help-33

[astral kite]

it's just the slope

[mellow frigate]

How do I calculate it?

[astral kite]

find the eq for the parabola

[mellow frigate]

How?

It’s not given

[astral kite]

y=ax^2+bx+c

so all we have to do is find 2 poinst on the parabola

[mellow frigate]

But the numbers are not given

Wait really?

[astral kite]

find the 0'5

[mellow frigate]

So (-5,0) and (3,0) ?

The middle?

-1?

[astral kite]

find the x-int

and plug it in

[mellow frigate]

I’m so confused

[mellow frigate]

These are the x intercepts

And vertex -1

What do I plug where?

[astral kite]

oh

oops

try

(-5,0)

and (7,-6)

plug them in

[mellow frigate]

How did you get 7?

[astral kite]

looked at graph

[mellow frigate]

Ohh

And how do I plug them in

What goes to a b and c

[astral kite]

nothing

y=ax^2+bx+c

plug in x as 7

and y as -6

[mellow frigate]

Ohhh

[astral kite]

lmk what u get

[mellow frigate]

So 0=-5^2+(-5)+0

[astral kite]

yeah

[mellow frigate]

25-5=20

20?

[astral kite]

no

[mellow frigate]

??

[astral kite]

25a-5b=0

then make the second equation

49a-7b=-6

then solve

(sorry a bit tedious)

[mellow frigate]

I’ve never seen this before

Do I use the quadratic formula now

[astral kite]

no

solve the 2 variable eq

[mellow frigate]

How??

Idk any of this

[astral kite]

ok i'll solve 4 u

[mellow frigate]

But I want to learn🥲

[astral kite]

ok

basically

you make one equal to the other

so 5a=b

[mellow frigate]

What does that tell us

[astral kite]

that every b is 5a

so

49a-7(5a)=-6

[mellow frigate]

14a=-6?

[astral kite]

yeah

a=-3/8

then we plug it in

5*-3/8=b

[mellow frigate]

Plug it in what😭

[astral kite]

b=-15/8

so our equation si

*is

actually we forgot the c

😭

anyway

basically we find the eq of the parabola using the points and solving an equation

and then we derivitave it

sorry i gtg

[mellow frigate]

How do you even know what’s c

I’m so confused

This seems way more complicated than anything we’ve done there must be an easier way

[mellow frigate]

<@&286206848099549185>

[tight kite]

wheres the original question

[mellow frigate]

well is this correct?

and im stuck here

[tight kite]

d/dx (1/2)x - 3 = 1/2 so seems right

[mellow frigate]

I just took two points and used the slope formula

[mellow frigate]

Idk what’s this😅

Derivative with respect to x?

[tight kite]

you have the roots and the vextex of the parabola, you can find the equation for the parabola with this information

[mellow frigate]

How so?

[tight kite]

the derivative of the equation of the line

[mellow frigate]

It’s

So (-5,0) and (3,0) ?
And-1

[tight kite]

make a system of equations

[mellow frigate]

How do you know what’s the equation?

[mellow frigate]

Wut

What’s that

Wdym by system?😅

[tight kite]

rise over run you find the slope, and y interscept gives the constant term

[mellow frigate]

Ohh

[tight kite]

the equation for a parabola is ax² + bx + c, you are given 3 points of the parabola, which means 3 equations for 3 variables (a,b,c)

[mellow frigate]

Oh it’s what the other guy attempted to do I have no idea what he did or how that works tbh

What do you plug where

I just know how to use the quadratic formula

Is c the vertex? Which point goes to a and b

[tight kite]

the vertex of a parabola happens when x = -b/2a

and y is at (c - b²/4a)

[mellow frigate]

I have never seen any of this before

Isn’t it just x1+x2/2

That’s how I’m used to finding the vertex

[tight kite]

thats an arithimetic average

[mellow frigate]

Yea isn’t the vertex always in the middle

Of a parabola

[tight kite]

the vextex is the minimum or maximum point of the parabola

[mellow frigate]

Which has to be in the middle of the two x intercepts

[tight kite]

ohh x1 and x2 are the roots

[mellow frigate]

Ye

[tight kite]

yes thats right

[mellow frigate]

So we got (-5,0) and (3,0) as roots and -1 as vertex no?

[tight kite]

yes

[mellow frigate]

How do I use these now?

[tight kite]

the factored form of a quadradic is y = a(x - r1)(x - r2) where r1,r2 are the roots

[mellow frigate]

Where did r even come from🥲

[tight kite]

-5 and 3

the roots

[mellow frigate]

What is a?

[tight kite]

a is the coefficient of x², as in ax² + bx + c

[mellow frigate]

Ohh

I see

Usually I just see it as y=(x-number)(x+number)

So here it’s y=(x+5)(x-3)?

How do we know what’s the coefficient or if there is one

[tight kite]

yes a(x + 5)(x - 3)

you know the vextex point

so you can use it to find the value of a

[mellow frigate]

Ohh!

Oh I thought that is a

How do you use it to find a

[tight kite]

y = a(x + 5)(x - 3)

you know when x = -1, y is 2, so you can just replace for x and y

[mellow frigate]

Wait how do you know y is 2

[tight kite]

thats the vextex

(-1, 2)

[mellow frigate]

Oh yea! From the picture

So 2=a(-1+5)(-1-3) and solve for a?

[tight kite]

yes

[mellow frigate]

2=a(4)(-4)

2=a(-16)

-16a=2

a=2/-16?

So -1/8

[tight kite]

yes, which simplifies to -1/8

[mellow frigate]

What does that tell us haha

[tight kite]

so y = -(1/8)(x + 5)(x - 3)

[mellow frigate]

Now I just multiply it out and take the derivative?

[tight kite]

yeah

or use product rule

[mellow frigate]

So y=-1/8(x^2-3x+5x-15)

(X^2 +2x -15)

[tight kite]

yeah

[mellow frigate]

-1/8x^2 3/8x -15

?

[tight kite]

why 3/8

[mellow frigate]

Oops multiplied by 3

-1/4

-1/8x^2 1/4x -15

[tight kite]

what about the -15

[mellow frigate]

Oh right!

Ugly numbers🥲

[tight kite]

using product rule would be easier

[mellow frigate]

-1/8x^2 3/8x + 15/8

[mellow frigate]

Hahah true

[tight kite]

3/8?

[mellow frigate]

I copied the wrong one🫠🫠

-1/8x^2 1/4x + 15/8

Finally🤣

[tight kite]

dont forget the minus sign on 1/4

[mellow frigate]

1/4x+1/4

Should’ve done it one a notebook

[mellow frigate]

Wait so it’s negative actually omg
-1/8x^2 -1/4x + 15/8

Derivative:
-1/4x -1/4

[tight kite]

yeah

[mellow frigate]

How do I know how to draw it haha

[tight kite]

find two points

and connect a straight line between them

[mellow frigate]

Just plug anything to x right?

[tight kite]

any two value of x

[mellow frigate]

When x is 2 y is -3/4
when x is 4 y is -5/4

[tight kite]

yeah but these are fractional

would be good to find where they are integers

[mellow frigate]

How do I know what will give whole numbers?

[tight kite]

like -1 and -5

give whole numbers

[mellow frigate]

Oh yea!

X=-1 y=0

-5,1

[tight kite]

yeah

[mellow frigate]

So like this?

[tight kite]

yeah

[mellow frigate]

Damn that took way longer than I thought🥲

[tight kite]

isnt there a straight line drawing thing?

[mellow frigate]

The next one is x^3🥲🥲🥲

[mellow frigate]

Oh yea there is!🤣

This is even more of a headache to find right?

🫠

Wait that’s more than x^3

It intercepts in more

[tight kite]

gtg now, but thats not polynomial, its trig, either sin or cos

[mellow frigate]

Thank you for your help

I’ll wait for another helper🤣

How do we know if it’s sin or cos they both look the same

[mellow frigate]

<@&286206848099549185>

[unkempt pike]

That's a sine graph

[mellow frigate]

How do we tell?

[unkempt pike]

You can tell because Sin (0)=0

In other words the sine graph starts at 0

Cos (0)=1

[mellow frigate]

Ohh!

That’s smart

So basically the derivative is cos right?

[unkempt pike]

Yessir

[mellow frigate]

But how do I draw it again😅

I start at 1 and make waves?

Like this?

[unkempt pike]

Let me figure out how to explain it 🤔

Sine and cosine waves have an amplitude of 1 without transformations. So they only go as high and low as 1 unit.

[mellow frigate]

Ohh right!

So the derivative should behave the same no?

[unkempt pike]

Exactly because the derivative of the Sine function is the cosine function

[mellow frigate]

So is what I drew correct?

[unkempt pike]

The thing that's missing is that the cosine function, when x =0, then y=cos (0)=1

[mellow frigate]

Damn

So the waves should be the opposite?

[unkempt pike]

You would start drawing the graph at the y-value=1

[mellow frigate]

Like this?

[unkempt pike]

It follows the same wave pattern

Yeah something like that.

[mellow frigate]

Good enough 🤣

[unkempt pike]

I don't know how precise it has to be but yeah.

[mellow frigate]

Professor is nice so I think he’ll take it

How do I tell?🥲

[unkempt pike]

To graph it more precisely takes a few more things.

[mellow frigate]

I hate these questions I want numbers😫😫

[unkempt pike]

Well what occurs to me is to observe the graph in red and try to figure what function does it sort of look like.

[mellow frigate]

But it’s not any main graph like x x^2 or x^3

It’s just random

[unkempt pike]

From the looks of it, it kinda resembles $y=2sqrtx$

[mellow frigate]

Oop

[elfin berry]

catonwolf

[unkempt pike]

Bruh

Lmfao

[mellow frigate]

🤣🤣

[unkempt pike]

Y=√x

It looks like that one

[mellow frigate]

Ohh

Damn we never studied that one

I only know asymptotes for the special ones

1/x

[unkempt pike]

And so if you look closely, it looks more specifically like y=2√x

When the X=4

y=4

When x=1, y=2

[mellow frigate]

Wait 4=sqrt4?

How

Ohh 2sqrt4

Ohhh

[unkempt pike]

Yeahp

Exactly

[mellow frigate]

Omg that makes sense haha

So basically

2x^-1/2?

No

1/2

Not negative

[unkempt pike]

It's just a hunch, not a very rigorous statement but it does seem like the graph is what I'm saying.

[mellow frigate]

The derivative will be negative

[unkempt pike]

Yep

Well no

[mellow frigate]

Like the exponent I mean

[unkempt pike]

Ah yep

[mellow frigate]

So basically the derivative is just x^1/2?

Or sqrt x

[unkempt pike]

You'd have to take the derivative of y=2√x

[mellow frigate]

Ye it’s sqrt x

Cuz 2 times 1/2 is 1

Ohh wait

X^-1/2

[unkempt pike]

y'=1/√x

[mellow frigate]

So it’s 1/sqrtx

Yup!

[unkempt pike]

Yep

[mellow frigate]

How th do I know what it looks like🤣

[unkempt pike]

Well let's look at its implications!

So the derivative is positive or negative?

[mellow frigate]

Well the number can’t be negative

Inside the root

So x>0

[unkempt pike]

Yeah exactly

[mellow frigate]

And also it can’t be 0

[unkempt pike]

Exactly

[mellow frigate]

Cuz it’ll be 1/0

So it’s no a

B c and d are positive

[unkempt pike]

If we were to think about how it will look like as x gets bigger

Does the y value become bigger or smaller?

[mellow frigate]

It’ll decrease!

[unkempt pike]

Exaaactly

[mellow frigate]

No

Oh yea

1/2 then 1/4 then 1/8

[unkempt pike]

Yeah because you have the X in the denominator

[mellow frigate]

Yuss

[unkempt pike]

So you should have a decreasing graph

[mellow frigate]

So it’s b

[unkempt pike]

Yeap

I think so

[mellow frigate]

Let’s gooo

Oh god

My brain

[unkempt pike]

Gotta go but I hope you can continue on!

[mellow frigate]

Thank you for your help!

Will wait for the 8th helper today!🤣

This channel should be permanently mine at this point

🤭

[mellow frigate]

<@&286206848099549185>

[peak turret]

ok

[peak turret]

bottom left\

[mellow frigate]

How do you know?

[peak turret]

top left is a joke

top right matches the stationary point but th gradient is negative for the part of the diff function that is positive so that doesn't match, the bottom left one has the correct stationary point, the linear gradient up till the parabola and then a positive gradient coming onto the stationary point (max of graph)

quesiton like this should take less than 5 seconds

[mellow frigate]

I have no idea what you just said tbh

What is gradient

What is diff function

[peak turret]

bruh

[mellow frigate]

What is stationary point

What is linear gradient

[peak turret]

you're doing calculus no?

you should know this

[mellow frigate]

No context to any of that lol

[mellow frigate]

🤷‍♂️

[peak turret]

its basic terms

what grade are you in?

[mellow frigate]

I know critical and inflection points

[mellow frigate]

Second semester of college

[peak turret]

what are you studying?

[mellow frigate]

Majoring in cs

[peak turret]

I'm in grade 11, this is the stuff they tought us when we first started calculus

[peak turret]

computer science?

[mellow frigate]

Yup

[mellow frigate]

I know whatever they teach me, beyond that is not dependent on me lol

[peak turret]

critical point is stationary point, its when the gradient of the graph (rate of change) is 0

[mellow frigate]

So slope?

[peak turret]

this is basic maths tho

yes gradient = slope

use gradient its more professional

[mellow frigate]

That’s all you had to say

[mellow frigate]

No point in doing so

[peak turret]

diff function is the derivative function

[mellow frigate]

Idc about any of this I study calculus cuz I’m forced to not cuz i want to

I just need the grade

[peak turret]

thats the wrong outlook

that is what will keep you at low levels in school and also in life

[mellow frigate]

I’m at 4.0 GPA

I’m fine

[peak turret]

Even if you don't want to do something, if that thing will bring you closer to your goal (attaining your degree I assume) then you need to implement the discipline to do it

[mellow frigate]

Saying gradient instead of slope won’t bring me closer to my goal

[peak turret]

mine is 6.7

[mellow frigate]

It’ll just make me sound arrogant and condescending and will waste my time whenever I say it to someone cuz then I’ll have to explain that it means slope anyway

[peak turret]

1% changes exponentially affect your life

[mellow frigate]

Idk any other scales than 4 scale so that doesn’t tell me a lot

[peak turret]

Good. The people who think saying gradient is arrogant shouldn't affect you, seriously.

[mellow frigate]

What exactly tells you this is 1% better

[peak turret]

I'm almost top of every subject in my school

[mellow frigate]

Again, it’ll be a waste of my time trying to explain it to them that it means slope

[peak turret]

its a small change that doesn't feel like much now but will affect you by a greater magnutide later down the track

[mellow frigate]

How so?

[peak turret]

Took me one sentence to explain it to you

[mellow frigate]

I’m literally never gonna touch this subject in my life again once I’m done with it

[mellow frigate]

1% of time wasted is 1% you’ll never get back

[peak turret]

It will boost your vocabulary knowledge and will make you seem more intelligent and insightful into the subject of mathematics, if the right people are listening and the majority of your cohort uses 'slope' you will stand out, yes even one word can have this effect. Who knows, it could get you employed or some shit

[mellow frigate]

Instead of simply saying the more known term which is slope

[peak turret]

Do you have any more questions you need help with?

[mellow frigate]

I don’t want to be seen as more intelligent I don’t care what others think of me or how they see me

[peak turret]

Have a positive outlook towards learning, always seek to improve by 1% each day

[mellow frigate]

And I don’t want to do 9-5 so I don’t care about getting employed

[mellow frigate]

Yes with things I care about

And things that will benefit me

[mellow frigate]

Ye, but I still don’t understand how to solve this one

[peak turret]

We all say that, but lets be real external perception (not in an arrogant or obnoxious way) is very important. Would you wear a shirt that said "I dont shower" for a job interview or a nice outfit? Exactly.

[peak turret]

Fine let me make a full tutorial

hang on

[mellow frigate]

Should I wear one that says it in a high level language or what’s the analogy here

[peak turret]

It will benefit you, its like when you were a kid and your parents said 'eat your vegetables' you never knew why it was a good thing, did you do it? Probably.

What do you not understand about the question?

[mellow frigate]

Vocabulary doesn’t matter to me, it’s just a way for humans to communicate

Efficiency is what matters in language

[peak turret]

Ok lets stop with that

[mellow frigate]

All of it tbh

I know how to find the derivative when I have the function

[peak turret]

I know what matters to you! SOLVING THE QUESTION RIGHT?

okay

[mellow frigate]

But these problems which I’m stuck on for 20 hours are all visual graphs

[peak turret]

sure

20 hours?

[mellow frigate]

That’s correct

[peak turret]

actually how much time does it take?

[mellow frigate]

Yup waiting in this channel for each helper to come

They usually left after 1 question and then it took 1-3 more hours for another person to come around lol

[mellow frigate]

I mean if you don’t know how to do it there is no set time

You just can’t do it

[peak turret]

until you give up

[mellow frigate]

Well it’s graded so I won’t

I’ll just keep waiting until someone teaches me

[peak turret]

ok

im eating

[mellow frigate]

Enjoy your meal

[peak turret]

can u wait like 4 min

[mellow frigate]

Of course

[peak turret]

ok

the graph on the left (blye one) is f'(x) when the red ones are f(x)

do you follow

[mellow frigate]

Yes

It’s the derivative

Then you need to choose the function

[peak turret]

do you know what an antiderivative is?

its not necessary

but do you?

[mellow frigate]

Yes we didn’t get to that yet

Integrals

[peak turret]

ok

yeah

do you know what a derivative function represents?

[mellow frigate]

The slope of the tangent line

[peak turret]

sure

I'll use your vocabulary even though it pisses me off

what is the tangent line?

what does it represent?

[mellow frigate]

I only know “rate of change” but was never explained to me what it actually means

[peak turret]

oh fuck

[mellow frigate]

But I’d guess it’s like the rise over run thing

[peak turret]

you are right though

do you know rise on run?

[mellow frigate]

Like the rate at which y increases or decreases compared to x

Is that correct or

[peak turret]

think about this

[mellow frigate]

Oh the slope formula ye

[peak turret]

are you familiar with it?

[mellow frigate]

Yea

[peak turret]

ok

[mellow frigate]

But it only works for linear functions no?

[peak turret]

not necessarily

the tangent will always be linear yes?

[mellow frigate]

How?

What if it’s x^3?

Tangent will be x^2

[peak turret]

take this function for example

[mellow frigate]

Ye here it’ll be linear

Cuz -1

But what if it’s to the third

[peak turret]

just focus on what i'm saying

the above image is the AVERAGE rate of change between those two points

where x is any value and x+h is that x value + any value h

[mellow frigate]

Ohh yea that formula

[peak turret]

h cannot be negative for this case (lets keep it simple)

[mellow frigate]

The long derivative way thingy

[peak turret]

if we bring h closer to zero .... watch what happens

[peak turret]

first principles

[mellow frigate]

Yea we did that at the beginning of the semester

And then there are sec lines until we get to the tangent line

Which touches only one point

Locally

Smth like that

[peak turret]

I hope this isnt too complicated to see

[mellow frigate]

What are all these lines

[peak turret]

the red line is when h is higher, then lower at purple, then lower at blue, then at green it is at 0

[mellow frigate]

Oh so the secant lines?

[peak turret]

except for green, that is tangent

[mellow frigate]

Yea and then they get closer until it’s tangent

[peak turret]

imagine basically that with the rise over run formula we are using two points so SUPER SUPER close together that the distance between them is basically 0 and we find the INSTANTANEOUS RATE OF CHANGE at that point

[peak turret]

yes!

that is rate of change

now looking back at your original question

the straight horizontal line in the blue graph indicates that a section of the original function will have a gradient that is positive (as that blue section is in the positive y -axis) and that it will have a constant slope (like the line seen in the rop right and bottom left red functions

at this point in time either of those could be the original function

[mellow frigate]

So basically because it has a constant line the curve has to have a linear section?

Because x^1 -1

Ohh then it’s the linear function that goes down cuz that section of the parabola faces down?

[peak turret]

then we see the blue graph show us a line which goes down (like a linear function) since this is a diferential function after all, this shows us the slope of that section of graph for the original function... it tells us that the slope is slowly decreasing. When the line crosses the x-axis (when y=0) the slope is 0... this is our critical point (as the slope of that tangent at the critical point is = 0)

[peak turret]

yes

[mellow frigate]

Ouuuu

[peak turret]

no

not necessarily

I do have to go help someone else in a couple of minutes

[mellow frigate]

Haha see it’s always one question then hours of waiting🤣

But fair enough step closer to finishing this

Now I’ll be stuck here for a few hours🤣

[peak turret]

but basically we know that it cannot be the top right graph because the slope of that graph is increasing (I know THE GRAPH ITSELF is going down but it's SLOPE is slowly increasing, which is not correspondant to the blue graph, so it must be the bottom left

[mellow frigate]

Isn’t it increasing

The graph

It has a local minimum

[peak turret]

bye bye

[mellow frigate]

Thank you lol I’d still love an explanation for this tho

[mellow frigate]

<@&286206848099549185>

[peak turret]

really?

be proactive and go on youtube instead of asking every question you dont understand here

if its something that you're having a hard time with and you havent been able to find any videos on then come here

[peak turret]

i made a mistake, the last line should dip below the x-axis before reaching 0 at the critical point and then going up

still a straight positive line though

its the same principles I discussed with you before

[mellow frigate]

What makes you think I didn’t try that

They always have the function in the video

I can’t find videos who do it based on the visual graph alone

[peak turret]

I mean... it's not relavent but ok

[mellow frigate]

How do you know where it’s drawn tho?

Like why is it x=1

At the beginning

[peak turret]

https://www.youtube.com/watch?v=O3vRXeE-iBI

[mellow frigate]

I’ll watch

[peak turret]

gradient of line = 1 🤦‍♂️

I feel like if you're in university you should know this basic stuff

just keep trying

[mellow frigate]

So basically you start by finding the critical points?

[mellow frigate]

Like I said i know whatever I’m being taught

Yea this doesn’t make any sense

How does a parabola stay a parabola

It’s supposed to be linear function

He basically drew the same graph

?????

What’s the logic here wtf

[marsh citrus]

@mellow frigate Has your question been resolved?

[peak turret]

Just look where the points are in relation to the scale

[hallow finch]

neither of those are parabolas

a parabola is a quadratic

[mellow frigate]

I don’t even know what that means

I thought a curve becomes linear

Other than that no idea what’s the logic

[old marten]

What was the question?

[old marten]

True, however I prefer to think on my own. And thanks to this, I have better and deeper understanding and insights of the concepts.

[thorn shuttle]

to sketch a derivative graph:
if f(x) is increasing (going up) then it is above the x-axis
if f(x) is decreasing (going down) it will be below the x-axis
if f(x) is neither decreasing or increasing then it touches the x-axis/stationary becomes x-intercepts
inflection points become stationary points

[thorn shuttle]

You’re thinking about a parabola

A parabola becomes linear when sketching its derivative

[marsh citrus]

@mellow frigate Has your question been resolved?

[marsh citrus]

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[dark acorn]

number 13

[dark acorn]

<@&286206848099549185>

[marsh citrus]

Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.

[dark acorn]

<@&286206848099549185>

[still temple]

?

[dark acorn]

upp

.close

[marsh citrus]

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[marsh citrus]

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[subtle swan]

I have two lines in 3d space , they are in the pic,
I have found and verified that the minimal distance between them is 17/sqrt(29)
I want to find the point C (in l1) which is the nearest point to l2
how can I do that?

[subtle swan]

.close

[marsh citrus]

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[subtle swan]

.reopen

[marsh citrus]

✅

[limber hearth]

im not getting what you mean by distance between since you want the closest point C

[subtle swan]

d( l1, l2 )

distance between the two lines: l1 and l2

[limber hearth]

ok

so if it's constant the they are parrallel no ?

[subtle swan]

no that is the minimal dist

[limber hearth]

as you said its 17/sqrt29

ok

[subtle swan]

if this helps

A : (1,5,-1)

[limber hearth]

i don't get any key solution mate

[subtle swan]

ok ty

.close

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[marsh citrus]

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[fallen sky]

prove using mathematical induction

[marsh citrus]

Please don't occupy multiple help channels.

[fallen sky]

im stuck on the induction step, again

[tight kite]

show what you did so far

[fallen sky]

i cant

i dont have discord on my phone

i can tell you where i got with this tho

i got pretty far with the induction step

but got stuck on sqrt(k) + 1/sqrt(k+1) >= sqrt(k+1)

[tight kite]

why did you get stuck here exacly?

move some terms around

[marsh citrus]

@fallen sky Has your question been resolved?

[fallen sky]

what

how

[tight kite]

like multiply by sqrt(k + 1) to clear the denominator, etc

[fallen sky]

read again now

[tight kite]

read what

[fallen sky]

!

[tight kite]

what about it?

[fallen sky]

what do i do

😔

[tight kite]

.

[marsh citrus]

@fallen sky Has your question been resolved?

[fallen sky]

OH

[fallen sky]

no

u cant reopen @dark acorn

this is mine

[marsh citrus]

@fallen sky Has your question been resolved?

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[undone hemlock]

I know its a simple question but i just can't visualise why this is a plane normal to k at all points. Any help?

[marsh citrus]

@undone hemlock Has your question been resolved?

[undone hemlock]

<@&286206848099549185>

[marsh citrus]

@undone hemlock Has your question been resolved?

[marsh citrus]

@undone hemlock Has your question been resolved?

[main pumice]

idk anything about this <@&286206848099549185>

[marsh citrus]

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[undone hemlock]

.reopen

[marsh citrus]

✅

[undone hemlock]

<@&286206848099549185>

would be greatly appreciated

[marsh citrus]

@undone hemlock Has your question been resolved?

[marsh citrus]

@undone hemlock Has your question been resolved?

[main pumice]

<@&286206848099549185>

[still temple]

what?

[marsh citrus]

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[novel juniper]

So here's my work :

$x = \frac{-y \pm \sqrt{48-3y^2}}{2}$
\
\
$\frac{dx}{dy} = \frac{-1}{2} \pm\frac{ 6y}{4\sqrt{48-3y^2}}$
\
$0 = \frac{-1}{2} \pm\frac{ 6y}{4\sqrt{48-3y^2}}$
\
So $y=2$

[elfin berry]

A dense set

[novel juniper]

,w solve (-1/2) + 6y/(4\sqrt{48-3y^2})=0]

[elfin berry]

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[novel juniper]

yeah, what am I doing wrong

[sand fable]

dx/dy = 0 is where the graph has vertical tangents, not horizontal

[novel juniper]

oh yeah

[sand fable]

you want horizontal to find the 'highest'/'lowest' points

[novel juniper]

shoot

This is the graph, it doesn't have vectical tangents at x=2, does it

[novel juniper]

sorry how does it have vertical tangents at y=2

[hushed egret]

observe

[novel juniper]

ah shoot

yeah

make sense

thanks

.close

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[still temple]

i can solve for equations like x + y + z = N where N is any positive integer

but idk i am not sure how i would think about a problem like this

where you have something like ax + by + cz = N

[valid cape]

this is a diophantine equation

a linear one at that

[still temple]

idk whats that

i am studying this under

a topic

permutations and combinations

[smoky stream]

i can help with that question

you have to try making different cases

[still temple]

uh

they would be too many

isnt there a shorter way

or a way with lesser cases

or maybe can we use multinomial theorem here ?

[smoky stream]

no not really most of the PNC questions require counting

let me think once again

[still temple]

but the whole point of pnc is to avoid counting

lol

minimize it

[smoky stream]

sometimes you dont have an option

[trail hamlet]

You can make cases for x

When x = 0
The following can be true for (y,z)
(21,0) (18,2) (15,4) (12,6) (9,8) and so on till (0,14)
Y decreases by coefficient of Z and Z increases by coffecient of Y

Do it similarly from x = 0 to 42
Answer will be a sum of AP
So no real need of counting

[honest creek]

can you start with (mod 3) both sides?

x + 2y (mod 3) = 0

you can list them all out with the given constraints on x,y >= 0 and x + 2y <= 42

also maybe (mod 2) as well

[still temple]

i see

[still temple]

not familiar with the notation , sorry

[trail hamlet]

it means x+2y leaves remainder 0 when didided by 3

[still temple]

oh

hmm

that can be a way to think about it

[still temple]

anyways ill try this

thanks

.close

[marsh citrus]

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[vestal mist]

for part a can i use the fact that

[smoky stream]

i am not sure if your school will allow you to use that but there is nothing wrong with that equation

WAIT NO

sorryyy my bad i didnt see it is a 3x3 matrix

there is a different equation for a 3x3 matrix

that equation you have given is used for a 2x2 matrix

[vestal mist]

oh

[smoky stream]

yes

[vestal mist]

is there a way to do part a without an equation

bc i never learned the one for a 3x3

[smoky stream]

i can tell you the equation but lets see if we can do it without one

so what you have to find is the determinant and trace of that matrix
and its characteristic polynomial's roots are given

so if the roots of the characteristic polynomials of a matrix A is given

you can find its determinant by simply multiplying all the roots

and to find its trace you can simply do that by adding all the roots

so in this case it will be
det = -2 x -2 x 5
tr = -2 + -2 + 5

@vestal mist

[vestal mist]

wait how did you find that though

[smoky stream]

it is um theory......

your textbook must have mentioned it

[vestal mist]

whats it called

[smoky stream]

properties of characteristic polynomials/equations of an matrix?

[vestal mist]

let me check

[vestal mist]

really?

[smoky stream]

yes

[vestal mist]

wow i didnt remember that

tysm!

wait but do you think you could help me with part b

[smoky stream]

the second answer

should be um very easy

if you look at it

[vestal mist]

its just 0 right?

factor out n from each row

and det of a matrix with all 1s is 0?

[smoky stream]

yes sir

[vestal mist]

wait why is the det of a matrix with all 1s 0

i just found out by trying it in woflram alpha

oh wait i see why

thank you for your help!

.close

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[little prism]

Does anyone know how I could check if I did this correctly, I’m unsure I could use wolfram alpha for that

[marsh citrus]

@little prism Has your question been resolved?

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[brisk totem]

13 a

[brisk totem]

What am I doing wrong?

And how am I supposed to determine whether a solution “can be found” or not

[marsh citrus]

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@brisk totem Has your question been resolved?

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[orchid fractal]

Do Fourier transforms work even if I don't want to integrate over all reals? For example from 0 to infty instead

Let's say I have a function u(x), can I use this kind of transform?

$\hat{u}(k) = \int_0^\infty u(x) e^{-ikx} dx$

[elfin berry]

gudmundr

[devout mauve]

well you shouldnt call it fourier transform

but sure you can do that. it might just not have all the nice properties the fourier transform has

[fervent rampart]

it is called a one-sided fourier transform

[orchid fractal]

thank you ! Is there any resource where I can see which properties hold and which ones don't?

[fervent rampart]

this may provide some insight: https://math.stackexchange.com/questions/1719754/unilateral-laplace-transform-vs-bilateral-fourier-transform

[marsh citrus]

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[proper cliff]

Could someone walk me through a solution?

[red nimbus]

Have you done Ax = 0?

or rather did you find the solution vector

[proper cliff]

I augmented the matrix after removing the bound var columns, but I think I’m off base on the whole problem

[red nimbus]

Say we have a b c d e.
Last row tells us b = -2c-3d-e.
First row tells us a = -3b-6c-3d+e = -3(-2c-3d-e)-6c-3d+e = 6d+4e

So basically you end up with 3 5-dimensional vectors

c(0,-2,1,0,0) + d(6,-3,0,1,0) + e(4,-1,0,0,1)

[proper cliff]

So the basis would be the set of these vectors?

[red nimbus]

yes

[proper cliff]

Okay thank you

.close

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[restive hound]

1a

[pearl haven]

do you the range of trigonometric functions?

[restive hound]

no, im not familiar with the content in this unit

its the first lesson

[pearl haven]

did they teach you that sinx ranges from -1 to 1?

[restive hound]

yes

[pearl haven]

so just use that,

-1 < sinx < 1

add 4

3 < sinx + 4 < 5

max would be 5 and min would be 3

[restive hound]

and then for b i would do the same

but -5?

[pearl haven]

no worries

cosx also ranges from -1 to 1, so

-1 < cosx < 1

subtract 5

-1-5 < cosx - 5 < 1-5

-6 < cosx - 5 < -4

[restive hound]

sounds good, thanks or the help

[pearl haven]

youre welcome mate

[restive hound]

.close

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[hushed crest]

How to show that the function
$f(x)=\frac{1}{\tan{x}}$ if $\pi/4 \leq x < \pi/2$
and $f(x)=0$ if $x=\pi/2$ is differentiable on the closed interval $[\pi/4,\pi/2]$

[elfin berry]

BeeReallyYum

[delicate cypress]

Ok

So you have to prove it is continuous first

[hushed crest]

yes I think I can do that

do i have to tho?

like is there a way to prove it’s differentiable without necessarily proving it’s continuous

[hidden dawn]

continuity is necessary

[hushed crest]

Okay, then what?

[hushed crest]

Also could you please tell me why

Like in some cases you would just directly show it’s differentiable right?

Without necessarily going through continuity

I know that a function can only be differentiable if it’s continuous but in some cases we would just skip it

For example if a function is differentiable on the open interval ]a,b[ then to show it’s differentiable on the closed interval [a,b] we would just show it’s differentiable at the right of a and on the left of b

[delicate cypress]

Man I wish I revised differentiability

But listen

You have to prove continuity

Say |x|

Or absolute value function

[hushed crest]

yes

[delicate cypress]

It is not diffrentiable at x=0

But it is continuous over R

Because left hand derivative and right hand reivative of |x| is different

[hushed crest]

I think this example is a bit different because the interval stops at the jump

So I can’t really talk about left and right derivatives

[delicate cypress]

Perhaps you have to prove limit x approaches c f(x)=f(c) for all c in (a,b)

If my memory serves me right

[hushed crest]

Do you mind if i call helpers?

Just to check

[delicate cypress]

No I don't

Please do

[hushed crest]

<@&286206848099549185>

[hushed crest]

.

[marsh citrus]

@hushed crest Has your question been resolved?

[hushed crest]

<@&286206848099549185>

[still temple]

You need to do that theorem

[hushed crest]

Which one?

[still temple]

I dont think it has a name but if a function is differentiable over a interval then its continuous over that interval

[hushed crest]

Yeah but I am trying to prove it’s differentiable not continuous

[hushed crest]

<@&286206848099549185>

[main idol]

Did you try using the definition of the derivative and show the limit of the difference quotient exists

[hushed crest]

Yes but would it be enough?

Because the limit would be a real number (0 in this case)

But does it necessarily mean it’s differentiable? I mean what if the jump discontinuity was 1 instead of 0 then it wouldn’t be differentiable

So there has to be something to do with the value of 0

[main idol]

it is if you prove the limit exists for all values in your domain

the only parts you have to be careful about are the endpoints

[hushed crest]

Yes so rn I know that f is differentiable on [pi/4,pi/2)

I’m just trying to close the interval

[hushed crest]

Yeah so I get stuck at the endpoint pi/2

I can’t understand how to proceed logically here

And I looked it up online but couldn’t find anything

[main idol]

why? just follow the definition

[hushed crest]

So basically calculating the limit at pi/2-?

But then what’s the result i should expect

To conclude

Does it only have to be a real number or does it have to equal to smth

[main idol]

do the limit and find out

[hushed crest]

Yes I did it and it’s 0

But how should I conclude

[main idol]

,w derivative cot(x) at x=pi/2

[elfin berry]

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[hushed crest]

I don’t think that’s right

Because it’s the left hand derivative at pi/2

[main idol]

,w lim x to pi/2 (cot(x) - 0) / (x - pi/2)

[elfin berry]

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Requested by rie.mann

[hushed crest]

The limit is 0 tho

At pi/2 on the left

Because tan(x) approaches infinity as x approaches pi/2 from the left so the limit of 1/tan(x) is 0

[main idol]

your question is about the derivative of cot(x), not cot(x) itself

[hushed crest]

Ohh yes you’re right

Sorry it’s late here I forgot about that

So what to do after finding the derivative

[main idol]

.

[hushed crest]

Yes so the limit exists here

Because it’s a real number

But what I’m having trouble understanding is how can I conclude that it’s actually differentiable at pi/2

f(pi/2)=0 but what if it wasn’t

Like what if f(pi/2)=1 so there has to be something to be done with f(pi/2)=0 right?

[marsh citrus]

@hushed crest Has your question been resolved?

[hushed crest]

<@&286206848099549185> anyone please?

[stiff jewel]

sorry i dont know

[hushed crest]

that’s alright

i didn’t think this question would be that difficult

[marsh citrus]

@hushed crest Has your question been resolved?

[marsh citrus]

@hushed crest Has your question been resolved?

[novel bear]

11837.20 i've had this and round this to 11837.00 but it's not right from the seems

[marsh citrus]

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[novel bear]

.reopen

[fast cove]

How do i find x?? Im doing circle theorems

[fast cove]

I think this angle here is 23

Wait, I got the other questions wrong too 💀

I'm so confused

[jolly seal]

AOC is 92

[fast cove]

Oh shi

Howd u get that???

[amber birch]

you misremembered the formula

the angle at the center is larger than the one at the circumference

twice as large as you know already

[fast cove]

Ohhhh

I got it opposite

[amber birch]

mhmmmmmm

[fast cove]

Daym I got it right now

x is 44 but I think I did the wrong formulas for the other questions too now

[jolly seal]

just...estimates with eyes bro

which is bigger

[amber birch]

bad idea in general

but yeah I guess

[fast cove]

Wym

I though they'd be equal

[jolly seal]

because its qualitative in this case rather than quantitativr

[amber birch]

fair

I suppose no geometry diagram would lie to you that hard

just that I see too many people interpeting a diagram as to scale when it isn't

[jolly seal]

lol when i cant get something right i use ruler and penpass to get an accurate graph and measure

[amber birch]

try moving the 43 degree angle to the left

like clockwise around the circle

[jolly seal]

trying is better than surrendering

[amber birch]

and then redraw the diagram

[jolly seal]

then angle on the circumferrence is always equal if its based on the same arc or arc of the same length in the same circle

[fast cove]

Thanks guys got it

.close

[marsh citrus]

Channel closed

Closed by @fast cove

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[wild hemlock]

I’m stuck at this step

[wild hemlock]

Oh wait is it cuz cos(whatever) limit is 1, so it’s gonna be limit of 1/(n+1) which 0?

[wary kite]

well this isn’t true

limits at infinity of sin and cos diverge

as in, they don’t converge to anything

they just oscillate

[nimble prairie]

cos(…) is bounded above by 1

So just use a comparison test instead by taking absolute value

Or are u forced to use ratio test?

[wild hemlock]

This is a Ratio Test exercise so yeah kinda

[nimble prairie]

I see

[static quarry]

you have the minor problem that cos(n pi/8) is zero if n = 4 + 8k for some integer k...

[nimble prairie]

Yeah if they’re aren’t allowed the lim sup version then this is a problem

Simply bcuz there isn’t a limit

Very crappy exercise when it’s perfectly solvable by a simple comparison test with absolute convergence

[marsh citrus]

@wild hemlock Has your question been resolved?

[marsh citrus]

Channel closed

Closed due to timeout

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[marsh citrus]

Available help channel!

Send your question here to claim the channel.

Remember:
• Ask your math question in a clear, concise manner.
• Show your work, and if possible, explain where you are stuck.
• After 15 minutes, feel free to ping <@&286206848099549185>.
• Type the command .close to free the channel when you're done.
• Be polite and have a nice day!

Read #❓how-to-get-help for further information on how to ask a good question, and about conduct in the question channels.

[still temple]

h

?

uh

.close

shits buggin

[static oxide]

Hello, i need your help for the differential equation of second order. I would like to know why there is a 2 (arrow) in my example exercise. Where does it come from please ?

I hope someone can help me 🙂

[wintry gale]

Why do non trivial solutions have determinant zero?

[still temple]

he

[static oxide]

which ?

[wintry gale]

Umm which what?

[static oxide]

i dont have many technical terms (like trivials), so could you write your questions by quoting of my picture ?^^

[real gulch]

Hello, I need some help in this differential equation. The method to be used is inspection method, and I am stuck on what to do with this.

[static oxide]

.close

[crisp kraken]

this is multivariable calculus with constrained variables.
He's explaining how to differentiate a function with another function as a constraint using 2 methods, differentials and chain rule.
He said that we often cannot or don't want to solve the constraint equation in terms of the variable we use either one of these 2 methods.
With chain rule I don't understand how to find the da/dtheta at the end of the equation, from the constraint.

is this working?

[real gulch]

i find it quite hard to understand

[crisp kraken]

i think those help channes are marked as available but they are not really closed ? <@&268886789983436800>

[quaint elm]

oh no the bot broke again

[novel juniper]

Oh no

😭

test

[real gulch]

I will find a way to this solution then, thanks

[opal schooner]

welp

uhh is this channel free for a question

does $\frac{1}{2} \times \frac{3}{4} \times \frac{5}{6} \times \frac{7}{8} \times \dots$ converge to 0

i think i have an answer

[oblique quest]

I dont think its opening to new channels Idk why

[opal schooner]

but i wanna see what other ppl do

[elfin berry]

ark

[opal schooner]

i forgot to say "to 0" skull

it obviously converges cause monotone and bounded below

[hoary pier]

I'm on limits right now I'm stuck on understanding the first step (in this image) here

It's tripping me up a the moment

[jolly seal]

you wanna get rid of the abs value

check line two right hand side for info

[hoary pier]

Ohh i get it now

I had to take myself back through the definition of absolute value again

[jolly seal]

well...kinda what was expected to be known when you get to limits already but...hang in there bloke

[hoary pier]

thank you

[halcyon surge]

hey i wanna help on this, the teacher said we can solve it with a table but im little confused

[tired oxide]

what is there to estimate here

you can just find the derivative exactly

[quaint elm]

i think the idea might be to evaluate f(1.0001) - f(0.9999)

and divide by the difference etc

[halcyon surge]

i dont think so

[lone heart]

I think you can use whole numbers, the result is a nice number

[halcyon surge]

i should solve it with a table but i forgot how

[quaint elm]

in the limit you'll see a whole number but any numerical estimate will be slightly over that whole number

[quaint elm]

do you have an example you can refer to?

[lone heart]

First, do you know what the derivative represents? And second, do you know how to make a table of values for f(x)?

[halcyon surge]

like
f(1)=3(1)^2-2(1)+1
x | y
1 | 2

[halcyon surge]

yeah ik but the teacher put it in another question

[lone heart]

That doesn't answer my question

At least the first half

[quaint elm]

well yes that's a (small) table of f(x). Do you have an example problem similar to this one, to refer to?