) = 20

[latent plaza]

okay thank you

[mystic saffron]

You're just giving them the answer

Bro learnt nothing!

[latent plaza]

no i did

hello

[toxic monolith]

i wrote a hint above

[latent plaza]

distance formula of A’B’ is 20

multiplied by 2 is 40

.close

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[latent plaza]

help

[mystic saffron]

Find the starting one

Find the last one

find the ratio to get from the first distance to the second distance

thats ur scale factor

[latent plaza]

how do i do that

[mystic saffron]

it says it

[latent plaza]

6?

[mystic saffron]

14 is P

p' is after

after is 6

so what do you multiply 14 by to get to 6

[odd edge]

@latent plaza Has your question been resolved?

[latent plaza]

wdym

[odd edge]

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[kindred torrent]

z(x,y) is the number of ways to get from (0,0) the origin to (x,y), using only North, east, and north east movements. each movement is an increment of 1 in either direction, north east is +1 in both m and k directions obviously. how would i create a recursion relation that recurrs on m and n to count z(x,y)?

[odd edge]

What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above

[kindred torrent]

.close

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[twilit spoke]

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

[clever fjord]

Merineth

[twilit spoke]

I’m trying to integrate this

I want to utilize the arcsin formula for it

So I rewrote it as

$\int \frac{1}{\sqrt{1 -(x-1})^2}$

[clever fjord]

Merineth

[twilit spoke]

However now I’m stuck

1 can be written as sqrt(1) ^2

To make the formula usable but

Then what? :(

[fresh ocean]

You already solve

change variable t = x-1

[twilit spoke]

You mean x-1?

Oh

To slow haha

But if I substitute it with t

Do I have to divide with t after?

Is that the rule

[fresh ocean]

No

You have to figure out what is dt

[twilit spoke]

I’m not too familiar with how that part works

Why dt?

[fresh ocean]

Because every integral have dx after integral and you want to have only t when you substitute x-1 you have to do it with dx too

[twilit spoke]

Okay, but what does that mean to replace dx with dt?

[fresh ocean]

dx=d(t+1)=dt

[twilit spoke]

Haha that doesn’t tell me much

What steps do I need to do to solve it?

This is the formula im going to utilize

x is my t and a is sqrt(1)

[tardy lagoon]

have you learnt u substitution?

[twilit spoke]

No, I haven’t :(

All I heard about it, is that it is reverse chain rule

So I assumed it was going to be division but I have no idea tbh

[tardy lagoon]

ok so

we let u = x-1

and we want to find du

[twilit spoke]

Ok!

[tardy lagoon]

to turn the integral with respect to u

first find du/dx

then multiply dx to the right side

[twilit spoke]

So

$sin^{-1}(\frac{u}{\sqrt{1}}$

[tardy lagoon]

???

why sin

[twilit spoke]

Because of this?

[tardy lagoon]

oh you skipped to the solution

ok

[twilit spoke]

Well I just used the formula

[tardy lagoon]

wait no

I'm not sure if you understand the steps of u sub

[clever fjord]

Merineth

[twilit spoke]

Yeah I do not understand the steps of sub u

[tardy lagoon]

ok lemme show you the steps

u = x-1

du/dx=1

du=dx

substitute u=x-1 and du=dx to get $\int \frac{1}{\sqrt{1-(x-1)^2}}dx=\int\frac{1}{\sqrt{1-u^2}}du$

[clever fjord]

WhereWolf(ping if needed)

[tardy lagoon]

important part here is that du might not be dx

maybe du=x^2dx or something

for other integrals

say we have $\int 3x^2\sin{(x^3)}dx$

[clever fjord]

WhereWolf(ping if needed)

[tardy lagoon]

we let u = x^3, then du/dx=3x^2 so du=3x^2dx

our integral becomes $\int sin(u)du$

[clever fjord]

WhereWolf(ping if needed)

[tardy lagoon]

btw you also need to turn u back to x

[twilit spoke]

Makes sense

But

Shouldn’t it be like that?

Why do you remove 3x^2?

$\int 3x^2sin(u)du$

[clever fjord]

Merineth

[tardy lagoon]

no

[twilit spoke]

Why

[tardy lagoon]

you don't just turn dx to du

[tardy lagoon]

using this we turn 3x^2dx to du

[twilit spoke]

$\int 3usin(u)dx$

So we have this

[clever fjord]

Merineth

[twilit spoke]

And now we want to make it du?

[tardy lagoon]

no

wth

[twilit spoke]

Like can we do it stepwise

[tardy lagoon]

ok

lemme rewrite a bit to make it more clear

we hava $\int sin(x^3) 3x^2dx$

[clever fjord]

WhereWolf(ping if needed)

[tardy lagoon]

substitute x^3=u and 3x^2dx=du

to get sin(u)du

[twilit spoke]

Alright

$-cos(u)du$

[tardy lagoon]

it's easy to integrate and remember to turn u back to x

btw when doing definite integral the bounds would change

[twilit spoke]

So we get

$ -cos(u)du$

And now we hav etc turn back

du to dx?

[tardy lagoon]

bro you integrated already

why is the integral sign still there

[twilit spoke]

Habit sorry

[clever fjord]

Merineth

[tardy lagoon]

we get just -cos(u)

[twilit spoke]

Yes

And now multiply with du which is 3x^2?

[tardy lagoon]

I'm really confused right now

$\int \sin{(u)}du$ = $-\cos{(u)}$

[clever fjord]

WhereWolf(ping if needed)

[twilit spoke]

Yes

And what happens with du

Are we finished?

Or rather

u

[tardy lagoon]

u is just another variable

how'd you calculate $\int \sin{(x)}dx$?

[clever fjord]

WhereWolf(ping if needed)

[tardy lagoon]

just -cos(x) right

[twilit spoke]

Yes

[tardy lagoon]

so just -cos(u)

[twilit spoke]

Yes

[tardy lagoon]

dx and du disappear after the integration

last step is substitute u=x^3 back

[twilit spoke]

-cos(x^3)

So 3x^2 just disappears to nowhere? Lol

I tried using that method on my integral

But I’m getting the wrong answer

According to wolf

[tardy lagoon]

please show your work

[twilit spoke]

$\int \frac{1}{\sqrt{2x-x^2}} dx = \int \frac{1}{\sqrt{1-u^2}}du$

[clever fjord]

Merineth

[twilit spoke]

This becomes:

$sin^{-1}(\frac{u}{\sqrt{1}}) = sin^{-1}(\frac{x-1}{\sqrt{1}})$

[tardy lagoon]

wait a sec

[clever fjord]

Merineth

[tardy lagoon]

ok I checked

they are both correct

[twilit spoke]

So what I wrote is right?

[tardy lagoon]

they differ by a constant

[tardy lagoon]

yes

[twilit spoke]

$-2sin^{-1}(\sqrt{1-\frac{x}{2}})$

This is what wolf gives me

[clever fjord]

Merineth

[twilit spoke]

Is that equivalent to this ^?

[tardy lagoon]

yeah

by a constant

[twilit spoke]

What constant?

[odd edge]

@twilit spoke Has your question been resolved?

[twilit spoke]

Well I got two different answers when I put In my integral

<@&286206848099549185>

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[twilit spoke]

$\int (x-5)e^{\frac{x}{2}}$

[clever fjord]

Merineth

[fervent hound]

did u use integration by parts ?

[cold urchin]

damn

[cold urchin]

by parts you should differentiate x-5 instead of integrating it

[fervent hound]

distribute and split the integral then integrate by parts

[merry kestrel]

I’d suggest you, $\int (x-5)e^{\frac{x}{2}}dx $= $\int x e^{\frac{x}{2}}dx$ — $\int 5 e^{\frac{x}{2}}dx$

[clever fjord]

.doc

[merry kestrel]

he took the first function as e^{x/2}

[cold urchin]

Yes and he should take u = x-5 du = 1 and dv = e^(x/2) v = 2e^(x/2)

everyone should learn the tabular method

not sure what you mean by firt function tho

[odd edge]

@twilit spoke Has your question been resolved?

[odd edge]

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[hollow pine]

need help, making sure im right

is it 3417 and 342?

any1?

.close

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[ebon olive]

How would this be written if it were on paper

[noble forge]

[R1/R2]

floor function is what you would use i guess

[ebon olive]

.close

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[brisk garden]

hey

[brisk garden]

How to show that 2nCn = [nCO]^2 + [nC1]^2 +…+ [nCn]^2?

been thinking but have no idea

[coarse shuttle]

Show what

[brisk garden]

this

[coarse shuttle]

Yea

First of all

is the C refering to combinations ?$

And if not then what is it

[brisk garden]

Mhm

[modest lake]

vandermonde

[coarse shuttle]

you should know what C is to answer first yk

[odd edge]

@brisk garden Has your question been resolved?

[quasi sparrow]

Did you try induction

[odd edge]

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[young bluff]

Hey guys

I need help with this, please

I think I've made a mistake in my calculations so now I'm stuck.

It tells me to find angle B and I used the laws of sine ** (B = sin^-1 (b*sin A)/a**

But it's given that angle B is greater than angle C, so I don't think it could be 28° that I got it to.

[orchid torrent]

Recall that $$\sin(180^{\circ}-x)=\sin(x)$$

[clever fjord]

Civil Service Pigeon

[orchid torrent]

There is more than one possible solution for angle B

[young bluff]

I also did it like this, which tells that angle B is 102° but I need to answer it in 2 decimal-form

(I don't know if it's correct)

[orchid torrent]

The triangle isn’t necessarily isosceles

[hollow hearth]

law of sines should work? what did you input into the calculator?

[orchid torrent]

The point of discussion is why we reject the acute case from the law of sines

[young bluff]

B = sin^-1 (12*sin 39)/16

and got 28

[orchid torrent]

.

[young bluff]

how do I calculate it?

[orchid torrent]

Let’s say x was your first solution for B

180-x is also a solution for B

That’s all the equation is saying (in this context)

Actually I should rlly say identity instead of equation

[young bluff]

so since I got 28 first

could it then be: sin(180-28) = sin(28) ?

last question.

I tried calculating: is it 43,69° ?

[orchid torrent]

,w 180-(180/pi)arcsin(16sin(39 deg)/12)

[clever fjord]

Results provided by WolframAlpha

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

Requested by civil_service_pigeon

[young bluff]

nvm

thanks for your help, I appreciate it

.close

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[mystic saffron]

what is the matrxi for T1 relative to the ordered basis B = [1, x, x^2,x^3] for P_3 and the base B* [1] for R

[mystic saffron]

Where T_1 is linear transformation

I am not sure how to begin on this problem

[mystic saffron]

matrix?

[echo ginkgo]

what's the transformation T1?

you haven't told us

[mystic saffron]

What you can do first is apply the transformation T1 to each vector in the basis B

and then you can maybe express the results as linear combinations of the basis B*.

[mystic saffron]

linear transformation T1 applying to each vector in B

So I guess that would be

T1(1) = 1
T1(x) = 1 + x
T1(x^2) = 1 + x + x^2
T1(x^3) = 1 + x + x^2 + x^3

Correct me if I'm wrong

[echo ginkgo]

nah

they didn't specify the transformation

"find the matrix of T1 relative to bases .... and ....."

if you don't know what T1 is you can't answer the question

[mystic saffron]

i think i did it now

What you got?

but yes t1(p) = p(1)

i got the matrix to be [1,1,1,1]

idk if that is correc ttho

Mmmm okay. Mine is
[1, 1, 1, 1]
[0, 1, 1, 1]
[0, 0, 1, 1]
[0, 0, 0, 1]

hahaha anyways

[echo ginkgo]

yea seems good

[mystic saffron]

Let T-1 -> P_3 -> R be given by T1(p) = p(1)

I just insertet x=1 on in the basis

my main problem is linear algebra is that sometimes im not certain on what i am being asked

[echo ginkgo]

well the main thing in intro lin alg is understanding the question

then it's just solving a linear system or pattern matching a lot of the time

[mystic saffron]

Ah okay got it. So T1 takes a polynomial p(x) of degree 3 or less and evaluates it at x = 1. So yes you're right, T1(p) = p(1) 🙂

A question

It goes from P_3 to a real number

what do they mean by that

[echo ginkgo]

I ?

the inputs of T1 are polynomials

[mystic saffron]

it takes a four dimensional input

[echo ginkgo]

and it outputs a real number

[mystic saffron]

and returns one dimensional number

where is that number

[echo ginkgo]

p(1)

p(1) is a real number isn't it

[mystic saffron]

well p(1) was [1,1,1,1] or

when i insert it

[echo ginkgo]

no?

imagine I take the polynomial p=x^3 + 2x^2 + 4x +1

what's p(1)?

it's certainly not a vector

[mystic saffron]

x^3 + 2x^2 + 4x +1 = 1+2+4+1=8

[echo ginkgo]

yeah

a number

[mystic saffron]

yh ok

but i've written it this way

[echo ginkgo]

ok well that's the matrix of the linear transformation

different thing

[mystic saffron]

alright

The next is p(a)

so i replace the x's with a

[echo ginkgo]

yea

[odd edge]

@mystic saffron Has your question been resolved?

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[signal oar]

[z^4 = -1 + i \sqrt 3 = |z^4|(\cos(4 \theta) + \sin(4 \theta)i)]

[signal oar]

Could one notice the values of theta here?

Or would you use a different approach

To find z

[lavish falcon]

are you trying to find the fourth roots of z^4?

[signal oar]

Yes

Oh, wait

.close

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[light hawk]

so to get the vector of the normal line to the plane, I know it's lambda<2x - 1, 2y, -1 > = <-6, 4, 2 >. After doing some calculations, I know that the normal line vector is <3, -2, -1 >

but I also need a point to get the equation of a plane. I don't know how to obtain that point

[light hawk]

i can also expand on these calculations if need be

[sullen ferry]

apply the equation of the tangent plant to a surface :)

that should help you restrict and solve for your x_0 and y_0

[light hawk]

but the equation would be 3(x-x0) -2(y-y0) -1(z-z0) = 0, right?

[light hawk]

is there an equation that I am missing?

because the one i have has 3 unknowns

[sullen ferry]

z0 can be written in terms of x0 and y0...

and also you have tangent plant equation

https://tutorial.math.lamar.edu/classes/calciii/TangentPlanes.aspx

[light hawk]

here's the solution my professor did:

i understand how <-6, 4, 2 > can be used as teh vector for the normal, but why is the point (2, -1, 3)?

[sullen ferry]

well, they used the fact that the the gradient of the surface is also determined by x,y,z coordinates

[odd edge]

@light hawk Has your question been resolved?

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[devout plinth]

how do i find the discount rate with the value for original

so i just make one up??

[signal carbon]

what's the actual problem?

[devout plinth]

9. How much does a discount rate have to be so that the total (after 15% tax) comes to half the original price?

<@&286206848099549185>

[odd edge]

@devout plinth Has your question been resolved?

[devout plinth]

<@&286206848099549185>

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[mystic saffron]

Hello, I need some help trying to understand integrals, I am new to this but I know an integral of something is the area behind the curve of the function, so I tried to find the integral for the area of a right triangle, I got this function f(x) = a - ax/b, and now I am stuck integrating it integral from x=0 to x=b (a-ax/b)dx, like, what is dx? how I solve this? I have to much questions and not to many answers, so I wanted to see if someone knew a good video series on integrals or something

[signal carbon]

have you tried something like https://www.khanacademy.org/math/calculus-1 ?

[mystic saffron]

I am a kid, I have no money to pay a course :(

[zenith tartan]

bruh but isn't khan academy free?

[cold sage]

its free

[chrome lynx]

watch essence of calculus by 3b1b on youtube

[zenith tartan]

i might be wrong

lmao

[mystic saffron]

I ain't native to english so, when I saw something to paid I was like "ow", but it was for donations ( I don't know why it is the first thing that appears when you enter the page but, well, it is what it is ),

[zenith tartan]

haha lmao

[toxic monolith]

I've just goggled now: integrals for dummies , and there are tones of very elementar sources

[mystic saffron]

wait, dx is the derivative of x? or the derivative of f(x)? ( haven't checked anything yet, I will check what you guys recomended but I want to see if I get an answer )

[toxic monolith]

without mathematical formalism, for you, dx means with respect to what you integrate, = which is the integration variable, and other letters are constants like numbers

in your triangle case, x is a variable, but a is a constant

and if you wrote, assume, da, then a wud be a variable, but x wud be a constant

later you find out more

[odd edge]

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[tepid maple]

help, where do I get P(-B|A)? It's probably the last thing I need to complete the solution. or if I'm totally way off with my solution, please let me know

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@tepid maple Has your question been resolved?

[open onyx]

note the relationship between P(-B|A) and P(B|A)

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[chilly marlin]

help please

[chilly marlin]

I need help with all 5, I am so confused. The professor never taught this :c

[fervent hound]

suppose a = 0 then a=1

[south plume]

what lol

[fervent hound]

it's boolean algebra ?

if a = 0 : 0*0 = 0
if a = 1 : 1*1 = 1

[south plume]

for the first one recall the tables of AND (*) and OR (+)

if a=0, then a*a = 0
if a=1, then a*a = 1

[chilly marlin]

oh okay

this is so stupid, i was never taught any of this

I havent missed a single day either

smh

[chilly marlin]

if a=0 a*a = 0, what about OR? if a=1 a+a=2? right?

[south plume]

hm

that's not part of your task

[chilly marlin]

I could just use truth table?

[south plume]

but if a=1, a+a=1

[south plume]

ye

'2' isn't a thing

[chilly marlin]

oh ok

how would i write the truth table for this?

[south plume]

for OR?

[chilly marlin]

for AND and OR

I know how to write a truth table with PQR

is it the same thing?

[south plume]

this is AND and OR respectively

[chilly marlin]

ahh i see

and = multiply

or = +

[south plume]

yes

[chilly marlin]

but for my question

since there is only a

and no b

for the first one

would i just do

top 2 of each row?

like

let me write this

a = 0 1
a*a = 0 1
a + a = 0 1

?

[south plume]

huh?

[chilly marlin]

i confused

[south plume]

let's go slow

first question

[chilly marlin]

okay :)

[south plume]

you have ONLY multiplication

[chilly marlin]

so a*a

[south plume]

yes

[south plume]

it doesn't talk about + at all

consider a=0
a*a = 0*0 = 0 = a
consider a=1
a*a = 1*1 = 1 = a

that's all

[chilly marlin]

uhm i would just write a= 0 a*a = 0, a = 1, at**a = 1?

[south plume]

yea

[chilly marlin]

oh wow

thats easy

[south plume]

maybe you want to reference the laws

[chilly marlin]

yeah

[south plume]

oh

and another thing

[chilly marlin]

yes?

[south plume]

don't forget to write =a

[south plume]

like here

[chilly marlin]

at the end?

[south plume]

yes

[chilly marlin]

ok :)

[south plume]

that's it for question 1

[chilly marlin]

oki :D

[south plume]

question 2

this is properties of + (OR)

[south plume]

look at the bottom 2 rows

both are 1

[chilly marlin]

yes

[south plume]

well

[chilly marlin]

so this would be

a = 1, a + a = 1 = a?

[south plume]

no

consider a=0

look at the question

[chilly marlin]

OHH

okok

a = 0, 1 = 0, a + 1= 0 + 0 = 0 = 1

[south plume]

1=0

[chilly marlin]

UHHH

[south plume]

you are overcomplicating it

consider a=0

a+1 = 0 + 1

what's 0 + 1

[chilly marlin]

1

[south plume]

yep

[chilly marlin]

oh okay

[south plume]

a+1 = 0 + 1 = 1

now the same thing with a=1

and that's it

[chilly marlin]

a = 1, a + 1 = 1 + 1 = 2?

wait but there is no 2

[south plume]

not 2!!!!

it's 1

[chilly marlin]

1 + 1 = 1?

[south plume]

yes

it might be easier understand if you just don't think of + as plus, but rather as OR

1 OR 1 is 1

0 OR 1 is 1

in any case

this question is done

[chilly marlin]

oki

[south plume]

copied so i don't have to scroll

[chilly marlin]

wait let me try the next one

[south plume]

3 is interesting

[south plume]

yes

hint: truth table

[chilly marlin]

a = 0, b = 0, a + b = 0 + 0, (a + b) * a = 0*0 = 0 = a

:D

[south plume]

yes

[chilly marlin]

a = 1, b = 1, a + b = 1 + 1, (a +b)* a = 1*1 = 1 = a

:D

[south plume]

yes

but!

[chilly marlin]

YAY

oh

[south plume]

what about a=1, b=0

[chilly marlin]

ohh

because its 2 values

[south plume]

and a=0, b=1

[chilly marlin]

so we have do 4

[south plume]

yes

[south plume]

yes

but

it would be much more succinct in a truth table

and!

[chilly marlin]

yes?

[south plume]

do you have any notes from your teacher?

on laws of boolean algebra

[chilly marlin]

he told us to just read in the book and do it

hence why im here

[south plume]

okay, well, can you reference the book in the homework?

[chilly marlin]

the hw is not from the book, he makes the questions himself

[south plume]

i know

but like

[chilly marlin]

oh the chapters in the book

[south plume]

ye

[chilly marlin]

that the homework is for

oh yes

[south plume]

i'm just asking if i can send you some useful properties

[chilly marlin]

yes u can :)

please do

[south plume]

and whether you can use them

[south plume]

without proof

[chilly marlin]

i know absolutely nothing

and have finals coming up

[chilly marlin]

I shall save this

thank you

what is nb1?

[south plume]

$\vee$ is $+$\
$\wedge$ is $\cdot$

[clever fjord]

artemetra

[chilly marlin]

got it

[south plume]

not relevant to you, you can ignore

[chilly marlin]

oki

reminds me of de morgan

[south plume]

so Absorption 1 is the law you are supposed to use

[chilly marlin]

so i could just write

[chilly marlin]

according to Absorption 1, this is true

[south plume]

see idk

technically yes but these names aren't fixed

if you could reference what the book uses it should be fine

but you can prove it on your own too

[chilly marlin]

Its ok because in one of the exams we had a question, and i guess it was de morgan theorem. I just wrote " This is true because it follows de morgan theorem" and he gave me a 100 for that answer

[south plume]

it's not difficult

[chilly marlin]

oki :D

[south plume]

let's make a truth table

[chilly marlin]

i will go through the book too

just to make sure

[chilly marlin]

okay

[south plume]

for a and b

[south plume]

,tex
\begin{center}
\begin{tabular}{ c c c c }
a & b & (a+b) & (a+b)\cdot a\
\hline
0 & 0 & ? & ? \
0 & 1 & ? & ? \
1 & 0 & ? & ? \
1 & 1 & ? & ? \
\end{tabular}
\end{center}

[south plume]

precisely

[south plume]

notice that the a row is the same as the (a+b)*a row

and that's it

[chilly marlin]

yes :D

nice

so if it was a b and c

would it be 8 instead of 4?

[south plume]

yes

for n variables, the number of rows is 2^n

[chilly marlin]

oo okay

[south plume]

it's easy to see why

[chilly marlin]

oh wait so 6?

[south plume]

$2^n$

[clever fjord]

artemetra

[south plume]

no

[chilly marlin]

OHH

u meant 2 to power

okok