help-27

[stark anvil]

Did permutations come first

Or combinations

Because this generalized newton formula can be proved using permutations

So I assume permutations?

[crisp niche]

yes permutations, but that also depends on constructionof the elcture, but i was showing always permutations first

that is great book:

[visual hazel]

can confirm this book slaps^

[devout snow]

@stark anvil Has your question been resolved?

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[slim sail]

Help 😅

[slim sail]

Got no idea how to start

[devout snow]

@slim sail Has your question been resolved?

[slim sail]

Actually I don’t even get the question, what does “w=/0 is purely imaginary” mean

[late orbit]

It is saying w is not equal to zero

[late orbit]

Take z=x+iy and expand the expression for w.

@slim sail

[devout snow]

@slim sail Has your question been resolved?

[slim sail]

Now what? @late orbit

[late orbit]

Now rationalize and find for imaginary.

[slim sail]

Is the question saying that w is purely imaginary?

[devout snow]

@slim sail Has your question been resolved?

[slim sail]

I rationalised and got this

I’m not sure what to do with it 😔

[heady tulip]

whats locus

[devout snow]

@slim sail Has your question been resolved?

[tender cobalt]

youll need that w is "purely imaginary"

so write w = bi = (z - 2)/(z - i)

then solve for z based on this b

[tender cobalt]

if you plot the solution of z, the locus is the geometric shape you see
so the locus of y = x^2 is a "parabola" and the locus of |z| = 1 is a "circle"

[tender cobalt]

remember that you can cross-multiply

[devout snow]

@slim sail Has your question been resolved?

[devout snow]

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[smoky sentinel]

i got this question right but i am having trouble understanding why dv/dx = .8

because i thought the derivative was the rate of change at a specific point

[smoky sentinel]

this is the question

[lunar harbor]

final answer is right and the work seems fine at first glance

[smoky sentinel]

this is my question

[lunar harbor]

oh I didn't see that part

it should be dV/dt, no?

dV/dt = 0.8, find dh/dt

[smoky sentinel]

ye i know but why is dv/dt = .8

[restive river]

Rate of change of volume = volume of water being added

[lunar harbor]

filling stuff adds volume

[shrewd thistle]

dv = the change in volume

dt = the change in time

it's stated that the change in volume is .8 as time changes

[lunar harbor]

why are there 3 people here

[shrewd thistle]

Because people aren't answering OP's question

😄

[smoky sentinel]

lol its all good

[restive river]

I didnt find anything easier

No offence everything else is just over my level

[lunar harbor]

you can have #help-45 if you want

[restive river]

Yea sure

[shrewd thistle]

Wasn't supposed to be directed at you, Eragon

[smoky sentinel]

can u explain derivative to me

[lunar harbor]

@shrewd thistle I'll let you keep this one lol

imma dip

[shrewd thistle]

So.. in short, they're the slope at a specific point on a curve

[restive river]

Somebody else took over me

[smoky sentinel]

so for my question the slope is .8 for all points?

[restive river]

I mean get a feel for the question

Its easier that way

So here we have two things

[shrewd thistle]

In this case, because the thing being changed is a uniform shape

[restive river]

Filling water at 0.8 ft³/min
Base area is 12ft²

[shrewd thistle]

But that's not a usual case

Take the slope of a line for instance...

y=(1/2) x + 3

What's the derivative of this equation?

[smoky sentinel]

1/2

[shrewd thistle]

And if you graph it, that is the slope across the entire function

[smoky sentinel]

ah

[shrewd thistle]

it's dy/dx, where y is always changing at the same rate, and x is always changing at the same rate

The bathtub has a uniform cross sectional area, so the rate of change is always going to be the same, like a stack of cards where each card is the same shape as you move up the stack

If the bathtub were an odd shape... oval, pyramidal, etc.

the same would not hold true

[smoky sentinel]

oh

i think am starting to unserstand better

i am just learning related rates so maby i just keep working on it i will understand better

thx tho u helped alot

[shrewd thistle]

np

[smoky sentinel]

.close

[devout snow]

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[devout snow]

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[cloud burrow]

Triangle ABC，vectorCM=vector2MB，vectorAQ=vectorCQ，BQ crosses AM at point N，Prove:N is the midpoint of BQ

[cloud burrow]

the rule like: BN=xBM+(1-x)BA is not allow here

😦

[devout snow]

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[true anchor]

How do i simplify this

[true anchor]

To look like this

My work so far

Going into nowhere with this

[cinder coral]

I suggest

factor 1/2 out

and then go as you're going

[true anchor]

You mean like this? @cinder coral

[cinder coral]

you did one thing wrong

it'll be 1-3x in the numerator

not -3x

[true anchor]

Oh yeah i forgot

[cinder coral]

!nosols

[devout snow]

As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.

[cinder coral]

@atomic idol

[atomic idol]

Oh oops

You didn’t see anything

[true anchor]

what happened?

[cinder coral]

nah nothing

keep going

[true anchor]

Quite stuck rn

[cinder coral]

do you see

that you can factor 2 from numerator

[atomic idol]

Same for the denominator

[cinder coral]

and we already have a 2 in the denominator

[true anchor]

I found it

[cinder coral]

good job

[atomic idol]

Congrats

[true anchor]

thanks yall

[atomic idol]

Btw I sent this picture in here because I didn’t know that you’re not allowed to give solutions lmao

[atomic idol]

That’s what happened🥲

[true anchor]

ohh

alr

really appreciate the help btw

.close

[devout snow]

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[devout snow]

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[smoky sentinel]

Is my solution correct?

[cloud burrow]

Could U type it because im using a translator plugin

thx a lot

[smoky sentinel]

Wht do u mean type it

Like type my solution?

[cloud burrow]

no,type the math problem

Sorry for my poor english

[smoky sentinel]

Ur good

But its a word problem is that ok

[cloud burrow]

ok

[smoky sentinel]

Assume that the radius r of a sphere is expanding at a rate of 90cm/min. The volume of the sphere is v = (4/3)pir^3 and its surface area is 4pir^2. Determine the rate at which the surface area is chaning with respect to time at t = 2min, assuming that r = 10 at t = 0

[cloud burrow]

f(x)=4pir^2=4pi(10+90t)^2

yes?

[smoky sentinel]

what do u mean yes?

[cloud burrow]

im not sure if i understand the problem correctly

[smoky sentinel]

oh

maby call another helper?

am also very confused

[cloud burrow]

first we need the rate at which the surface area is chaning

so ignore the volume

[smoky sentinel]

ok

[cloud burrow]

then from"Assume that the radius r of a sphere is expanding at a rate of 90cm/min"and"assuming that r = 10 at t = 0"

we know that r=10+90t

[smoky sentinel]

yes this makes sense

so do i just do the derivative of s = 4pir^2

[cloud burrow]

yes

combine the two

and we get 4pi(10+90t)^2

[smoky sentinel]

ok i will try

[cloud burrow]

Tell me if ur all right

waiting for u

[smoky sentinel]

i got 144400pi

[cloud burrow]

no where the t?

[smoky sentinel]

t = 2?

[cloud burrow]

no thats the last step because we are looking for rate

so just dont replace the t with 2

[smoky sentinel]

oh so i should find the derivative of 4pi(10+90t)^2 first then replce t with 2

[cloud burrow]

yes

[smoky sentinel]

ok

[cloud burrow]

brilliant

[smoky sentinel]

THX YOU

i got it correct

[cloud burrow]

yeah

[smoky sentinel]

.close

[devout snow]

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[devout snow]

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[slim sail]

So bi = (z-2)/(z-1)

Not sure how to continue

[devout snow]

@slim sail Has your question been resolved?

[tender lotus]

z is the complex number that represents the point M

[woven radish]

Adam Chebil

[slim sail]

Oh ok that’s an interesting way of putting it

I wish I knew what to do next sorry

[woven radish]

Adam Chebil

Adam Chebil

[tender lotus]

C is the circle with diameter = [AB]

also z≠i cause for w to be defined the denominator must be ≠ 0

so M≠(0,1)

[devout snow]

@slim sail Has your question been resolved?

[slim sail]

Why does it imply that they are perpendicular?

[slim sail]

Sorry I meant to ping you @tender lotus

[tender lotus]

Honestly idk why that is true 😅
I was too lazy to understand the proof behind it back in high school

But there's another method if u want

That doesn't require that "theorem"

[slim sail]

Yeah maybe that’s a good idea, I’ve never seen that theorem before

[tender lotus]

so let z=x+iy and forget about w

you don't have to replace w with anything

[slim sail]

Yeah

[woven radish]

Adam Chebil

[tender lotus]

Replace z with that

And write the fraction in the form : X + iY

Here's why we're doing all of this:

• you were given : w is imaginary
• one way to solve this : is to write w in the form X + iY and then set X=0 and Y≠0

[slim sail]

Yeah got it

But now the question remains how do I find the locus of z from this

[tender lotus]

You set Re(w) = 0

[slim sail]

Yeah okay, but then what does that equal? I’ve got an expression, Im(w), but what do I do with that?

[tender lotus]

Wdym ?

Just set Re(w) = 0

You'll get an equation of a circle

Then exclude the points that verify Im(w) = 0

Cause w≠0

Remember that x and y are the coordinates of z not w

W is the whole expression

[slim sail]

Ohh yeah thanks so much @tender lotus

.close

[devout snow]

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[devout snow]

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[paper sentinel]

Hi :( can someone help me with understanding this type of question 🥲 I have no idea what any of this means ** I am only talking about questions 29-32** I have no idea what I’m doing. I hope I’m doing this asking for help thing right 😅

[paper sentinel]

My book also isn’t giving any good examples on this either so I have no idea

[radiant anvil]

the pairs (x,y) such that, x and y are in Z (integers), y=x-3 and -4<=x<=5

putting 29 to words

[paper sentinel]

Where is 29 coming from what

[winter torrent]

problem number 29

[paper sentinel]

Oh

[whole oyster]

I believe that it's asking for domain and range

[paper sentinel]

How are u supposed to know what Z is if ur not given numbers

[radiant anvil]

Z is all integers, ..., -3, -2, -1, 0, 1, 2, 3, ...

[paper sentinel]

So what does like the -4< thing mean

Is that just like the numbers are from -4 to 5?

[radiant anvil]

yes, in this case, x can take on the values in, {-4, -3, -2, -1, 0, 1, 2, 3, 4, 5}

[whole oyster]

In this case, would we fromulate a graph under x and y to check for repeating x variables?

[paper sentinel]

So it only tells you x? Like u got one of them but how would u know what Y is or are they the same -4 to 5

[winter torrent]

there is no such restriction on y, but we know that y = x - 3

[paper sentinel]

Wait so like I got the domain and I need the range so would I just put the integers into x to find Y or would that not work?

[whole oyster]

It is saying that x is greater than or equal to -4 and also less than or equal to 5

So y= (greater than or equal to -4, and less than or equal to 5) -3

[paper sentinel]

So there’s no Y at all ?

Where’s the range then

D: I’m so confused

[winter torrent]

essentially yes this will work

[paper sentinel]

So like it also says say if it’s a function but like what is a function and how do I know if it is one my professor never said he was just like “it’s also a function” and then moved on 🥲

[winter torrent]

is this picture taken from a textbook?

[paper sentinel]

Yeah, my test book isn’t really giving me anything like at all

So like if x doesn’t repeat it’s self it’s a function ? Like every x value is different

[whole oyster]

Yes that means its a function

[paper sentinel]

oh

Well u kinda answered the question tho

Unless ur wrong

😭

[whole oyster]

Were you able to find the domain and range?

[paper sentinel]

I think?

Idk I haven’t looked at the answer yet

Hold onn

Okay i got it right but like -4 to 5

Ur repeating 1 2 3 and 4 but some are just negative

How is that a function cause ur repeating x

Their the same numbers

Technically

[whole oyster]

I think it's saying that x can be between -4 and 5 that's a limit to the function, not necessarily that is defined specifically to all variables in between those limits

so domain of x = (-oo, -4] U [-4, 5] U [5,oo)

[paper sentinel]

Istg that literally says -4 to 5 either way even if you can go more numbers you would still repeat numbers just with a negative sign so how is that a function if functions are when x is not repeating its self

LMAO I don’t get it

Cause [-4,-3,-2,-1,0,1,2,3,4,5]

They are repeating are they not??

😭

[whole oyster]

So if x=-4 then y = (-4)-3

and if x = 5 then y = (5)-3

[paper sentinel]

Yeah

5 doesn’t repeat

But 1234 do

It’s just negatives

Am I wrong

What

[whole oyster]

Why would they repeat?

[paper sentinel]

I’m so confused

Wdym

[whole oyster]

You're plugging in a variable between -4 and 5 to find y

[paper sentinel]

They show up twice

[whole oyster]

-1 is not the same as 1

[paper sentinel]

Oh

[whole oyster]

different numbers entirely

[paper sentinel]

Oh…

[whole oyster]

Think of it like a number line

-1 is before 0

1 is after

different points of the line

[paper sentinel]

So it wouldn’t repeat itself

[whole oyster]

So your domain for X is within -4, -3, -2, -1, 0, 1, 2, 3, 4, 5 which are all different variables

And Y is simply X - 3

[paper sentinel]

So Y doesn’t actually have a number

[whole oyster]

Not until you plug in X

[paper sentinel]

Oh

[whole oyster]

if X = 0 then Y = -3

[paper sentinel]

Ohh

Okay

[whole oyster]

Your range is all the possible values of Y, so -4 through 5 minus 3

so that's -7 to 2

[paper sentinel]

So I’m only plugging in -4 and 5

[whole oyster]

You can plug in every variable between those two points

[paper sentinel]

And then it’s kinda not hard to figure out what is in between

[whole oyster]

Since its less than or equal to, it makes it a square bracket

and greater than or equal to also makes the number included (square bracket)

So your X is defined as well as your Y we simply plug in the limitations to Y to find the function

[whole oyster]

^

Your X is your independent variable while Y is dependent on X. Y = F(x)

X = input

[paper sentinel]

Okay hold on

Cause I get this

[whole oyster]

If it were -4 < X < 5 without (equal to) that makes your X infinite both towards negative infinity and positive infinity

[paper sentinel]

Bro I’m gonna get such a good mark on this test now

If I don’t I might just cry a little

[whole oyster]

I believe in you !

I think you understand it now

[paper sentinel]

I think so

I hope

[whole oyster]

It's not very complex

[paper sentinel]

It’s not but like it looks messed

[whole oyster]

It looks complex with the epsilon Z

Which just means that it's in the grouping of Z which is basically irrelevant to what we're trying to figure out

[paper sentinel]

It’s not as like organized as like the ones above in the picture cause it just shows u the points

[whole oyster]

Now we know what this question is asking and how they've made it look more complex than it really is

[paper sentinel]

What are natural numbers

Like

[whole oyster]

That's one thing about math is that it can be over complicated while the solution isn't as complex as we think

[paper sentinel]

Are they just numbers with no negative signs?

[whole oyster]

Yes natural is beyond 0

[paper sentinel]

So what would R be?

[whole oyster]

Are you referring to questions 31 and 32?

[paper sentinel]

Yeah

Specifically 31

I can prob figure it out if I know what R means

[whole oyster]

So basically ε is our expectation of a random input

R is (x,y)

[paper sentinel]

Wdym

3x+y=2

So I would just rearrange that

To like y=-3x+2

And then just put in whatever x is for this question

Or can be I guess

Cause it says x is -3 to 3

So it would be like y=-3(-3) +2

Right?

Or no

[whole oyster]

Give me a minute to check

Yep y=-3x+2

And your domain for x is (-oo,-3)U(-3,3)U(3,oo)

[whole oyster]

R is your X and Y, so input and output together

So for this scenario, (-3,11) = R

[paper sentinel]

Wdym

So it would be like

[whole oyster]

or (-2,8) = R

[paper sentinel]

-3y= -3(-3) +2

Huhh

Wouldn’t u need y by itself

[whole oyster]

Take out that -3 cause you moved it to the other side

[paper sentinel]

Wdym

[whole oyster]

y = -3x+2

[paper sentinel]

Didn’t u just say it’s also y

[whole oyster]

Correct

[whole oyster]

No, when we plug in -3 to X Y = 11

y = -3(-3)+2

[paper sentinel]

wth is XY

[whole oyster]

It's just a point on your graph

[paper sentinel]

oh

[whole oyster]

X is the X axis

Y is your Y axis

[paper sentinel]

Yeah

[whole oyster]

So if your point above X is at -3 X then your Y will = 11

[paper sentinel]

There’s only one question where we have to graph it

I’ve taken this test before I just failed it 💀

[whole oyster]

I think you'll do much better this time

basically a function is just a line on a graph

[paper sentinel]

I sure hope I do 😭

[whole oyster]

X is your input to the graph, while Y is your output

[paper sentinel]

I think last time I got 1/45

LMFAO

[whole oyster]

haha been there done that

trial and error right?

[paper sentinel]

Truee

[whole oyster]

glad you're taking the time to fully understand it, lots of people just fake it through math and it catches up to them

[paper sentinel]

So ignoring the XY thing you said

[paper sentinel]

I need this math to be a certain grade to even be considered to get into my paramedic program so I can’t even fake it 🥲

[whole oyster]

You got this!

I am studying computer science so I'll have to do a lotttt of math before I graduate

[paper sentinel]

So when I do this my answer should be 11?

[whole oyster]

(-3,11) is your point on the graph for that input

(x,y)

[paper sentinel]

Where is 11 coming from tho

[whole oyster]

Our equation Y=-3X+2

That's our function

[paper sentinel]

Yeah

[whole oyster]

also can be seen as f(x)=-3x+2

[paper sentinel]

No

Other way

I don’t like this way

LMAO

[whole oyster]

hahaha you'll want to relate your Y = mx+b to f(x)=mx+b

it's interchangable

f(x) just means function of X or output of X

[paper sentinel]

He only shows us y=mxtb

[whole oyster]

Yeah that's okay, just another way to look at it as a function

[paper sentinel]

I don’t think he would ever show us that he has never even mentioned the fact that it can be shown that way

[whole oyster]

Don't worry about that

Focus on what you've got

[paper sentinel]

Yes

[whole oyster]

I'm glad you've figured it out!

[paper sentinel]

I HAVENT

[whole oyster]

WDYM lol

[paper sentinel]

where did the 11 come from

😭

U didn’t tell me

[whole oyster]

Y = -3x + 2

Replace X with -3

then solve

[paper sentinel]

So it’s the answer

U should get 11

When I solve it

[whole oyster]

Yes you get Y = 11

[paper sentinel]

Hold on

Okay

I got 11

[whole oyster]

so that means (-3,11) when X = -3

(x,y)

[paper sentinel]

So that’s it?

Like that’s all

[whole oyster]

Yep, it's a function, what does the answer say?

[paper sentinel]

Which one did we do?

[whole oyster]

That one was 32 I believe that we were just discussing

It also says it wants us to find the domain and range

[paper sentinel]

Oh I don’t have the answers to 32

[whole oyster]

Oh shoot

[paper sentinel]

I only have odd answers

😭

[whole oyster]

Thats whack

[paper sentinel]

Yeah :/

[whole oyster]

Do you want to go through 31?

[paper sentinel]

I think I got it

Gonna try 31 my self later so I can actually see my self doing it

[whole oyster]

Range = All possible variables of Y in interval notation

[paper sentinel]

Yes

[whole oyster]

Domain = All possible variables of X in interval notation

Which usually has multiple intervals

[paper sentinel]

And negative and positive are not the same numbers

😭

[whole oyster]

Correct

And positive numbers are natural

[paper sentinel]

So would they ever ask for unnatural numbers

Or is that not a thing

[whole oyster]

Yes however with the epsilon symbol (the little E) I believe that is limiting us to positive numbers

Notice how it shows ---> ε R

Meaning we want our function above 0

[paper sentinel]

Yeah

So if there’s no e

[whole oyster]

Because none of the random inputs will be below 1

[paper sentinel]

What does that mean

[whole oyster]

If there's no ε then we aren't limited to positives

I want you to ask your professor about that symbol asap because it becomes more important later on

It's called epsilon

For now, it's arbitrary because we're just checking if it's a function and finding the domain and range

[paper sentinel]

That’s all we do with those types of things

We don’t actually ever see that type of thing again

[whole oyster]

Yeah, so you're good (:

[paper sentinel]

Omg slay

[whole oyster]

Okay, I'm going to close this channel now since I think we've solved your problems

Feel free to shoot me a DM with any math questions anytime

[paper sentinel]

Yeah we did idk how to do it

[whole oyster]

.close

type that

[paper sentinel]

.close

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[devout snow]

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[strange arch]

for the taylor series' of sin and cos

[strange arch]

can I use them as alternating boundaries for the functions themselves?

e.g. sin(x):

sin(x) < x

sin(x) > x - x³/3!

sin(x) < x - x³/3! + x^5/5!

sin(x) > ...

[half birch]

u trying to use the squeeze theorem?

[strange arch]

not necessarily, just wondering whether it applies for sin & cos :)

[half birch]

alr

but isnt it based on guesswork?

[strange arch]

that's why I'm asking if this pattern applies for arbitrary degrees of the taylor polynom

[half birch]

hmm

[heady tulip]

sin(0) < 0 is false

[strange arch]

sry for x>0

otherwise it's always 0 = 0 ofc

[heady tulip]

so only x > 0

[strange arch]

ys

is there some proof that shows that it applies

[heady tulip]

idk

[strange arch]

ahh

yea it's strictly decreasing

and increasing from the other side

so they're sandwiching sin and cos

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[inner ibex]

excuse me

[devout snow]

Please don't occupy multiple help channels.

[inner ibex]

can i make a circle 100x100?

or is thato nly squares

[tender cobalt]

what

[twilit comet]

??

if you have a rectangle/square

and it's 100x100

that means it has a length of 100 and width of 100

[inner ibex]

fark for my school thingy i had to make a 100x100 coaster for a drink

and i made it a circle

[twilit comet]

a rectangle that's 56x7 has a length of 56 and width of 7

[tender cobalt]

this question is getting screenshotted and going in the nonsence folder

[twilit comet]

...

[twilit comet]

fr

[inner ibex]

NOOOOO

[tender cobalt]

what shape is a 2x4 piece of plywood

[twilit comet]

RIP

[tender cobalt]

aint no way its a cylinder

[twilit comet]

nah bruh it's a triangle

[tender cobalt]

why is it vertical

[twilit comet]

F

[inner ibex]

i had to make a 100x100 coaster and i did this LOL

[tender cobalt]

this just means to say the dimensions can be 100x100

[twilit comet]

"The Geometric Grind" 💀

[twilit comet]

nice dam

[tender cobalt]

Id do a circular coaster too

[inner ibex]

oh what

so it can be 100x100

[tender cobalt]

its just that its diameter is 100

[inner ibex]

wait so how would i describe it

[tender cobalt]

think of it like the width and height of an image

[inner ibex]

cause im doing a report rn

[tender cobalt]

just say you chose to use the 100x100 dimensions to draw a circle

[tender cobalt]

if that doesnt work, use a square and then make the background around the circle plain

[inner ibex]

okok

thank u

[tender cobalt]

np

[inner ibex]

.close

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[inner yarrow]

can someone explain the difference in approaches for part c of each question, why did we treat X as exact same for one but not for other?

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[cursive stirrup]

How do iget the answer guys plz help

[lethal oxide]

Are you asking how to find x?

[cursive stirrup]

yes

adjacent side

Im stuck i dorgot

[lethal oxide]

Tan(75˚) is a number, but I assume that is what confused you

[cursive stirrup]

yes how do i get the x

[lethal oxide]

Do you agree that these triangles have the same angles?

[cursive stirrup]

yes

[lethal oxide]

So that means the ratio for the adjacent side and the opposite side is the same too

Let's say that angle C and angle F are 75˚

Their tan values will always be the same

So we usually have a big list of all the angle ratios somewhere of a triangle

If you look it up, tan(75˚) = 3.73205080757

You can't really get this value by hand, you'll need a calculator unfortunately since it has that stored in its memory

[cursive stirrup]

THANK YOU SOO MUCH

[lethal oxide]

No problem!

I do what I can <3

.close

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[signal crag]

can you solve for the poi between a plane and a line using matrices

[signal crag]

i'm trying to construct some sort of matrix but cant visualise the equations

i know for the plane we must use the cartesian form

but for the line in r3, i cant think of what equations i would use

[devout snow]

@signal crag Has your question been resolved?

[acoustic leaf]

in this case, it may be more useful to think of a line as the intersection of 2 planes

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[split dune]

Can someone explain this solution? I proved it by moving everything to the left and then by definition+example using a-b>0, but this is a significantly shorter proof.

[radiant dune]

Are you aware of how quadratic polynomial graphs look like ?

[split dune]

Well yeah, parabolas.

[radiant dune]

And their dependency on the leading coefficient?

[split dune]

yep

concave down and allat is easy but i just dont unerstand ho they used it for the proof

[radiant dune]

When it's concave down, you get the maximum value of the quadratic polynomial at the vertex

And all other points are lower than that point

It's value decreases in both directions

For example if a downward opening parabola has vertex at x=5

Then as you go from x=5 to x =infinity, f(x) will keep decreasing

And as you go in the reverse direction, i.e. start from 5 and go towards negative infinity

The same will happen

This can be easily visualized if you draw the graph once

[split dune]

I'm aware of these properties, but a concave down parabola doesn't have minimuums, so what do they mean here?

[radiant dune]

When you restrict the domain

It will have a minimum value then

In this case they're restricting k from 1 to n

[split dune]

So 1 & n would become minimums for this graph

[radiant dune]

So you only have access to the graph from x=1 to x=n

[radiant dune]

One of the two

That's something you have to manually check

Like here

Let's say we restrict the domain

From the red one to the blue one

Then either of the two will give me the minimum value

But since the problem they've given you is quite symmetric, it looks like this:

Restricting the domain is just like chopping off the remaining graph

[split dune]

so in the case we restrict the graph

would the minimums become the restrictions?

[radiant dune]

Yes because they've told you to by saying 'Suppose 1≤k≤n

[split dune]

So 1 & n would be the minimums

Okay I wasn't aware of that property seems like

[radiant dune]

Points of minima

It isn't a property

It's just analysing your graph

They're are asking you how low can f(k) get if k varies between 1 and n

[split dune]

I wasn't aware restricting the graph would give you both minimums and maximums*

[radiant dune]

And you know how the graph looks hence the end points should be checked because all points between the end points will only have a higher value

[radiant dune]

As I've illustrated here

All points between the red and blue one have a higher y coordinate

Than the minimum

[split dune]

yep

would the minima be both restrictions, or the x point with a smaller output?

[radiant dune]

The one with the smaller output

[split dune]

subbing in either 1 or n for k gives you n regardless

therefore n is greater than or equal to n

QED

[radiant dune]

Yes

[split dune]

I proved it by moving everything to the left and then factorising and some long shit

Worked but not efficient

thank you very much

[radiant dune]

You're welcome

[split dune]

.close

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[verbal grail]

Any of these calculus subtopics allow u to make exponential functions?

[sage burrow]

can you elaborate a little bit? at the moment the question does not mak e any sense (for me).

[verbal grail]

do the calculations and formulae in these subtopics of calculus give you the ability to make exponential functions @sage burrow

I want to make an exponential function to model population growth

but not sure if I can use calculus to do so

and this is what we have learnt so far, so I'm checking if I'm just too dumb to realise I can apply what I've learnt to an exponential data set

[sage burrow]

sorry, i do not understand. y=e^x is an exponantial function. i "made" it. i didnt need a chain rule, a second derivative, stationary points, or anything else.

[verbal grail]

damn bro i think this may be true then

cheers

[cobalt void]

What exactly do you want to do?

Like population grows 10% every year?

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@verbal grail Has your question been resolved?

[devout snow]

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[restive river]

Segment MN is the midline of triangle ABC. Line MD intersects the continuation of side AC at point D, and side BC at point E (see figure).
1.What fraction of the area of triangle ABC is the area of quadrilateral AMNC?
2.Prove that Triangle MNE ~ triangle DCE.

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[sinful robin]

Integrate (1-x)^1/2

[sinful robin]

What am I doing incorrectly

[cloud burrow]

sqr(1-x?)

[sinful robin]

Correct answer is -2/3(1-x)^3/2

[sinful robin]

Well that's the correct part... 4/9 is supposed to be 2/3

[dense lynx]

in 2nd line the -3/2 is supposed to vanish

but u write it as 1/(3/2) in 3rd line

[sinful robin]

According to the formula in the top right it should be n+1

Which is 1/2 + 1 = 3/2

that's where it comes from

Like this

[dense lynx]

3/2 divided by 3/2 is 1

transition from 1st line to 2nd is wrong

[sinful robin]

How is it supposed to be done

I thought the 3/2 in the first line should dissapear

<@&286206848099549185>

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@sinful robin Has your question been resolved?

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[still crown]

how to solve this

[cloud burrow]

draw it

[shrewd thistle]

Have you drawn it out yet?

[cloud burrow]

draw triangle ABC on your paper

[still crown]

ok

[cloud burrow]

then 1:50000 means the actual length=50000*paper length

[still crown]

ok

[shrewd thistle]

Label your sides and find the length of the diagonal in your paper triangle (unless you recognize the special right triangle you're looking at)

then it's just a matter of multiplying, like Cuno said

[still crown]

diagonal is 10cm

[shrewd thistle]

just remember, it asks for the final length in km, not cm

hence, the answer you have penciled in 😉

[still crown]

its 5km

[shrewd thistle]

Do the same for the sides in part B for your area

[still crown]

but

the area of the rectangular plot is '

48 right?

[shrewd thistle]

you're multiplying 6 * 8

[still crown]

yes

[shrewd thistle]

You're skipping the conversion by a factor of 50,000

[still crown]

i mean the rectangle on the paper

model

[shrewd thistle]

Sure, the rectangle on your paper is

but the area of the real plot of land is not

[still crown]

12 right

[cloud burrow]

yes

maybe correct,i didnt calculate it

[shrewd thistle]

that's what I got

[still crown]

i got the answer, thank you for your assistance

.close

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[solar latch]

Hi. I have the probability of a team winning, team losing and game drawing in a match. It has come to my attention that the game will go on and there won't be a draw. How can I find the new probability of team winning without the game drawing. what is the formula/approach I should use for this?

[solar latch]

Is anyone there 😔

[devout snow]

@solar latch Has your question been resolved?

[devout snow]

@solar latch Has your question been resolved?

[solar latch]

<@&286206848099549185>

[short bone]

Does it mention how many games are played?

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@solar latch Has your question been resolved?

[devout snow]

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[restive river]

binomial theorem

[devout snow]

Please don't occupy multiple help channels.

[restive river]

i am not

i dont see my previous channel

[viral sluice]

close this one

and go to #help-17

@restive river

[restive river]

how to close?

[viral sluice]

write ".close"

[restive river]

.close

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[dense lynx]

this seems like trick question lol

first try to find restrictions on values of c that work

[molten pier]

hm

So from what I have so far on this question, 0 and 2 should be one of the restrictions, right?

wait no not 2

Uh

Nevermind, i'm just completely stumped now

[orchid laurel]

Can I ask what are you studying?

[molten pier]

AP Calc AB rn

[orchid laurel]

ok, just curious because it looks very interesting to me

I say this as a noob, but what if you differentiate it?

then you will see where it's not continuous

and those would be your restrictions?

[molten pier]

Yeah I've differentiated and I got 0 and 2, but my smarter friend says the way I found it isn't correct or something along those lines

[orchid laurel]

I'm looking at how this is differentiated and it's not familiar to me so I don't know

this is a multivariable function right?

like f(x,y)=

https://www.youtube.com/watch?v=ZI8jF5A-VWc

[molten pier]

Uh

ye

[orchid laurel]

you will understand this better than me

should take a look

[molten pier]

Yeah i understand this, I'm just not really registering how (c,c) fits into the equation

[orchid laurel]

what is the derivative?

[molten pier]

I don't know, I'm probably just having my brain not click

It's uh

[dense lynx]

for now just assume c is a positive number

[molten pier]

y ' = xy lny - (y ^ 2) divided by xy ln x - (x ^ 2)

[dense lynx]

the point ur given is (c, c) which is a point on the graph where x = y = c

what do u notice about these points c

[molten pier]

They're all positive numbers right?

[dense lynx]

ye

[molten pier]

Hm

[dense lynx]

but what can u say about which positive numbers which and which don't

[molten pier]

which positive numbers which?

[dense lynx]

which positive numbers are possible values of c

[molten pier]

uhhh

i'm not quite sure

[dense lynx]

well

we have x^y = y^x

this becomes c^c = c^c

[molten pier]

mhmm

[dense lynx]

it immediately follows that all positive values of c work

because c^c = c^c is just an equality

[devout snow]

@molten pier Has your question been resolved?

[molten pier]

yeah i'm still stumped on this question, I'm not really getting the reasoning behind this problem

<@&286206848099549185>

[chrome valley]

ya

@devout snow

yaya

[molten pier]

mhmm

So we would end up at something like

xy lny - (y ^ 2) / xy lnx - (x ^ 2)

before putting in (c, c) right?

[woven radish]

BotToes

[molten pier]

Hm

Ohhh

I see where I did it wrong

So if I understand correctly, I'd get

ln c + c (1 / c) y' = (1 / c) c + ln c

right?

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[dim horizon]

This is the problem I am struggling with

my work is as follows, give me a minute to translate from paper to paint (*-64 in the solution is meant to be -64u)

I am fairly sure my work is fine in this, I checked with a video that goes over the same problem but my solution differs from that in an integral calculator (I used symbolab to check my work)

This is the solution of symbolab

oh nvm

I think I figured it out

finally

holy crap

took me 3 hours for this one

.close

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[devout snow]

@sweet cloak Has your question been resolved?

[sweet cloak]

<@&286206848099549185>

on blud no ones gonna help me?

.close

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[glacial python]

I think I can prove the first implication by saying a projection is the closest point and since x is in U the projection would be x

Not sure how to prove the other one tho

[magic pine]

what is your definition of proj_U(x)

[stone stump]

the projection being the "closest" point is a bit imprecise and handwavy. I imagine this is in the context of a linear algebra class and not analysis

[glacial python]

ye its lin alg

[glacial python]

was what i said above be enough or nah

the closes thing

[magic pine]

no

they must have given you an exact definition of proj_U(x)

[magic pine]

use the definition they gave you

[glacial python]

im guessing this is the one

[magic pine]

yes

[glacial python]

lmao im stuck could u help me out a lil

[magic pine]

where are you at

[glacial python]

determining how these result in x

[magic pine]

let x be in U. let {f_1, ..., f_m} be an orthogonal basis for U
what can you say about x in terms of that basis

[glacial python]

X can be written as a linear combination of the vectors f1….fm

[magic pine]

it can be written uniquely

as a linear combination of the basis vectors

[glacial python]

Alr yeah

[magic pine]

note that proj_U(x) is a linear combination of the basis vectors

[glacial python]

Yes

But how are we sure that this linear combination is the one that corresponds to the unique one for x

[magic pine]

ok im not sure of the actual proof, but i think there is a roundabout way to do that

[glacial python]

What is it

[magic pine]

you can note that proj_U^2(x) = proj_U(proj_U(x)) = proj_U(x)
we then have that proj_U^2(x) - proj_U(x) = 0, and so
proj_U(proj_U(x) - x) = 0, so proj_U(x) - x is in the kernel of proj_U
either proj_U(x) - x = 0, or proj_U(x) - x is orthogonal to every f_i (why?)

so we basically consider that proj_U(x) - x is orthogonal to every f_i, but this must mean that proj_U(x) = 0, so either x = 0, or x is not a linear combination of the basis vectors, a contradiction

thus proj_U(x) - x = 0, and proj_U(x) = x

[glacial python]

Mmmm ok

I think it might be easier to say since x is in U the projection will be x cause the projection finds the closest vector

Tricky part is proving the reverse implication

[magic pine]

the problem is that this isn't rigorous

you started out by saying that and we said it wasnt rigorous enough

[magic pine]

the reverse is the easy part and what i was hinting at above

proj_U(x) is a linear combination of basis vectors for U

[glacial python]

that is true ngl

[glacial python]

alr

[magic pine]

so x is in the span of the basis of U

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[hard cosmos]

I dont know where to begin

[hard cosmos]

Sorry, I didn't completely clarify my confusion. The first part, before "a" doesn't make sense to me

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[autumn fjord]

<@&268886789983436800>

[weak cove]

this was unnecessary

[winter torrent]

you've been here for over a year. you know that math requires you to put in effort, and to me it looks like you're refusing to do that and are just waiting for people to spoon feed you everything. please reëvaluate this opinion.

[hard cosmos]

Sorry for being disrespectful @/moosey (didn’t want to ping you) and not realizing you are here to help. I also realize that I’m in the wrong for getting upset at someone who volunteers their time when they could be doing other things, and for me it’s easy to forget that. I became almost — egotistical to the idea that I deserve some type of help when this server could just not exist and I wouldn’t receive any help. Once again, I apologize Moosey.

Thank you to those who held me accountable.

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[brave latch]

Hello

[brave latch]

Idk if the other part is wrong as in the percent or the other part of the function that I put

And I’m struggling cuz I was in the hospital for a while so i could not attend

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@brave latch Has your question been resolved?

[brave latch]

Absolutely not

He-he-hell nah

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[coral wraith]

i have the idea for how to solve this, but im not sure how to make it convenient

[coral wraith]

each layer is a triangular number

its a series of triangular numbers

but how could i represent the sum of n triangular numbers

because the sum 30 triangular numbers is a tedious number to calculate

[soft umbra]

n(1+n)/2 represents the numbers of balls of a certain layer, do you agree?

[coral wraith]

yes

[soft umbra]

set sequence <an> = n(1+n)/2

<b> = sigma an

If you have three layers of balls, you'll get three first layer(which is 1 ball)

[coral wraith]

wdym

[soft umbra]

By this logic, you'll get 1* 3 + 2* 2 + 3 * 1 balls

[soft umbra]

how many balls will you get with three layers?

[coral wraith]

1+3+6

=10

[coral wraith]

oh i think i understand this

[coral wraith]

how does thsi help?

[coral wraith]

@soft umbra how will this be easier to solve when n=30

it would be something like

how does it scale

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[wooden narwhal]

How do I put polynomials into the quadratic formula

[wooden narwhal]

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[hardy condor]

im on question 3

my status is 1

<@&286206848099549185>

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[reef swift]

How would I find the limit assuming P_0>M and P_0, M, alpha are some constant

[severe nest]

This looks like a job for L'Hospital

[reef swift]

Would I take the lim outside the ln or inside?

[green kelp]

You can take it Inside, just divide P in numberator and denominator

[severe nest]

Neither. After you apply L'Hospital the ln goes away

[green kelp]

you'll have M+P0/P numerator, p0/p approaches zero as p is going infinite so just M remains in numerator

Similarly u do it for denom

Don't need l hospital

[reef swift]

Ah

Thank you

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[eternal musk]

is. (0,0) a local max/min

[vapid nest]

(x,f(x)) is a local maximum if there is an interval (a,b) with a<x<b and f(x)≥f(z) for every z in both (a,b) and the domain of f

[copper hatch]

Have you tested the concavity?

You can use this by testing the gradient on each side or using the double derivative.

[coral pecan]

^

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[uncut yew]

https://media.discordapp.net/attachments/1173766169048666194/1213723051703795792/IMG_9667.jpg?ex=65f682ad&is=65e40dad&hm=da143f661c46e765c006700cb2e2637f7cdf5c00fe75656241316da9494e2e0c&=&format=webp&width=1672&height=1364

[uncut yew]

How do i do c?

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@uncut yew Has your question been resolved?

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@uncut yew Has your question been resolved?

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[eager cairn]

<@&286206848099549185>

i did the second part not the firsrt part

[chrome pilot]

look at the triangle DFB

that's all

forget point A

[sweet ridge]

are you familiar with thales' theorem?

[eager cairn]

no

[eager cairn]

then what?

[chrome pilot]

wait, did you know thales' theorem?

[eager cairn]

no

[chrome pilot]

lolllllllllllll

[eager cairn]

?

[eager cairn]

OH NVM I DO KNOW

i didnt know the name