help-13

[cedar kiln]

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[olive geode]

can I use l'H rule on this to test for divergence?

[olive geode]

or should I use a diff test?

[frigid sinew]

LH works just fine here no?

[hollow trail]

nth term test / test for divergence is a good pick for this one

and l'hopital's works fine for computing that limit

[olive geode]

ok thanks just checking because every work I see online uses ratio test

[hollow trail]

it's also worth "just knowing" that exponential functions outgrow polynomials

[olive geode]

quick question idk if im tripping though i have to switch the n to x correct?

[hollow trail]

l'hopital's would require you find the limit for real numbers rather than integers, yes

[olive geode]

ok thanks

.close

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[cedar kiln]

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[olive geode]

this correct power rep?

[kindred venture]

erm yes

valid for $|2x^2| < 1$

[wraith dagger]

Obotron

[kindred venture]

wait where did the 'x' go on the outside

[olive geode]

tbh i only do it because my professor said it makes it easier to see.

does it matter?

[kindred venture]

i think you did it incorrectly

or im confused idk

$x(2x^2)^n\neq (2x^2)^{n+1}$

[wraith dagger]

Obotron

[crimson sedge]

Ye

[olive geode]

i’m confused now lol

[cedar kiln]

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[bright sorrel]

why did this happen?

[bright sorrel]

why is it 2/sqrt(3) in the denominator?

shouldn't it literally be sqrt(3)/2 ??

[cedar kiln]

@bright sorrel Has your question been resolved?

[bright sorrel]

<@&286206848099549185> ??

[glad peak]

what is the original integral

looks like 1 / (x^2+x+1)^2?

[bright sorrel]

that's right

yes

why is it the reciprocal at the denominator

to my very logical understanding

ah nvm

I understood

.close

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[cedar kiln]

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[white frigate]

where did they get the underlined value from?

[shy widget]

thats the proability that Suzi will be take more than 45 minutes

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[wicked jetty]

Is the x - int of a rational function calculated on just the numerator ?

[cedar kiln]

Please don't occupy multiple help channels.

[wicked jetty]

I closed the other

<@&286206848099549185>

[unique valley]

i dont understand this question

maybe give some context

[median lantern]

Yes

So

If we have

Y = f(x)/g(x) we want f(x) to be 0

Unless there is a + constant

[wicked jetty]

Okay

What if there is a + constant

[oblique marlin]

Timmy getting his hw done for free

Lmfao

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[exotic hollow]

hello
https://cdn.discordapp.com/attachments/830511422005968957/1235202123013296138/-2147483648_-211196.png?ex=6633831f&is=6632319f&hm=708eca8f62c1f572421b445fc82604e84cc74f9f50099787522a5dfc1afb07f8&

[cedar kiln]

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[frigid sinew]

aer they correct?

all of them are a_n > a_n+1 for me

[lyric narwhal]

Recheck 3rd and 4th

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[indigo root]

im having a real hard time with this integral, i cant find how to integrate the square root

[cedar kiln]

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[carmine bronze]

Try numerical integration using Simpson's Rule or the Trapezoidal Rule.

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[cedar kiln]

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[velvet crystal]

How do i solve this?

[leaden radish]

im guessing solve the integral and use a C that connects the points

[velvet crystal]

does this seem correct?

I dont know what im doing :D

nevermind i found a tutorial

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[fierce basin]

Help explain this issue

[fierce basin]

<@&286206848099549185>

[cedar kiln]

@fierce basin Has your question been resolved?

[fierce basin]

😭

<@&286206848099549185>

[plucky perch]

What issues exactly

[cedar kiln]

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[bleak basin]

What am I missing, I cannot figure this out...surely not enough relevant data? It asks for area

[candid marsh]

You have enough

[unique valley]

it definitely is

[candid marsh]

I assume it mentions that they're both parallelograms?

[unique valley]

you need assume that tho, and that they are equal and split in half

[candid marsh]

You don't have to assume them equal

[bleak basin]

It doesn't explicitly say that, but I tried that approach

[unique valley]

but thats fair assumption to make

[unique valley]

so what did you try?

[bleak basin]

I tried to incorrectly assume the left could be calculated as a right triangle to get the hypoenuse

We know the answer is 40 from the riddle at the top

[unique valley]

yeah it is

do you recall the formula for the area of parallelogram?

[bleak basin]

Base x height

[unique valley]

do we know the base?

[bleak basin]

8

[unique valley]

yeah

[bleak basin]

And 5 and or 10

[unique valley]

notice that combine height of those is 10

[bleak basin]

I'm only confused because 8x10=80

And 8x5 would be the 40 but that's half the shape

[unique valley]

can you show the full question?

and that riddle at the top

[bleak basin]

Sure

[unique valley]

hmm

thats strange

i dont know how to help you

[bleak basin]

Yeah I don't get it

I appreciate it anyway

[candid marsh]

I'm confused too

[bleak basin]

I would think 80

[candid marsh]

Although the joke makes sense

[unique valley]

it doesnt seem possible it is 40

[candid marsh]

A typo maybe?

[bleak basin]

Must be

[unique valley]

maybe

[bleak basin]

Yeah, I mean that was the one we got stuck on

Rest worked

So clearly the intended answer

[candid marsh]

Although, did they just pull the joke off the internet??
Are people supposed to typically know that a gross is 12 dozen?

[bleak basin]

I think we're going with the assumption they intended you to apparently find only half the area

Thank you both

My daughter was getting upset and I could not figure out how it was 40

Have a good one

[unique valley]

u2 man

[cedar kiln]

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[crisp lintel]

how did they find 45 degrees?

[glossy creek]

135 - 90 degrees

[crisp lintel]

where did we get 135?

[glossy creek]

it's in the qeustion

question*

[crisp lintel]

this is the question

[crisp lintel]

this is the answer

[hollow trail]

they assumed that "northeast" meant exactly the direction halfway between north and east

[glossy creek]

the answer should be the magnitude of the red vector, I think the angle is given

or what's the precise question

[celest osprey]

This is the question

[celest osprey]

East and northeast tell you the directions of the vectors

[glossy creek]

yeah that's what I meant

[crisp lintel]

i see and thats half of 90 ?

[glossy creek]

general question, not what you have to do

[hollow trail]

yes, halfway between two directions at 90 degrees to each other is 45 degrees from each direction

[crisp lintel]

so its 90 + 45

thats how they got 135?

[glossy creek]

yeah assuming northeast = 45 degrees

[crisp lintel]

i seee good to knowww

so when they say north east they saying its on 45 degrees?

[glossy creek]

well, 45 degrees relative to the northern, or eastern direction, in this case relative to north

[crisp lintel]

ohh i see

[glossy creek]

it doesn't rly matter which one you choose it'll always be 45

[crisp lintel]

oooo

okk tyy

.close

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[cedar kiln]

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[steel needle]

Hello can someone help me w this problem?

[cedar kiln]

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[wintry rover]

I'm tasked with proving the following:

[wintry rover]

whoops

[digital cliff]

[wintry rover]

I'm tasked with proving the following:
[ \sum_{d\mid n} \sigma(d) = n\sum_{d\mid n} \frac{\tau(d)}{d}]

I'm famiilar with the proof that $\sum_{d\mid n} \sigma(d) = \sum_{d\mid n} (d \cdot \tau(\frac{n}{d}))$, but I don't know how I could adapt it into the given result (if that would be the right strategy to take. Any hints?

[wraith dagger]

cat_food_sounds

[mental trail]

have you thought about an index change?

[wintry rover]

I struggle with the concept, tbh

[mental trail]

well

if d is a divisor of n, then what other number can you find that is a divisor of n?

[wintry rover]

Though my prof loves doing that so I could totally see that being the strategy

[wintry rover]

n/d?

[mental trail]

you got it

[wintry rover]

Huh

[mental trail]

well

you got your index change

d' = ...

[wintry rover]

I'm lost 😭

[cedar kiln]

@wintry rover Has your question been resolved?

[wintry rover]

NEVER MIND I FIGURED IT OUT

.close

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[cedar kiln]

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[royal loom]

So far, I've been able to show that f(1/z) must have a pole at z=0 and be elsewhere holomorphic

I want to now use this to show f(z) must be a polynomial

but I'm unsure how

[cedar kiln]

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[royal loom]

.close

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[hollow osprey]

🔠

[cedar kiln]

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[spring jungle]

I am struggling with the basics of evaluating the derivatives of inverse functions, and don't know where to begin on a problem like this

[upper abyss]

For simplicity in notation, let g be the inverse of f. It must be true that:

f(g(x)) = x

Taking the derivative:
f'(g(x))g'(x) = 1

Gives us the inverse function theorem

You want g'(4), to be clear

[nimble mulch]

@spring jungle how would you apply this to your task? :]

[spring jungle]

well, im thinking through what was said, and I am confused a little about the taking the derivative part being equal to one, so im a little stuck on that

its using the chain rule which i understand

[upper abyss]

All I did was take the derivative of both sides.

Derivative of x is 1.

[spring jungle]

oh okay, gotcha

that makes more sense

[nimble mulch]

given f'(g(x))g'(x) = 1 and you seek g'(4), we can isolate g':
g'(x) = 1/f'(g(x))

g is f^-1 as Kaynex mentioned

[spring jungle]

okay, makes sense so far

[nimble mulch]

ok so let's plug in 4:

g'(4) = 1/f'(g(4))

so all you require to get the value is g(x) and f'(x)

well f'(x) should be simple since it's a polynomial

[spring jungle]

okay, and i can get f(x) by just taking the derivative

[nimble mulch]

Yop

[spring jungle]

yeah

and what about g(x)

[nimble mulch]

and g(x) by switching x/y and solving for y

meaning since
f(x) = x²+4x-5

x = g(x)²+4g(x)-5

and you want to solve for g(x)

[nimble mulch]

Since that's the meaning of the inverse

[spring jungle]

agreed, i attempted to do so and got lost in the algebraic component, specifically at the step x + 5 = g(x)^2 +4g(x)

[nimble mulch]

It's PQ-formula

since you just solve it like a regular quadratic

[spring jungle]

quadradic formula?

gotcha

[nimble mulch]

yep

Consider x as a constant

[spring jungle]

but doesnt that require one side to be 0, like how I have solved polynomials in the past?

[nimble mulch]

You just bring the x+5 over

[spring jungle]

okay, so i would have 0 = g(x)^2 + 4g(x) - 5 -x

[nimble mulch]

Ys

[spring jungle]

and then make x a constant? does it matter what constant? should i use the one in my problem

[nimble mulch]

g(x) = -4/2 ± sqrt((4/2)² - (-5-x))

[nimble mulch]

Yup as mentioned regard x as constant

[nimble mulch]

But note that this would be the general solution to the quadratic formula, we can't have ± in our definition of g(x)

since it can't map to two values at once

[spring jungle]

yeah, i just take the value within the restricted domain

is the PQ formula the same thing as the quadratic equation? im afraid ive never used / heard of the PQ formula

[nimble mulch]

Yeah same principld

PQ European I think

and the other version is American I believe

[spring jungle]

gotcha gotcha

[nimble mulch]

and yes you got it right, find domain & range of f

flip them to get domain & range of g

[spring jungle]

okay got it, thank you!

.close

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[quasi cradle]

i need to show step by step how to do this

[quasi cradle]

erm

[hollow osprey]

Let me ask my sir @royal loom what to do

[quasi cradle]

ok thanks

[hollow osprey]

What is your doubt?

[quasi cradle]

i dont know how to solve it

i know i have to use subtitution but idk how to go off from there

[hollow osprey]

Try multiplying top and bottom by e^10x

[royal loom]

Multiply top and bottom by

e^10x

[quasi cradle]

ok

[hollow osprey]

🔠 my sir is the goat

[royal loom]

And then perform what my sir calls a u substitution

[hollow osprey]

Your writing sir?

[quasi cradle]

like this?

[royal loom]

Yes

[royal loom]

Yes now perform a u substitute

[hollow osprey]

Yes

[quasi cradle]

so is u = 10x?

[hollow osprey]

No

There is a better one

[quasi cradle]

i include e?

so e^10x?

[hollow osprey]

Yes

🔠

@royal loom W?

[quasi cradle]

ok let me try that

like that?

[royal loom]

W doubt clear

now just solve it @quasi cradle

you have everything you need

[quasi cradle]

okay

am i on the right track? i feel so lost

[quasi cradle]

i dont know how to solve it

[royal loom]

this is 1

[quasi cradle]

oh

[quasi cradle]

am i suppose to use a formula to solve this before plugging u back in?

[royal loom]

I suggest you restart the problem

with all the hints in mind

[quasi cradle]

alr

im going to sleep

thanks for the help

.close

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[pseudo merlin]

why is speed equal to the magnitude here

[digital cliff]

by definition speed is the magnitude of velocity

its velocity without the direction

[pseudo merlin]

oh ok

thx

.close

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[ionic wolf]

idk Abt those numbers in front idk if I can condense it with that there

oh wait

I can put it back in right

with like exponents

[slow jewel]

Yes

[ionic wolf]

help

[cedar kiln]

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@ionic wolf Has your question been resolved?

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[light adder]

How do I solve this? I hate this! Help would be appreciated.

[tranquil hamlet]

so $y = 3x-1$

[wraith dagger]

e_waste

[slow jewel]

What have you tried

[light adder]

Wow

[tranquil hamlet]

and $3x - 5(3x-1) = -10$

[wraith dagger]

e_waste

[light adder]

Really fast respond

[tranquil hamlet]

so$3x-15x+5=-10$

[wraith dagger]

e_waste

[tranquil hamlet]

I'm not gonna tell you how to do it next

do it by yourself

[light adder]

Thats where I got stuck

I already got to that point

[tranquil hamlet]

Ok then

so $3x-15x = -12x$

[wraith dagger]

e_waste

[tranquil hamlet]

and you move 5 to the right it becomes -5 right

-10 - 5 = -15

so 12x=-15

[light adder]

Hmm

[tranquil hamlet]

oh yes

then -12x=-15

I actually told you how to solve the problem directly...

[light adder]

XD im not good with this

[median lantern]

U could use cramers rule

[light adder]

Hmm

[median lantern]

Just use the things above and isolate a variable

[light adder]

Im in high school idk if thats worth learning

[zenith cosmos]

hello i need help who can explain me how to solve it f(x) = 3 ^ (x + 1)

[median lantern]

It's just a formjla

[cedar kiln]

Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).

[light adder]

How does the formula work ?

[zenith cosmos]

yes i know

[median lantern]

It's just elimination but in a formula

Search it up

I'm on mobile so it's harder

[light adder]

Okkay

[zenith cosmos]

Show me the formula pls

[wind sentinel]

what specifically do u need help with

like r u tryna find the derivative

or

evaluate it at a certain point

or whag

be specific

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[rich pendant]

bro 💀

[rich pendant]

.close

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[sudden heart]

Here is my work

Can someone check it

<@&286206848099549185>

[void glen]

yeah that looks correct

[sudden heart]

Ok thanks

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[vague musk]

how do I solve this

[onyx aurora]

definition of continuity, do you know the relation?

or formula idk the term

[vague musk]

idk

what is it

[onyx aurora]

$\lim {x\rightarrow x{0}^{-}} f( x) =\lim {x\rightarrow x{0}^{+}} f( x) =f( x_{0})$

[wraith dagger]

! Giovyx90

[vague musk]

oh I know this

yeah

[onyx aurora]

The function f(x) changes analitic expression at x=-1

so thats probably a discontinuity

[left quartz]

What's up

Bro is cr7 fan

[onyx aurora]

$\lim _{x\rightarrow -1^{-}}\frac{6}{1-x}$

[left quartz]

Calls messi pedsi

[wraith dagger]

! Giovyx90

[onyx aurora]

$\lim _{x\rightarrow -1^{+}}\frac{x-2}{x^{2} -4}$

[vague musk]

Why is the discontinuity at x = -1

[wraith dagger]

! Giovyx90

[onyx aurora]

Because it changes analytic expression.

[vague musk]

Wdym it changes the analytic expression

[onyx aurora]

it changes equation, function, idk how to say sorry

first it is 6/1-x

then its x-2/x^2-4

[upper abyss]

There are multiple discontinuities. x = -1 might be a discontinuity, as it's the "cutoff between two functions"

Worth investigating that point

[onyx aurora]

also x=2 might be because the denominator gets 0

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[sinful field]

Integral of k^x

[sinful field]

I know it's k^x / ln k

But why, I can't find proof online or like an explanation

If someone could help me that would be great

[weak otter]

Take ln of both sides I think

Oh wait no what am I saying

$k=e^{ln(k)}$ so $k^x=e^{ln(k)*x}$

[wraith dagger]

coak

[weak otter]

So integral is $\frac{e^{ln(k)*x}}{ln(k)}$

[wraith dagger]

coak

[weak otter]

Which simplifies to what you wrote

[sinful field]

Oh wow

They really be doing everything in math

[weak otter]

Simple

[sinful field]

OK what about e^x

Ik dumb question forgive me 😭

[idle tusk]

use the same formula

it's just that ln(e)=1

[mighty drift]

depending on your definition of e^x, this may or may not be directly from the definition

[sinful field]

Okay thanks

.closw

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[sand cradle]

#1195800467880558672 message

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[errant wasp]

in a vector field how do you know if the domain is simply connected

[dull oxide]

It needs to be path connected and have no holes

[cedar kiln]

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[sacred grail]

what's a hole

[dull oxide]

If @errant wasp is still around, I will answer that

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[sacred grail]

lol

good timing

if you put a "hole" into R^3, it's certainly still simply connected

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[pseudo merlin]

Hi

[pseudo merlin]

Is this vectors i component negative?

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[ancient lodge]

Do yk anything abt parallel line angles? This is a pretty direct application of that.

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[haughty cloud]

how would i solve for x to have 1 pivot position?

[haughty cloud]

in the solution they do x^2+10 = (3x+1)2

but why/

?

[bright fiber]

What context is this matrix in?

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[vapid rivet]

can someone explain what i am doing wrong

i keep getting zero so it should be inconclusive

using the limit comparison test

[hollow trail]

if you have a series $\sum a_n$ and are comparing it to the series $\sum b_n$ by computing the limit [ L = \lim_{n \to \infty} \frac{a_n}{b_n} ] then you have 3 conclusions:
\begin{itemize}
\item If $L = 0$ then $b_n$ grows faster than $a_n$. So if $b_n$ is convergent, so is $a_n$ (smaller than a finite number is finite'') (Note: inconclusive if $b_n$ is divergent) \item If $L = \infty$ then $a_n$ grows faster than $b_n$. So if $b_n$ is divergent, so is $a_n$ (bigger than infinity is infinite'') (Note: inconclusive if $b_n$ is convergent)
\item if $0 < L < \infty$ then $a_n$ and $b_n$ have the same growth rate, so they either both converge or both diverge
\end{itemize}

[wraith dagger]

cloud

[vapid rivet]

ok thanks

i wasnt old this in the notes she gave me lol

so i could just use direct compariso then?

[hollow trail]

you could in this case, but in general limit comparison can be used in more scenarios than direct comparison

[hollow trail]

since you're told to use the limit comparison test though, note that your limit falls into the first category

[vapid rivet]

ok

im gonna that to these notes

mine just say if l doesnt equal zero they both converge or diverfge

[hollow trail]

that's the most common case, but it's good to know the two edge cases as well

[vapid rivet]

for sure

thanks

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[winged schooner]

What is going on in the last line? How did we go from -(n-1)* that integral which is a product of 2 terms being split in a difference of two? What is going on there? And does the fact that the professor copied x^(n-1)*sqrt(1-x^2) part of it? To me it looks like he just forgot there should have been a + sign under the sqrt.

[silent tide]

Took me a while to get but, he just distributed the xⁿ⁻² and then split the fraction

$\int \frac{1+x^2}{\sqrt{1+x^2}}\cdot {x}^{n-2} \text{d}{x}$
$= \int \frac{{x}^{n-2}+{x}^{n}}{\sqrt{1+x^2}}\text{d}{x}$
$= \int (\frac{{x}^{n-2}}{\sqrt{1+x^2}} + \frac{{x}^{n}}{\sqrt{1+x^2}})\text{d}{x}$

[wraith dagger]

Jafar / جعفر

[winged schooner]

Jesus Christ he did that all in one go

and also split it with -(n-1)

[silent tide]

That's what happens when someone does many integrals

[winged schooner]

but then he must've forgot the first term in front

should have been sqrt(1+x^2)

[silent tide]

Yeah that's probably a typo

Or whatever the handwritten equivalent of a typo is

[winged schooner]

yeah he corrected it

thanks!

[silent tide]

np

[winged schooner]

.solved

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[humble gale]

how do I find the bounds?

how am I supposed to know what the correct bounds are

[red stone]

Is that calculus

[humble gale]

yeah

calc 1

[cerulean sail]

Where do those equations intersect?

[static topaz]

can get a rough sketch of the graphs, doesn't need to be exact

then set those equations equal to each other

you get the x value of where they intersect(two points)

those x values are your bounds

and then its just bigger function - smaller function

[humble gale]

so if I were to set those two equations equal to each other, the x values I get from that are my bounds?

[static topaz]

correct, but always get a rough sketch of the graph first

so you actually get an idea of what those points represent

[humble gale]

k

[static topaz]

i try to ignore translations when sketching quick graphs

for this, u just need to know if parabola is open downwards or upwards

and then draw a line with -1 slope intersecting it

[humble gale]

Yessirrrrr

got it

answer is A

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[errant loom]

Hey, could someone walk me through the algebra steps here to get the answer?

[cedar kiln]

@errant loom Has your question been resolved?

[torn island]

$((5-y)/2)^2=(5-y)^2/2^2$

[wraith dagger]

nrf

[torn island]

$=(5-y)^2 * 1/2^2=(5-y)^2* 1/4$

[wraith dagger]

nrf

[torn island]

@errant loom

[errant loom]

ah are you pulling out a 1/4 after multiplying 2^2?

[torn island]

yeah

[errant loom]

oh okay lol that's simple

thank you

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[elder pewter]

how do u use one to one property for 1/125 = 5^x-2

[elder pewter]

im stuck on making 1/125 equal to 5

[native rain]

$$ \frac{1}{125} = 5^{x-2} $$

$$ \frac{1}{5^3} = 5^{-3} $$

$$ 5^{-3} = 5^{x-2} $$

Now that both sides of the equation have the same base, we can apply the one-to-one property

[wraith dagger]

misha

[elder pewter]

oh i shouldve used the negative exponent

ok thx :)

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[royal loom]

I was hoping someone could check my proof for this question, I'm a bit wary since I never used that k|n at all

I've just realized it should be over multiplication instead so for proving its a homomorphism part imagine I wrote \cdot instead of + wherever there is a +

[cerulean sail]

Is your f well defined?

[royal loom]

it didn't ask me to show that :p

normally I would but the next question actually says to do that

so I feel like

omission here gives me an excuse

[cerulean sail]

I find you not guilty

But you really should show it's well defined, I'll let you off with a warning this time

[royal loom]

let [x]_n=[y]_n
then f([x]_n)=[x]_k
f([y]_n)=[y]_k
want that
[x]_k=[y]_k
this is if and only if
[x-y]_k = 0
i.e that
k| x-y
but
[x]_n=[y]_n
so
[x-y]_n=0
so
n | x-y
but k|n
so
k | n | x-y
k | x-y

Q.E.D well-defined

[cerulean sail]

Yep, without that divisibility property you can run into real problems

[royal loom]

I see

What kind of monster would ask me to prove something that's not even well-defined is a homomorphism?? Proof by I thought you weren't a horrible teacher

[cerulean sail]

"hey, you were supposed to notice that it was poorly defined and my questions made sense, not my fault you assumed I was telling the truth"

e.g. in Z4, [1] and [5] are the same, in Z8, [1] and [5] are distinct, so if you tried defining a map from Z4 to Z8 just by inclusion then it would be a poor definition

[royal loom]

that's too good a response to be off the top of the head

you went through this before

@cerulean sail did you think the rest of the proof was fine then?

[cerulean sail]

Yea think no complaints, seems alright to me! Just adding in the well defined part

[royal loom]

okay thank u very much, I did add it 😎

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[lucid shell]

for v i got -C cos(thetha)x - (C-S)sin(theta)y but I dont get how to make that into an ode

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<@&286206848099549185>

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[silver crater]

hi, i have to demonstrate |sinx - x| <= |x|³/6, probably using series expensions ( the previous question is "write taylor's series of sin(x) at order 2", and i can't get it, i've tried a few things using the triangular inequality but the rest of the series blocks me

[silver crater]

that is the best i've got so far

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[silver crater]

<@&286206848099549185>

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[hollow zodiac]

For each value of b with 0 ≤ b ≤ 100, points A(0 | 0) and B b( | 0) are given, as well as point C. Point C has the x-coordinate b and lies on the graph f(x)
Determine the value of b for which the area of triangle ABC is maximal, and provide the corresponding area.

[hollow zodiac]

Can anyone help me?

I know that the Fuction for that will be:

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[crimson sedge]

I have a question, when I get my eigenvalues, at what order do I put them on so my eigen vectors matrix is correct?
from the largest to smallest or smallest to largest

[dawn junco]

you're doing diagonalization or something ? @crimson sedge

[crimson sedge]

yes
lets say P is the matrix that has all my eigen vectors
at what order do I put them on?

which goes first, second n third

[dawn junco]

the order in itself doesn't matter

you just have to keep the same order for D and P

so if you start with 5 as your eigenvalue in D

the first eigenvector in P has to correspond to the eigenvalue 5

[crimson sedge]

Ahhh, beautifully explained, thank you so much!
one of my teachers said that I have to put them from smallest to largest without explanation, this makes more sense

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[dawn junco]

now if you're doing other stuff like SVD for example, it's a hugely followed convention that you start with the largest eigenvalue, and go in decreasing order

but technically you could swap them around still

[crimson sedge]

since I don't know what's SVD, I guess that it doesn't matter that much
Thank you so much!

[cedar kiln]

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[silver arrow]

I need some help

[silver arrow]

Soooo

Task 3:

When the blades of a wind turbine rotate, a circle is formed whose diameter is called the rotor diameter.

Figure 4 illustrates that a wind turbine's effect increases the larger the rotor diameter.

Rotor diameter 27 ma Power 225 kW

Rotor diameter 33 ma Power 300 kW

Rotor diameter 48 ma Power 750 kW

Rotor diameter 54 mi Power 1000 kW

Figure 4
The file Vindmoeller shows a dataset for the correlation between rotor diameter (measured in m) and power (measured in kW) for selected wind turbines.

a) Plot the data in a coordinate system. The relationship between rotor diameter r and power p(r) can be described by a power development of the form p(r)=b.r

b) Determine a and b. A newer wind turbine has a power of 12000 kW.

c) According to your model, what is the rotor diameter of the wind turbine? According to the wind turbine industry, doubling the rotor diameter will quadruple the effect.

d) Decide whether the claim fits your model.

Figure 4

I have finished tasks a and b

How do I do tasks c and d?

In problem c, we have to solve the equation, if I'm not mistaken

[plain ridge]

yeah basically

[silver arrow]

yes and can you give me a start?

I have the a and b values

[plain ridge]

$p_r = br$

[silver arrow]

should it not be inserted into this formula:

[wraith dagger]

b

[silver arrow]

YES!

1200 kW is b?

what is a?

r*

[plain ridge]

you're trying to find r, you have p(r)

[silver arrow]

what is r? 2?, because the radius of a circle is 2

[silver arrow]

yes how

[plain ridge]

substitute into your equation

replace p(r) with its value
and replace b with its value

[silver arrow]

okay se here

p(r) = 12000 * r

should we divide 12000 on both sides?

[plain ridge]

yes

[silver arrow]

and we get

p(r) = 12000

mhm

is this correct?

is a bit unsure

ehh

Do you know what the a value is?

I don't have the graph with me right now

@plain ridge

[plain ridge]

wait what do you mean by "a value"

[silver arrow]

don't know if you can make the graph and see it for yourself, am on mobile right now

😂

[plain ridge]

oh ok

so you should know p(r), b, and a
and you want to find r

[silver arrow]

yea, but if you make the graph on excel or another tool, you will get the b and a values

you know rotor is your x values ​​and y is power

[cedar kiln]

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[crimson sedge]

How I do this

[silver forge]

opposite sides are equal

so 6a-45 = 4a+5

[crimson sedge]

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[calm raptor]

any one can help

[cedar kiln]

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[silver arrow]

@

[silver arrow]

@plain ridge

can u helpp

[silver arrow]

.

I still need help

[cedar kiln]

@silver arrow Has your question been resolved?

[silver arrow]

.

Ehm

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[cold bridge]

where is this sin and cos coming from ???

[modern sparrow]

which specific sin and cos are you talking about? The sinx in Bgsinx, or the substitution x = sint?

[cold bridge]

the substitution

why are we substituting x=sint thats really random and I’m very lost

[modern sparrow]

it's just an integration technique, because normally we can't integrate $\frac{x^2}{\sqrt{1-x^2}}$ directly, but making the substitution $x = sin(t)$ can make the integration much simpler (well, doable)

[wraith dagger]

lgkoo

[modern sparrow]

it's a very common substitution to use x = sin(t) or x = cos(t) when you see $\sqrt{1-x^2}$ because we know that $\sqrt{1-sin^2(t)} = cos(t)$, and similar for sin(t). So it will result in nice simplification

[wraith dagger]

lgkoo

[cold bridge]

oh alright I wasn’t really familiar with this technique

so I can always use x=sint for cases like that if I can’t solve it?

[modern sparrow]

well it depends on the integral, there's no guarantee that a certain substitution will work for all integral/cases. It'll come down to try and error, and building experience with choice of substitution (if substitution is the right technique to use in the first case)

[cold bridge]

okay thank you

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[vapid dove]

just want to check if this is the right integral form for finding total charge from charge density

i think its similar to total mass but cannot remember if its a double integral or single

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[vapid dove]

<@&286206848099549185>

[crimson sedge]

that stuff is hard

[vapid dove]

yea lol

i tried to google the formula

it brought me some physics formulas which are half true ig

are mostly true

gauss's law

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[tawny drum]

Optimisation Problem: A geometry student wants to draw a rectangle inscribed in a semicircle of radius 8. If one side must be on the semicircle's diameter, what is the area of the largest rectangle that the student can draw?

[high coyote]

What have you tried?

[tawny drum]

idk how to form the first equation

i cant find any information that i can use

nvm.i solved it

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[tawny flare]

Hello If we had an integral 1/[ u^2 * (u+1) ] u^2 would be repeated two times

A/u + B/u^2 + C/u+1. But why didn't we write B as Bx+K

[hollow trail]

they treated it as a repeated linear factor rather than a quadratic factor

[tawny flare]

Oh so it's like (u)^2 similar to how (x+1)^2 is treated as linear?

[hollow trail]

yes

either way works here, it's mostly preference

[tawny flare]

I got it. Thank you

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[coral jewel]

Given a square with side length 8cm and a circle with radius 5cm, stacked on top of each other such that the circle center aligns with the square center. Calculate the volume of the object created after revolving the shape around the axis XY.

[coral jewel]

not sure how to approach this

maybe i should illustrate the shapes as functions?

[cedar kiln]

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[hidden vapor]

Well if it's possible to get upper half of function then we can use integral pi y^2 dx to calculate the volume

[distant rain]

by the way if anyone joins the stream recently calc stands for calculator i’m just using slang

[hidden vapor]

My bad

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[coral jewel]

hm actually yeah, thanks

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[hidden vapor]

Hold on, how did you get the function of the cat-ear?

[coral jewel]

its not a function per se

its just the y=4 and y=sqrt(25-x^2)

[hidden vapor]

We will take union area of both circle and square and then rotate about xy right?

That figure is itself like cat-ear

For integrating that, function isnt required?

[cedar kiln]

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[ocean parcel]

idk how to do q5 b part I and ii

[atomic tendon]

Google Combinatorics

[ocean parcel]

this is supposed to be permutations and combinations

[cedar kiln]

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[ocean parcel]

<@&286206848099549185>

[cedar kiln]

@ocean parcel Has your question been resolved?

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@ocean parcel Has your question been resolved?

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@ocean parcel Has your question been resolved?

[cedar kiln]

@ocean parcel Has your question been resolved?

[young roost]

,rotate

[wraith dagger]

Couldn't find an attached image in the last 10 messages.

May 5, 2024, 11:10 AM

[cedar kiln]

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[ocean parcel]

,close

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[restive ingot]

bro use this formula

[ocean parcel]

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[tight topaz]

Explain "expansion by C1"

[slate lintel]

c1 is column one

[tight topaz]

Ya but what did bro do

[slate lintel]

have you taken the determinant of a 3x3 matrix before?

[tight topaz]

Yes I have but not by the expansion method

I always forget that method

[slate lintel]

wait what other way is there

[tight topaz]

Cross method?

[slate lintel]

oh. sure. well anyway yeah it's the expansion method

[tight topaz]

Hm

[tight topaz]

So he basically took the determinant by expansion method?

[slate lintel]

yes

[tight topaz]

Ohhh

[slate lintel]

and two of the entries were 0

[tight topaz]

Ic uc

[tight topaz]

Oh yea that makes sense

Aight thnx

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[worthy basalt]

hell#

[worthy basalt]

hello

I have a very difficult exercise to deal with

the point P(6t^2,12t) lies on the parabola C with equation y^2=24x

the point x has y coordinate 9 and lies on the directrix of C

which I found

is -6

cuz the cartesian equation of a parabola is y^2=4ax

the tangent at the point b on C goes through point X

find the possible coordinates of B

?????

how do I solve this

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[worthy basalt]

no

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[worthy basalt]

.reopen

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✅

[earnest geyser]

Equation of tangent of parabola

$y=mc+c$ where c is $\frac{a}{m}$

[wraith dagger]

Monarch of Eternal Night

[earnest geyser]

Now just put the value of X in x,y to get m

And that's all

Btw a is from y^2=4ax

@worthy basalt

[worthy basalt]

yes

so

6

becuz its directrix

-6

Tell me first step

what do I do

I know x is (-6,9)

after???

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[paper trellis]

I am struggling to find the quotient of 3x^4 - 4x^3 + 4x - 8

[kindred venture]

this doesn't make sense. A quotient is the result of a division

what are you dividing that function by?

[paper trellis]

mb

x - 2

[kindred venture]

ok

what have you tried?

[paper trellis]

Ive put it through long division and got 0

which is good

but when I put it through the remainder formula I get 12

right now im looking for the correct remainder

if that doesnt make sense I could reword it

[kindred venture]

well you get the remainder by just substituting x = 2

which should get you 16, not 12

[paper trellis]

shouldnt it be 0 considering thats what I got from the division

[kindred venture]

that division must be incorrect

remainder theorem never fails

substituting x = 2 should always give you the correct remainder assuming you didnt make mistakes

[paper trellis]

this was my working

[kindred venture]

the last division

one sec

[paper trellis]

i see it

[kindred venture]

oh good

was about to draw on it lol

[paper trellis]

wait

nvm i dont lmao

sorry

[kindred venture]

np

[paper trellis]

carry on

[kindred venture]

you didnt bring down the 4x

it shouldve been $4x^2 + 4x - (4x^2-8x)$

[wraith dagger]

Obotron

[kindred venture]

which gives u 12x +8 for the final division

[paper trellis]

im a bit lost

I do bring it down

then I do -8x - 4x

which turns out as -4x

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@paper trellis Has your question been resolved?

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@paper trellis Has your question been resolved?

[sand cradle]

What do you mean?

-8x - 4x is -12x

[cedar kiln]

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[true jetty]

shouldnt the xz plane be:
-x+z
and not,
-x-z

also how would i be able to determine if a plane is an empty set?

[lyric narwhal]

yes, that's a typo

it does say it's a hyperbola

the x-y plane is an empty set since square numbers are always non negative

[true jetty]

so my understanding for xy plane is that (since it does not fit any conic section unless multiplied by -1 to both sides) it becomes
x+y=-1
instead of,
-x+-y=1

however sum of k number of squares is always positive so it is impossible thus empty set?

[lyric narwhal]

correct

i assume you mean x^2 and y^2 when you say x and y

[true jetty]

yes tyvm!

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[elder elm]

Is anyone familiar with graphs and spanning trees?

[lyric narwhal]

!da2a

[cedar kiln]

No need to ask “Can I ask…?” or “Does anyone know about…?”—it’s faster for everyone if you just ask your question! See https://dontasktoask.com/

[elder elm]

Sorry

I've seem to stumble upon a conceptual misunderstanding here

Here I have a complete graph

It is meant to model this: Social media platforms like Facebook rely on a network of data centers to provide seamless user experiences. These data centers need to be interconnected with redundancy to ensure that users can still access the platform even if some network links fail.

This is the setup, the nodes/vertices are the data centers at Facebook or some social media platform, and the edges are the network links in between each data center (which is again, a node/vertice).

From there, I extrapolate this

Every node is connected to every other node, implying multiple paths between any 2 nodes in graph

If path between 2 nodes fail via a direct edge AB, then the network link between AB can still be established via ACB...

But I want to introduce the topic of spanning trees

This is a spanning tree of the original graph

Isn't it the case that if data center A cannot transmit data to datacenter B directly because their direct link is compromised or something, then the spanning tree becomes a suitable alternative because we can then take a different route to relay that data from data center A to B, and thats through data center D?

[elder elm]

I'm just a little lost on how spanning trees can increase efficiency in a network example like this, relative to the complete graph.

[cedar kiln]

@elder elm Has your question been resolved?

[elder elm]

<@&286206848099549185>

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