help-13

[gaunt meadow]

Thank u🙌

.close

[cedar kiln]

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[crimson sedge]

Yumm

[cedar kiln]

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[cinder jay]

A function for a bungejumper. h is the height and x is the time in seconds from when it starts to extend.

[cinder jay]

How do find the point they are the closest to the water and how long it took

[nova frigate]

One would translate it into a minimization problem

Study the variations of the function to ensure it admits a minimum, this minimum is to be found when the derivative is equal to 0

[cinder jay]

how would i start?

[nova frigate]

Derivate h

[cinder jay]

then put it equal to zero?

[nova frigate]

yes

it would give you an extremum, to be sure it is a minimum you have to study the variations of h (show that it is convex for example)

[cinder jay]

I think it gave me the wrong output. it said x was either 3 or infinity . it was supposed to be 0

[nova frigate]

If you know it is x= 0, you can argue that every other x>0 satisfy f(x) ≥ f(0)

[cinder jay]

i wouldnt have known it since thats the solution

so like the whole derivate part was indeed not needed to solve it .

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[dire geode]

No

Just google your question

[fallen trail]

Use gpt or SMT

[mighty shuttle]

!nogpt

[cedar kiln]

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

[twilit escarp]

This channel make no sense

[coral jewel]

[fallen trail]

😔

[opaque root]

Not a math question.

.close

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[eternal pollen]

"Verify by direct substitution that the given function is a general solution"

[eternal pollen]

I don't understand how to do it

[hollow trail]

take the two derivatives of the given function and verify that if you plug into the left side of the equation you get 0

[eternal pollen]

I tried

[hollow trail]

could you explain how you took the derivatives?

[eternal pollen]

Idk what the rule is called

But I just found the derivative of the trig functions

[hollow trail]

but it's multiplied by e^x

[eternal pollen]

And let e^(x) alone

Do I need to use the chain rule?

[hollow trail]

so you have the product of two functions

[eternal pollen]

Yeah

[eternal pollen]

?

[hollow trail]

why would the chain rule apply?

[eternal pollen]

Isn't that used when you multiply two functions?

Product

[hollow trail]

no, that's for the composition of two functions

the product of functions uses the product rule

[eternal pollen]

Product rule is what I meant

U' • V + U • V'

[hollow trail]

yes

[eternal pollen]

Okay I'll try

[eternal pollen]

Is this how you do it?

[hollow trail]

yes

[eternal pollen]

Okay

Is there a way to make it shorter?

[hollow trail]

you can simplify the second derivative

[eternal pollen]

That's what I thought

I'm just struggling to see how

(I'ma go eat brb)

[cedar kiln]

@eternal pollen Has your question been resolved?

[eternal pollen]

I'ma open a new chat when I am back

[eternal pollen]

Thank you for your help ^^

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[naive bluff]

this is the problem

[naive bluff]

i know u have to use chainrule

and like do it step by step

but i keep getting the wrong answer

this is the solution

i undesrtand the step one

<@&286206848099549185>

[cedar kiln]

@naive bluff Has your question been resolved?

[naive bluff]

<@&286206848099549185>

[odd verge]

,w diff cot(ln(ln(√(arcsin(3x²+1)))))

Gimme a minute

[sage epoch]

what "step 1"?

but it's really all work outside to inside pretty much

[odd verge]

It seems correct tho

[naive bluff]

yeah

the thing is

this part

to this

like the root is still staying as a root

shouldnt it have the minus 3/2 thungy

likle even if we were to assume the next step must have it

[sage epoch]

no but like

[odd verge]

Try substitutions first

[naive bluff]

its not there

[sage epoch]

you're differentiating the logarithm in tha

the last term is what actually differentiates the root

[naive bluff]

i dont understand

[odd verge]

Show your steps

[naive bluff]

okay imma send it

[odd verge]

From where you can start to wherever you stop

[sage epoch]

i think this is really a case of "it's harder to read than do"

[odd verge]

Yh

[sage epoch]

heic lol

[odd verge]

Oh no

You aren't supposed to differentiate the previous term

[naive bluff]

huh

[odd verge]

You are supposed to differentiate the input of the functions after making necessary substitutions

[wraith dagger]

Aero

[naive bluff]

yeah

no i did all that

and then i came to that step

[sage epoch]

right

so lets walk through it shall we?

[naive bluff]

yes please

[sage epoch]

how do you differentiate du/dx here

by applying the chain rule again

same procedure, pick a u

u = inside function

[naive bluff]

yeah

[sage epoch]

so what should be the u

[naive bluff]

the first u has to be the ln whole thing

[sage epoch]

or maybe lets call it v to avoid confusion

meaning?

[naive bluff]

no i cant vc

[sage epoch]

huh? who said anything about vcing

[naive bluff]

oh

sorru

i read it as vc

[sage epoch]

Sooo

what's happening

[naive bluff]

im not understanding it

its fine forget it

[sage epoch]

I mean i think its just the amount of nesting is confusing you, but it seems like you conceptually understand what's happening

[naive bluff]

yeah

[sage epoch]

do you understand that the chain rule is dy/dx = dy/du * du/dx?

[naive bluff]

yeah

[sage epoch]

ok how about this

here is a good idea

let's work inside-out

[naive bluff]

okay

[wraith dagger]

Aero

[sage epoch]

how would you do this

try to use the chain rule by picking a suitable u

what should u be in this case

[naive bluff]

afk

[sage epoch]

okay

[cedar kiln]

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[young haven]

i was solving this question and while finding adj A i got the same thing as pic 3 except the 12 and othe other 12,180,336 are negative

[young haven]

question

most of the solution

i dindnt get how they have 12 12 180 and 336 +ve when i got mine -ve

[cedar kiln]

@young haven Has your question been resolved?

[young haven]

<@&286206848099549185>

[dim plaza]

Co factor

[young haven]

isnt adj calculated with minors?

[dim plaza]

Yes it is

[young haven]

correct me if im wrong but cofactor is the +- minor thingy

[dim plaza]

Yes it is

[young haven]

but the minors are all negative

[dim plaza]

Exactly

[young haven]

we put minor to calculate the adj right?

or cofactor?

[dim plaza]

Co factor

[young haven]

oh

ic

thank you

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[twilit escarp]

You probably given a percent of tax

And a salary

[cedar kiln]

@night wigeon Has your question been resolved?

[nimble veldt]

i guess you need a tax consultant.

[pallid trout]

Or he can do by visit government official for regrading taxes...

[dire geode]

<@&268886789983436800>

[twilit bison]

The rules are complicated

[solar vector]

Don't be hostile like this

[twilit bison]

That's why people make a living doing that

Also withholding is different from how much you actually owe

[solar vector]

People don't owe you an answer.

There's no point in getting upset with people over this

On top of that, mathematics and tax law are very different. So the assumption that people aren't helping you just to be jerks as opposed to them not wanting to give you bad info due to them just not knowing is silly.

[vague sparrow]

I am trained in nordic tax law and I withheld this information

[nimble veldt]

questions about a national tax systesm for a country (which is not under the hundred greatest countries) arent typical qeustions for a math server. expecting to get answers for that is like hoping for a lottery win.

[solar vector]

They are joking

Even if people can help you with this, there's a certain amount of risk on your end accepting help from unqualified strangers and in said strangers giving you help.

[nimble veldt]

well, if you had a link how to do that, what do you need then? 🤦

[solar vector]

That whole scenario makes it unlikely for people to want to help at least on the basis of not wanting to fuck up your money situation.

Uhh referring to my last comment I mean

[nimble veldt]

good luck for your next questions.

[twilit escarp]

most of the time govt are giving website to simulate tax also

[solar vector]

This isn't really going anywhere and you said you solved your own problem so I'm just gonna close the channel for now. Don't be hostile to helpers going forward though.

.close

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[vagrant flower]

Wondering what I have done wrong to be getting different answers between using tan, cos and a^2+b^2=c^2

[cedar kiln]

@vagrant flower Has your question been resolved?

[loud lintel]

Probably because with side b=6 and c=10 doesn't exist like this

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[waxen anvil]

this is already solved but i don't know how to solve it

[dusk goblet]

consider reading the screenshot you sent

[cedar kiln]

@waxen anvil Has your question been resolved?

[waxen anvil]

?

[waxen anvil]

<@&286206848099549185>

[thick bay]

Give me time

[waxen anvil]

ok waiting

[cedar kiln]

@waxen anvil Has your question been resolved?

[waxen anvil]

<@&286206848099549185>

[pastel vault]

What don't you understand

In the screenshot you sent

@waxen anvil

[waxen anvil]

2nd page b. and c.

[pastel vault]

Do you understand how the directional derivative is $\nabla f \cdot u'$?

[wraith dagger]

south

[pastel vault]

They've just used a different notation with the transpose

It's the same thing

[waxen anvil]

where did we get 36 and 25

[pastel vault]

u is defined as (-6, 5)

In the question

So that is the magnitude of u

[waxen anvil]

ok, and what we do the point [-2,-4]?

[waxen anvil]

cuz i don't see it being used

[pastel vault]

That is used at the very end

[pastel vault]

We need to arrive at the directional derivative for any x1, x2 first using this

Once you get the -18x1 - 96x2 - 40 thing

It says to use the 2nd directional derivative so you need to take the partial derivatives once again

And then that is the 2nd directional derivative for all x1 and x2

It's a constant

Doesn't depend on the point (-2, 4) at all

It's like if you want to find the 2nd derivative of x^2

It's always 2 regardless of the value of x

[waxen anvil]

i see thank you, how do i calculate it using the calculator

i have fx-82es plus

[pastel vault]

I don't know about that sorry

[waxen anvil]

ok question here, where did the 25 go?

[pastel vault]

Ah when you take the partial derivatives w.r.t to x1 and x2

It disappears!

Derivative of a constant is 0

[waxen anvil]

thank you!

[pastel vault]

Np!

.close

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[cosmic umbra]

what the difference between 130 and 131 can someone guide me 130 so that i can do 131

[cedar kiln]

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[cosmic umbra]

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[vestal bluff]

what have you tried

[elder sky]

Tired that but got stuck

[vestal bluff]

you differentiated v incorrectly

[elder sky]

Lemme try again and see

Am getting the same thing ln10/(xln10 lnx) so ln 10 cancel out and I get 1/xlnx

<@&286206848099549185>

[tropic jay]

seems a bit harsh to look at

you can do it with log base 10

you won't need natural log

It seems like it's messing up due to that, not because it's not possible but maybe some silly error

Log y = log x × log(log x)

Differentiate both sides

@elder sky

[elder sky]

Ok

wait how am I supposed to differentiate log directly

[tropic jay]

My bad

y = (log x)^(log x)

taking log both sides

so log y = log x × log(log x)

[elder sky]

my concern is I only learnt how to differentiate ln so thats why I converted it to ln first

[tropic jay]

ln and log

are same

not in the general sense but in calculas we do interchange them

d(ln y)/dx = 1/y(dy/dx)

d(log y)/dx = 1/y(dy/dx)

[elder sky]

Ohhh

[tropic jay]

They work the same way but you can't transform one into other

In this question from the start we're taking log base 10

we can't directly say log = ln and put it

but we can take the one we like to and do operations

as both work the same way

[elder sky]

so how is the first step going to look like

[tropic jay]

so log y = log x × log(log x)

LHS gonna be

1/y (dy/dx)

In RHS apply u×v rule

log x = u
log(log x) = v

v(du/dx) + u(dv/dx)

Till this step is it clear for you?

Or do you want me to do it

[elder sky]

Yes I'm actually writing on paper which makes more sense

[tropic jay]

then gimme a min

I'll write this step and show you

you'll understand in no time

[elder sky]

sure

[tropic jay]

@elder sky

[tropic jay]

applied this rule

[elder sky]

Does that mean u differentiate ln and log in the same way?

[tropic jay]

Yes

Both are logs just with different bases so the rules apply the same

[elder sky]

ok

[tropic jay]

Understood this?

[elder sky]

Yes

[tropic jay]

Any doubts you might think worth asking?

[elder sky]

I was confused how u differentiated log(logx) but I've seen it now

[tropic jay]

Yes with applying chain rule

[elder sky]

yes I've understood till that step now

[tropic jay]

Now simplify rhs

[elder sky]

ok

Logx(log(logx)+1) / xlogx

@tropic jay

[tropic jay]

Cancel out the log x

wait

[tropic jay]

See on the right side log x and
1/log x will cancel out

[elder sky]

yes

[tropic jay]

now send the y into rhs

[elder sky]

It is the same answer

[tropic jay]

and write the whole expression by putting y's value

[tropic jay]

Ohkay wait lemme see

[elder sky]

I've done that

[tropic jay]

there's this property you can search

(log x) = ln x/ ln 10

so I used that property on log(log x)

[worldly socket]

spicyl

[elder sky]

I've seen it

We are basically changing the base to e?

[tropic jay]

Yes

We arrived close to the answer not quite

I see a tiny conversion we missed

As we want to arrive in natural log

[tropic jay]

In the first step see d(log x)/dx

If it's normal then we write 1/x

As I did

But we need answer in terms of natural log

so

d(log x)/dx = 1/(x ln 10)

It's also a property

So instead of 1/x it'll be 1/(x ln 10)

[elder sky]

I thought we get 1/x when we differentiate logx now the natural log thing is a new concept which is a bit confusing

Coz it changes the derivative to sth else

[tropic jay]

It does

Conversion is overwhelming

Actually sorry but I've to go now I've my tuition you shall ping helpers they'll help you in a better way than I did

[elder sky]

Thanks man

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[elder sky]

.close

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[shy spoke]

I have these points that make a function and I want to find what the equation of the function is

[shy spoke]

this is what i have so far

I think the next step for me is to get this into reduced row ecehelon format

so that I can find what each variable represents

I think the first thing I'd do is to create an augmented matrix and then use a calculator to do rref

how do I create the augmented matrix and can someone explain the process

[plucky owl]

Augmented matrix is in the form of [A | b] where the A matrix is the matrix of coefficients and the b matrix is the solution matrix

If you are typing it into a calculator, you would ignore the |

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[sacred bolt]

.open

Happy new year + happieness =

[idle gulch]

Happy new year !!! 🎊

[void sand]

happy new year!

[azure horizon]

@​Moderators this is not on topic for a math help channel

[trim tendon]

Happy new years even if it's off topic

[sacred bolt]

.close

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[surreal geyser]

can someone help me here

[mighty shuttle]

I think you may have to use AM>GM>HM

[cedar kiln]

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[whole tree]

can someone please help with this approximation? I really can't seem to find the right way

[whole tree]

nvm i found it cuz im built different

.close

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[lost thunder]

whys last line true

[cedar kiln]

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[hollow quarry]

f:IR->IR, f'(x)=0 xeR, and f^2(x)+1=2f(2). Find f(x)

[hollow quarry]

its based on the constant function theorem

[weary ingot]

what does f^2(x) mean? f(f(x)) or f(x) * f(x)

[hollow quarry]

f squared

so f(x) times f(x)

[steel heart]

since this function is a constant

[weary ingot]

so in general since f'(x) = 0, f(x) is a constant

[steel heart]

f(2) = f(x)

[weary ingot]

Let f(x) = a.

and then just solve

[hollow quarry]

hold on

[weary ingot]

Is there something you dont understand?

I could try and re explain it

[hollow quarry]

im doing what you told me

i dont have f(x) though

[weary ingot]

f(x) is just a constant

[hollow quarry]

sure

[weary ingot]

so f(x) = a

[hollow quarry]

yes

[weary ingot]

and you just solve for a, and that will be the constant f(x) equals

[hollow quarry]

i have f^2(x)+1=2f(2)

f^2(x)=2f(2)-1

f(x)=+-sqr of that

[weary ingot]

since f(x) is constant, f(2) equals f(x)

[hollow quarry]

or we dont need that

alright

after that?

[weary ingot]

you can just solce for f(x)

f^2(x) + 1 = 2f(x)

[hollow quarry]

oh

then its a complete square

[weary ingot]

yep

[hollow quarry]

so f(x)=1

[weary ingot]

since its a quadratic you have 2 solutions

[hollow quarry]

its (f(x)-1)^2=0

[weary ingot]

Oh yea, then yes f(x) is 1

to make sure just plug 1 back into the original equation and check that f'(x) = 0

[hollow quarry]

it is

thanks

could you help with some more?

actually i dont need it

[cedar kiln]

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[shy spoke]

I have a function, that I plotted on geogebra, but when I solved for it and tried to sub it in, it is not giving me the same function. Anyone know why?

[shy spoke]

The blue function line is what i calculated

i can show you my work

[steel heart]

solution got lost in approximations ig

[shy spoke]

hm ok

what should I do?

since i am trying to calculate the surface area of the leaf

and that is for my upper function

[steel heart]

Use more precision

[shy spoke]

true, i approximated the points I used for A,B,C,D etc

so maybe if I didn't limit it it would be fine

can i just say in my paper that im using rounded vlaues for clarity and that the final result will not have approximations?

[cedar kiln]

@shy spoke Has your question been resolved?

[vivid tundra]

you just need more precision in doing the RREF process

try inputting your augmented matrix into wolframalpha

[cedar kiln]

@shy spoke Has your question been resolved?

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[gilded rapids]

For an mxn matrix A, is the degree of freedom of the matrix rank(A) = n - dim kernel A?

[cedar kiln]

@gilded rapids Has your question been resolved?

[flint plinth]

wdym by degree of freedom?

the rank is given by that formula yes

[cedar kiln]

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[twilit escarp]

Is it correct ? D = {(x,y): x > 0 y > 0 ; x+y < 1}

[dire geode]

I can't find any errors

[twilit escarp]

Perfect, thanks for the review !

.close

[cedar kiln]

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[mighty drift]

Yaku helpee arc

[hollow trail]

,w integrate e^(-x-y) over x from 0 to 1-y, y from 0 to 1

[wraith dagger]

Results provided by WolframAlpha

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

Requested by cloudedhorizons

[hollow trail]

👍

[twilit escarp]

On self learnt topic yeah i need some xd

[cedar kiln]

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[pallid parrot]

hi

[pallid parrot]

A light bulb placed below a liquid surface and at a depth of 12cm, if the radius of a tablet is placed on the surface of the liquid and is sufficient to block the light of the lamp is 16cm

Can't do the drawing

[cedar kiln]

@pallid parrot Has your question been resolved?

[pallid parrot]

<@&286206848099549185>

[low holly]

what is your question?

[pallid parrot]

find n

snells law

[low holly]

do you have a picture or smth of the entire question? do they say anything about the light bulb?

[pallid parrot]

No

[low holly]

well then your question is too vague i believe

[pallid parrot]

the answer is even more vague

wait

$$[
\tan \theta = \frac{16}{12}, \quad \quad
\theta = 53.13^\circ = \phi_c, \quad \quad
n = \frac{1}{\sin{\phi_c}}, \quad \quad
n = 1.2500016
]
$$

@low holly

[wraith dagger]

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

[cedar kiln]

@pallid parrot Has your question been resolved?

[cedar kiln]

@pallid parrot Has your question been resolved?

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[vagrant apex]

How should the isoquants of this function look like to highlight the local optima?

I have found the all the critical points and what the nature of each of them are (please do correct me if something is off):
(0,0) - saddle point
(0,5) - saddle point
(0.5,2.5) - local maximum
(1,0) - saddle point
(1,5) - saddle point

[cedar kiln]

@vagrant apex Has your question been resolved?

[cedar kiln]

@vagrant apex Has your question been resolved?

[vagrant apex]

.close

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[keen kindle]

i have these numbers and i need to decrease them all down to where they fit from 100-100000, and are all multiplied/divide/subtracted/added by something equally. like if i have 1, 2, and 8 and multiply 8 by 15 then all numbers should be multiplied by 15
numbers:
5,000
6,250
7,500
6,500
10,000
40,000
1,000
50,000
2,000,000
100,000,000
1,000,000,000
8,000
16,000
400,000
10,000,000,000
30,000,000,000

do not know how to do this one bit but it is something for a sort of personal project

[fresh quarry]

You mean a common operation is to be performed on all the number so that all the numbers lie in the range 100 to 100000?

[keen kindle]

yeah

...

i had a horrible explanation, sorry

its 2:30am i cant think

[blazing zephyr]

does it have to be 1 operation?

like hypotethically like subtract 1000 then divide by 2?

[fresh quarry]

If we try
Adding 100,000 to all numbers
Then dividing them by 1000

[blazing zephyr]

you can always divide by a large ammount and add 100

[keen kindle]

that would be fine
if its x123 /3 + 4 would be fine
doesnt just have to be x123
(not actually that, just an example)

[blazing zephyr]

say dividing by 1,000,000,000,000,000,000 amd adding 100 and that works for the list of numbers

it would lead to like 100.00000000001 or smth

[keen kindle]

i forgot to specify

these are percentages

and

it wont allow decimals

because its a multiplier

like, for the first one
its 50x multiplier
so 5000%

[fresh quarry]

Try this

Is it wrong?

[keen kindle]

one sec

no

since the 30,000,000,000 would then be 30,000,100

[blazing zephyr]

e9/1000 is e6 and the range is e2 to e5

[keen kindle]

which is way past 100,000

[loud gyro]

take log and add 100

[blazing zephyr]

its only +-×/ no?

[keen kindle]

yes

well

[loud gyro]

Oh

[keen kindle]

really i dont think it would matter

im only saying that because i would be able to know what youre saying

im half way through algebra

(or half way through the school year in which im learning algebra)

[blazing zephyr]

let f(x)=101
put all the numbers into the function

if the number is n you can do $\frac{n}{10^{v_{10}(n)}}$ (i am flexing dont mind me)

[wraith dagger]

skissue.in.a.teacup

[keen kindle]

okay so

i was lost at the v and after that

[blazing zephyr]

how did tiu even makw thos problem lol

[keen kindle]

...

"how did you even make this problem lol"?

this

[blazing zephyr]

sorry
racial skill..?

[keen kindle]

the numbers there are the actual numbers i just need to dumb them down to all fit between 100 and 100000

[keen kindle]

well
im editing the config files of multipliers in game
they for some reason set the limit as 100000

even though it could go up to 2,147,483,647 they decided to make it 100000

[cedar kiln]

@keen kindle Has your question been resolved?

[low briar]

So do you want to use 2 147 483 647 as your upper bound?

If so then you could simply divide the number you have by 2 147 483 647 and then multiply it by 100000

That way it all translates from min to max properly

Or rather multiply by 99900 and then add 100

[keen kindle]

1.) cannot include decimals
2.) must be between 100 and 100000

[low briar]

Though the'll get squished down a lot

Yeah, you can just cut of the decimals since they dont matter at that point

And your number is larger than the upper limit there so that causes issues

What's the largest possible value for this multiplier in theory?

Use that as your divisor

Basically the formula is (value / true upper bound) * (scaled upper bound - scaled lower bound) + scaled lower bound

This guarantees your conditions, just round up or down a number for the decimals

[keen kindle]

explain that please?

[keen kindle]

largest possibe value in the end? 100000

[low briar]

So let's say the true upper value is 999 and you have some guy at level 50, and you want to fit that on a scale from 10 to 100

What you'd do is

(50 / 999) * (100 - 10) + 10

[keen kindle]

so true upper value would be the largest number in the original (30,000,000,000) or the limit on the config (100,000)?

[low briar]

The largest possible original number, doesn't have to be included in the set

How strong can a character theoretically get basically

[keen kindle]

im... confused?

[low briar]

Suppose you have a character that's the very top of power, what would his original multiplier be before the conversion?

[keen kindle]

30,000,000,000

that is also the highest number in the original set

[low briar]

Right so your formula is then (value / 30 000 000 000) * (100000 - 100) + 100

For linear scaling

[keen kindle]

mkay

imma up that a bit to 50,000,000,000

[low briar]

Sure, play around with the numbers

Add SSG UI UE Gogeta at that level lmao

[keen kindle]

nvm scratch that keeping it at 30,000,000,000

[keen kindle]

damn they figured it out

the over all integer limit would be 2,147,483,647 since thats the minecraft integer limit and im changing the configs of a dragon ball minecraft mod

[keen kindle]

idk why the dev would not set all max to 2,147,483,647 in the configs

and i dont feel like changing the actual mod to make it the integer limit

[low briar]

Problably for the sake of simplicity or smth

Easier to deal with a set scale of clean numbers

[keen kindle]

oh the config says having the multiplier too high would cause bugs and glitches and stuff

that may be why

but who cares :3

[low briar]

Yeah minecraft do be like that

That game is built with duct tape

[keen kindle]

real

[low briar]

Did you... just use your catchphrase?

[keen kindle]

...

maybe

anyways i got my help

.close

[cedar kiln]

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[keen kindle]

.reopen

[cedar kiln]

✅

[keen kindle]

huh

ok...

imma break all my rules

and just make it if its like that to have it x10

oh wait

base

[keen kindle]

.close

[cedar kiln]

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[remote swallow]

Shouldn't these two be equal?

[twilit escarp]

How is rref defined ?

[remote swallow]

Row reduced echelon form? Are there multiple ways of defining it?

[twilit escarp]

Well there is one who made a upper triangle matrix and one that made it diagonal

So its diagonal

[proper mortar]

did you try to left multiply inv(A)?

[remote swallow]

This one is making it it diagonal (the identity, in fact) as you can see in the first ans = in the output

[twilit escarp]

Its probably a commutativity issue then

[remote swallow]

Hmm that works

Okay so uhm

If I'm trying so solve

XA = B

So I have to calculate B A(-1)

I cannot use the rref

I have to do it by columns I'm guessing?

[proper mortar]

yes

[twilit escarp]

Never seen this

[proper mortar]

rcef

[remote swallow]

Or the rref of the transpose

Would work too, right?

[proper mortar]

yes

[remote swallow]

Doing stuff by columns confuses me

[remote swallow]

I'm using it to calculate homographies

Okay, thanks you so much you both!

.close

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[remote swallow]

Note for future reference. It doesn't work by columns or with the transpose. I'll just stick to first calculating the inverse and then multiplying

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[coral jewel]

how can we verify that this is indeed a typical element of U?

[lyric narwhal]

Can you use the factor theorem?

p(x)=(x-a)q(x)+p(a), where q(x) is a polynomial of degree 1 less than p(x)

[coral jewel]

isn't that just PRT

[lyric narwhal]

What is that

[coral jewel]

polynomial remainder theorem

[steel heart]

Remainder theorem

[lyric narwhal]

Yeah it's the same thing with different names

If you've proved that already you can use it

[coral jewel]

i haven't proved that, but i think i can use it

[lyric narwhal]

You have p(z)=(z-2)q(z)+p(2), and as p(2)=p(5) it means q(5)=0

Now repeat the same thing again

[coral jewel]

i don't get it, how does this show this is a typical element of U?

but also, in order to show it really is a typical element, don't we kinda have to prove it spans U? (every element can be written in its form for some a, b, c)

so it's kind of circular

[lyric narwhal]

Because you'll get p(z)=(z-2)(z-5)(z-6)q(z)+p(2)

And since p is at most degree 4 it means q is of degree 0 or 1

[lyric narwhal]

We are showing that every element of U can be written in the way you wrote

But we're not using the linear combination thing

We're using the remainder theorem

[coral jewel]

i see

is there a way to show the list is a basis without proving it spans U?

using dimensions

[crimson delta]

you dont have rank nullity yet, right?

[coral jewel]

no i don't have it yet

[crimson delta]

unlucky

with rank nullity you can easily show dim U=3

[coral jewel]

is this what you're talking about?

it's like 3 chapters ahead

[sacred grail]

pain

[strange grove]

blake

[coral jewel]

well is there another way?

[crimson delta]

can't think of one right now

[coral jewel]

so i really have to prove span

[lyric narwhal]

If you really want you could take p(z)=az^4+bz^3+cz^2+dz+e and get the conditions required

[coral jewel]

that was my original idea, but it sounds like a pain

[lyric narwhal]

It is

[sacred grail]

gigantic pain to do

[lyric narwhal]

I don't see why you don't want to use the remainder theorem thing though

[sacred grail]

you ideally want to reframe the conditions as p(2) - p(5) = 0 and p(5) - p(6) = 0 or something similar

perhaps you could write down a map P_4(R) -> R^5 given by p -> (p(2), p(5), p(6), p(69), p(420))

then this is a vector space isomorphism and you prove the dimensions in R^5 instead

[coral jewel]

i haven't learned linear maps and isomorphism yet

[sacred grail]

skull

[coral jewel]

shouldn't the constant term be (p(2) + p(5) + p(6))?

since we divided it three times, the remainders stack up

[lyric narwhal]

why?

no

p(z)=(z-2)p1(z)+p(2), so p(5)=3p1(5)+p(2). since p(5)=p(2) it means p1(5)=0, so that p1(z)=(z-5)p2(z)

[coral jewel]

ah i see

[coral jewel]

how does this look now?

[lyric narwhal]

I'm not too sure what good vs bad formatting of answers looks like

i feel like you should explain how you got p(z)=(z-2)... in a bit more detail

[coral jewel]

what about now?

the text looks a bit clumped, but i don't know how to write it more concise than that

[lyric narwhal]

yeah looks good

[coral jewel]

alright, thanks

.close

[cedar kiln]

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[cedar kiln]

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[pastel vault]

try applying the mean value theorem

[mighty nest]

I could try it if x>0

But now that xER it confuses me

[mighty nest]

Yeah I know

But first I need to appear it

[pastel vault]

yeah so try rearranging this as f(x + 1) - f(x) < f(x + 3) - f(x + 2)

then you are using the open intervals (0, 1) and (2, 3) which are disjoint

that's the idea of the proof, then yeah ofc you need f'(x) strictly increasing

[hidden mural]

that just means x is an element of R

R = set of real numbers

[mighty nest]

I know that lol

[hidden mural]

what

oh alr

so u got a proof for x > 0?

[mighty nest]

I need to know it so I know how to change the inequality

Lemme try something south said

[hidden mural]

know what

bro just admit u've got nothing

yes what south said is correct

[mighty nest]

Lmao there are 2 sub queries I have done the first

[pastel vault]

extra hint: ||the maximum value f'(c) can be for c in (0, 1) is f'(1)
but the minimum value f'(d) can be for d in (2, 3) is f'(2)||

[mighty nest]

The second just confuses me

[hidden mural]

sub queries?

[mighty nest]

What confuses you

[hidden mural]

what sub queries ru talking about

there's only 1 question

[mighty nest]

Yeah cause I didn’t send the other one

[hidden mural]

right

[mighty nest]

Want me to send it ? And show u the solution of it

If u don’t believe me

[hidden mural]

no thanks

[hidden mural]

like what does this sentence mean

[mighty nest]

Damn bro idk why u be coming to chat and acting like that if u not trynna help

[hidden mural]

you could try it? then try it

[mighty nest]

Damn bro just leave it

[hidden mural]

k

[mighty nest]

South do I not need to show the denominator I can apply the theorem?

[pastel vault]

I mean yeah you have to apply it properly, like I started by assuming the proof statement, which you should never do in a proper proof

the key idea is that the denominator will be 1 for both of them

[mighty nest]

If I assume the statement and then lead to a senseless result , then isn’t that considered proper proof?

That may not make sense sorry for my English

[pastel vault]

yeah that doesn't make sense sorry

[mighty nest]

if I proceed with what I want to prove, and I conclude a conclusion that is valid then my initial consideration will be correct? Isn’t that considered a proper proof?

Makes sense?

[pastel vault]

no, that would be the definition of an improper proof

assuming what you want to prove

you need to start from something like (f(x + 1) - f(x))/(x + 1 - x) < (f(x + 3) - f(x + 2))/(x + 3 - (x + 2)) from MVT

and then you rearrange this statement

into the statement in the question

then that becomes a proper proof

[mighty nest]

Hold on I’m sending my work

Was translating it

Can that be considered correct proof?

Any1????

<@&286206848099549185>

[twilit bison]

didn't you swap zeta 1 and 2?

zeta2 > zeta1

[mighty nest]

Why would I

[twilit bison]

i don't know, just by mistake i guess

[mighty nest]

Wait where did I do that

[twilit bison]

in the last line

[mighty nest]

f’ is strictly increasing tho?

[twilit bison]

f'(z1) < f'(z2) iff z1 < z2

[mighty nest]

Oh yeah you are right

Still without this mistake , is it considered a correct proof?

[twilit bison]

yes it seems correct

[mighty nest]

Okk thanks 🙏

[twilit bison]

rather than "thus" i would say "because f' is strictly increasing"

[mighty nest]

Yeah I have wrote that in the paper just not on the translation

[cedar kiln]

@mighty nest Has your question been resolved?

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[main flower]

what is the formula for cos^2x

[unique nest]

1-sin^2x?

[coral jewel]

which formula

they're asking for cos^2x tho

[mental trail]

the half angle formula? pythagorean identity?

[coral jewel]

either way, please show original problem

i don't know which formula you need

[wraith dagger]

Pro_Hecker

Pro_Hecker

Pro_Hecker

[main flower]

how would you then solve cos2(x), could you re-write it as cos(x) ・ cos(x)?

[coral jewel]

wdym solve

there's nothing to solve

!xy

[cedar kiln]

Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.

[steel heart]

cos^2x or cos2x

[main flower]

cos2(x), how could you re-write it, i will grab the original equation one sec

[wraith dagger]

Pro_Hecker

Pro_Hecker

Pro_Hecker

[main flower]

im supposed to solve this without a calculator, and no values are given for x, given that everything seems to be a rewrite of itself, how should i begin?

[slate lintel]

get everything in terms of cosx

[main flower]

once in terms of cos, what would I do? I cannot determine where to go with it, would i need to do any more conversions?

[steel heart]

solve a quadratic in terms of cos x

see if you get acceptable answers, i.e. -1 =< cos x =< 1

[main flower]

what would that involve?

[cedar kiln]

@main flower Has your question been resolved?

[steel heart]

what have you gotten?

[main flower]

were you sugesting to get everything on one side and solving?

[steel heart]

yes

a quadratic equation

in terms of cosx

[main flower]

okay, give me a moment

thank you

[cedar kiln]

@main flower Has your question been resolved?

[steel heart]

what do you have? what have u worked out?

[main flower]

im sorry, you can close my channel, i got busy, thank you so much for your help

[steel heart]

.close

[cedar kiln]

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[frigid dust]

!original

[cedar kiln]

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[mighty nest]

#help-12

[cedar kiln]

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[stone bluff]

how do we pass from the first expression to the second one ? i dont get it

[blissful turtle]

the e^x was factorised out

and so was the (x-6)^m-1

[frigid dust]

$(x-6)^m=(x-6)^{m-1}(x-6)$

[wraith dagger]

Bonk

[cedar kiln]

@stone bluff Has your question been resolved?

[cedar kiln]

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[past sleet]

how do i work out question 16 a

[past sleet]

im wondering if this is correct?

just those first bits

[modern compass]

some of what you have just, isn't true.

[past sleet]

really?

[modern compass]

you can find the area from -6 to -4 using geometry

[past sleet]

yeah

[modern compass]

Then, sure integrate from -4 to k -x-4 to find k so that those two integrals add to 0

[past sleet]

oh you’re right

idk why i added 1

in that first integral

[modern compass]

it looks like you're trying to make the integral from -6 to -4 one polynomial and it isn't.

[past sleet]

let me try now

is this correct?

hm

it isnt

what do i do

i assumed k was between -1 and -2 just eyeing the graph

i mean

between -4 and -2

oh wait

I GOT IT

it’s correct

[cedar kiln]

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[past sleet]

.close

[cedar kiln]

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[woeful isle]

Ye

[blissful turtle]

big equation i guess

[woeful isle]

Solid size

[blissful turtle]

For the upper limit do i show it diverges?

[woeful isle]

Does it diverge?

[blissful turtle]

not sure

Cause if it diverges to infinity then its a nicer upper limit

[woeful isle]

||It doesn't ||

You can check

And then start working on it

[blissful turtle]

converges via root test

but man

[woeful isle]

Now, what does it converge to

?

[blissful turtle]

working it out is gonna be much harder

[woeful isle]

We see a cube in the denominator

Try and write out its terms

Few of those are written out in numerator

Use partial fractions

[blissful turtle]

im not too sure how partial fractions come into this

[woeful isle]

Need a hint?

[blissful turtle]

yes please

Ive written out the first 5 ters

but not making much progress

[woeful isle]

$(n+1)^3 - (n-1)^3 = ?$

[wraith dagger]

lex.in.a.teacup

[woeful isle]

And

$((n^2-1))^3 = ((n+1)*(n-1))^3$

[wraith dagger]

lex.in.a.teacup

[blissful turtle]

ohhh

this is huge undoubtedly

Will i get some stuff to cancel out?

[woeful isle]

Yup

Telescopic series

$(n+1)^3 - (n-1)^3 = 2(3n^2+1)$

[wraith dagger]

lex.in.a.teacup

[woeful isle]

$2\frac{(n+1)^3 - (n-1)^3}{(n+1)^3(n-1)^3}$

$2\frac{1}{(n-1)^3} - \frac{1}{(n+1)^3}$

See it now?

@blissful turtle

[blissful turtle]

I think i messed up my partical fractions

[wraith dagger]

lex.in.a.teacup

[blissful turtle]

Cause if i get to this i see the telescope

[wraith dagger]

lex.in.a.teacup

[woeful isle]

Ye

So

What do you get?

[blissful turtle]

I have

give me a sec new to Texit

[woeful isle]

Ok

[blissful turtle]

$\frac{1}{2(n-1)^3} - \frac{1}{2(n+1)^3}$

[wraith dagger]

Ferocious Frawg

[woeful isle]

2 in numerator

Oh

Wait

[blissful turtle]

let me double check my fractions

[woeful isle]

No you righ5

[blissful turtle]

oh cool

[woeful isle]

My fault

[blissful turtle]

no worries

i still think it telescopes

[woeful isle]

It does

Every 3rd term cancels out

So

You are left with

$\frac{1}{2}(\frac{1}{(2-1)^3} + \frac{1}{(2-1)^3} - \frac{1}{(\infty-1)^3} - \frac{1}{(\infty)^3})$

[wraith dagger]

lex.in.a.teacup

[woeful isle]

Which is just 9/16

[blissful turtle]

I think ive got that

just gonna make sure

[woeful isle]

Which makes the upper limit = 1

[blissful turtle]

got it

lovely

thank you so much for all the help

Im pretty bad at summations and Im definetely glad to practice them

[woeful isle]

Np

Let's keep going

[blissful turtle]

hyped

lower integral

is it evaluate the sum first

I'm looking at it and absolutely stumped on how to start

[woeful isle]

$I = \int_0^{1}e^x \frac{x}{(x+1)^2}dx$

[wraith dagger]

lex.in.a.teacup

[blissful turtle]

that integral looks pretty reasonable

[woeful isle]

Ye

Try it

Got it?

[blissful turtle]

trying parts rn

but im gonna have to try something new

if the next part doesn evaluate nicely

[woeful isle]

Ye

[blissful turtle]

parts was no good

Or maybe i picked bad parts

[woeful isle]

Hmm

Why not

What did you pick?

[blissful turtle]

I ended up with an e^x/(x+1)

which im pretty sure has no elementary antiderivative

[woeful isle]

.

[blissful turtle]

e^x and x/(x+1)^2

Im gonna try x*e^x

[woeful isle]

Try xe^x and 1/(x+1)^2

[woeful isle]

Ye

That's much better

Got it?

You should get e/2 -1

I think

[blissful turtle]

lovely

worked out perfectly