help-27

[severe prairie]

i lowkey forgot how to do these lol

how do i find a and b

[severe prairie]

statistics btw

[faint zinc]

I assume you haven't yet found a and b?

@severe prairie

[timber pebble]

looks like a test

o the url

[faint zinc]

I've never heard of past papers at Papa Cambridge, but from the title, I would reckon that this is from previously given tests

[devout snow]

@severe prairie Has your question been resolved?

[velvet coral]

I think you need to check for each of these properties

I feel like a, b coudl have multiple values tho

[velvet coral]

nvm yeah right continutiy

[velvet coral]

So you need to find b first

whcih will give you enouogh info to find a

To find b use fact that F needs to be right continuous

we know that F(6) = F(6+) = 1

so then b = 7/25

then use F is non decreasing to find other

[devout snow]

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[severe prairie]

no idk how to

[severe prairie]

its a past paper this was the exam from previous years

[severe prairie]

wait whaat

[devout snow]

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[devout snow]

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[obtuse stirrup]

Someone help

[obtuse stirrup]

Calculating turning point coordinates

<@&286206848099549185>

[sand pumice]

,rcw

[sand pumice]

what have you tried

[obtuse stirrup]

I know that gradient will be zero at turning point, so i try to do the equation y=x+9/x =0

[sand pumice]

did you not differentiate it first?

[winter prairie]

Hmm

[obtuse stirrup]

I tried but derive is too big

[sand pumice]

use the power rule

[obtuse stirrup]

I need to convert x/9 to xpower-9

Its -9x power -10

[sand pumice]

huh

[obtuse stirrup]

How to do it then

[sand pumice]

you have 9x^(-1)

how would you differentiate it?

[obtuse stirrup]

OHhHhh

Wait wait

Are you sure

Oh my

No i mean the derivaigobe

[sand pumice]

!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.

[obtuse stirrup]

Deriavitive

[obtuse stirrup]

Yes i agree

Now help me understand actually pls

Ok can you help me understand now

Thats easy just substitide

Ye but for b?

F of x?

Yes

Dy/dx = 0

[sand pumice]

the point is, the derivative at the turning points is zero, so you solve the equation f'(x)=0

[obtuse stirrup]

Is

Yes but power rule i was wrong?

9x power -1 will it be

[sand pumice]

you have to differentiate this

don't forget the other x you have btw

[obtuse stirrup]

So then -1×9 power -1-1?

After diffrentiate

😡

Waitt

It was x/9

So use power rule

Turns to

9 power

😡

Brother what is this

Yes

Then

Wait wait

oh yes

Your right

Howw

Whats the rule say?

I

Ok

No its ok

Why would you bracket out 9

Diffrentiate has propertys?

I didnt know that

How did you know that

Ok what do i search

Wheres my materials

Iam igcse edexcel

Iam just studying past papers..

No i only do past papers

Omg found it

Look

Thanks too much walnut

7 marks too, very big

Close

close.

[devout snow]

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[vocal hull]

How to find powered variables values
Like
$5^x=25$

[woven radish]

Weinnion

[topaz axle]

calculator

[upbeat seal]

so like u trying to solev for the equation?

[vocal hull]

yes

[upbeat seal]

so like tell me what is 25 expressed in powered form

[vocal hull]

5²

[topaz axle]

,calc log(25) / log(5)

[woven radish]

Result:

2

[upbeat seal]

rewrite that into the equation

[nimble dragon]

Genius

Wow

[glossy dew]

true enough

[vocal hull]

So you are telling me that \
$5^x=5^2$ (now we extract the base) \
Therefore, $x=2$ ??

[upbeat seal]

yea

[woven radish]

Weinnion

[upbeat seal]

or u can use logs

[vocal hull]

Wait, tell me that logarithms and exponents are inverse operations

[upbeat seal]

Yes

did you learn laws of logs yet?

[vocal hull]

So how can we find exponential value with logarithm?

[upbeat seal]

so like a^b = c<=> log_a c = b

[vocal hull]

Maybe we i will learn here

[vocal hull]

Uhh can you use latex

[upbeat seal]

I think u have to learn the basic of logs before doing that

https://www.youtube.com/watch?v=LRbi_pMX1DM

I suggest u watch this

its helpful

[vocal hull]

Bro i know man $\log_a(a)=1$

[woven radish]

Weinnion

[upbeat seal]

ya so basicly

log_a (x^n) = n log_x a

[vocal hull]

Oh wait

$\log_a (x^n) = n \log_x n$

[woven radish]

Weinnion

[vocal hull]

Did you mean this?

Also

[nimble dragon]

Base a on RHS

And x inside log

Not n

[vocal hull]

$a^x=b$ $\rightarrow$ $x = log_a(b)$

[nimble dragon]

Remember it as pulling the n out

[woven radish]

Weinnion

[vocal hull]

is this how you take into log?

[harsh stream]

Just note that comparing bases only works when the base is not equal to 1

[vocal hull]

Okay resolve this help channel

.close

[devout snow]

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[devout snow]

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[bitter sky]

ABCD is a quadrilateral where AB=AD and BC not equal to CD, CA bisects BCD, prove A,B,C,D are concycli

[vital edge]

!status

[devout snow]

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

[bitter sky]

well

2

i was usin

ptolemys extension

and somehow tryin to

find a contradiction

[vital edge]

Idk what ptolemys extension is but

Draw a diagram

And think about what exactly you wanna prove

Actually why are you using ptolemy

Use the angle property of cyclic quadrilaterals

[bitter sky]

i tried but

not really very useful

[bitter sky]

i did

[vital edge]

Would help to show what you've done so far then

[bitter sky]

well do you know about ptolemys theorem

[vital edge]

I realized, the product property yeah

[bitter sky]

well tbf i didnt write stuff down i js did stuff in my diagram and erased it

[bitter sky]

yea extension is just for

normal quadrilaterals

and product is > the sum

so i was tryin to assume

its a normal quadrilateral

and tryin to contradict

[devout snow]

@bitter sky Has your question been resolved?

[devout snow]

@bitter sky Has your question been resolved?

[devout snow]

@bitter sky Has your question been resolved?

[devout snow]

@bitter sky Has your question been resolved?

[lunar harbor]

Draw the circumcircle of BCD and ||consider the midpoint of arc BD that doesn’t contain C||

[bitter sky]

is that a hint

[devout snow]

@bitter sky Has your question been resolved?

[devout snow]

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[devout snow]

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[tardy ravine]

I need help with algebraic expression

[devout snow]

Asking the actual question right away is more likely to get responses.

Asking "Can I ask...?" or "Does anyone know about...?" doesn't give people enough information to decide whether they can help, and answering can feel like a promise to help with the actual question, which they might find themselves unable to.

@tardy ravine Has your question been resolved?

[granite island]

whats your question

is there a question you re struggling with

[devout snow]

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[crystal saffron]

I need help with this question I know I need to replace the x or y to find the intercepts but once I get to solving for x I am struggling

[still garnet]

Can u show ur work so far

[crystal saffron]

yes

the last part i wrote is where i was just attempting to do something because i was lost

im hoping to get redirected to the right path

[still garnet]

Even if u break it down u are not being able to find a real value of x intercept,are u?

[crystal saffron]

Im not sure what you mean

are you trying to say that there is no x intercept?

[still garnet]

Yes that when u draw a graph,u wont see the any intersection between the shape and x axis

[crystal saffron]

ohh okay

When i graph the x intercept is -5

[still garnet]

That's the abcissa of center

[crystal saffron]

Okay

So would the x intercept be -5 and the y would be 4?

[still garnet]

Do we know what intercept means?

[crystal saffron]

yes

I don't really know what abcissa is though

I know the intercept is where a graph meets the x or y axis

[still garnet]

(x,y) is a point where x is abcissa and y is ordinate

[still garnet]

Yes

[crystal saffron]

Ohh alright

so the x intercept would be written as (-5,0) correct?

[still garnet]

No the ellipse doesnt intersect x axis

So we dont hv an intercept for x axis

This is a rough drawing ,u see the ellipse doesnt intersect x axis at all

[crystal saffron]

Oh okay I see now

[still garnet]

When u try to gind x intercept, set the value of y as 0 and for y intercept x=0
If u find non real value,u will know physically it doesnt intersect the axis

But if u want to put values ,u can try inputting the complex number u get after u plug y=0

[crystal saffron]

Okay

do you think you could show me how I would go about solving for them through the equation?

thank you for the help btw

[still garnet]

Lemme try

[crystal saffron]

Thank you

I understand a lot better now!

[still garnet]

That's great

[crystal saffron]

yes

have a nice day or night

[still garnet]

U too

[crystal saffron]

.close

[devout snow]

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[devout snow]

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[bitter sky]

what do I do after considering the mid point of that specific arc ? Draw the quadrilateral from that point ?

[devout snow]

@bitter sky Has your question been resolved?

[bitter sky]

also it will be helpful if someone can help me solve it w

ptolemy

i really tried to use ptolemy but

didnt make a breakthrough

[devout snow]

@bitter sky Has your question been resolved?

[devout snow]

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[bitter sky]

dude wy does ts keep closin sob

[bitter sky]

my question

i tried that construction earlier, couldnt make muc use of it

[bitter sky]

and^

[devout snow]

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[devout snow]

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[devout snow]

Please don't occupy multiple help channels.

Channel closed

Channel closed due to the original message being deleted.
If you did not intend to do this, please open a new help channel,
as this action is irreversible, and this channel may abruptly lock.

[devout snow]

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[olive mural]

i know how to do base case and inductive hypothesis but no idea how to proceed from there

[gloomy aurora]

Please specify clearly where you are stuck.

[olive mural]

?

i proved where n=1

[gloomy aurora]

k

[olive mural]

i assumed n=k is true

and n=k-1 is true

and then im stuck

[gloomy aurora]

so you proved that if it is true for $n=k-1$ then it is true for $n=k$? Or do you need help there?

[woven radish]

Annie Maqionde

[olive mural]

no i assumed both

for the inductive hypothesis

[gloomy aurora]

do we have to use induction>

[olive mural]

yes

[gloomy aurora]

k ill give you a hint: consider the product $(x^k + \frac{1}{x^k})(x+\frac{1}{x})$ after assuming for $n=k,k-1$

[woven radish]

Annie Maqionde

[olive mural]

i asked chatgpt and it told me to do the same thing
how would i know to do that tho

[gloomy aurora]

I mean experience while solving problems mostly.

[olive mural]

is there nothing to look for

[gloomy aurora]

I mean patterns exist; forms like above when there's like 1) addition of powers 2) two inductive assumptions

So then yk its multiplication

you need to multiply something by something to add the powers

and we use the 1/x so that we can diminish the power(two inductive cases)

also for your proof to be rigorous, you'll need to prove for both $x=1$ and $x=2$

[woven radish]

Annie Maqionde

[olive mural]

i see

.close

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[devout snow]

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[orchid shuttle]

Help

[nimble dragon]

put x=z in equation 1

[orchid shuttle]

Am I correct

[nimble dragon]

huh

did you add -z to the left and +z to the right?

[orchid shuttle]

Yes

[nimble dragon]

thats not right

you have to add the same value on both sides

[orchid shuttle]

😿

[nimble dragon]

2=2

2+3=2+3

2-3 is not 2+3

[orchid shuttle]

So what do I do

[nimble dragon]

.

thats it

its given x is the same as z

[orchid shuttle]

If I add -Z to both sides will it be correct

[nimble dragon]

so it doesnt matter if you write x+y or z+y

[orchid shuttle]

We don't have to do any solution?

[glossy dew]

what about the euclid axioms

[nimble dragon]

yes

they didnt ask to state the axioms used

[orchid shuttle]

I have it

[nimble dragon]

you may state any of the five axioms

[glossy dew]

postulate..

[nimble dragon]

yea whatever

[glossy dew]

but like what about the axioms you used to prove z + y = 10

[nimble dragon]

yea thats right

[glossy dew]

writing the equations is half of it

[nimble dragon]

also dont put an equals sign before every line

[orchid shuttle]

Things which are added to the same thing are equal to one another

[nimble dragon]

its misleading and (slightly) triggering

you may use a two way implies sign instead

[orchid shuttle]

.close

[devout snow]

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[green crypt]

good handwriting

[orchid shuttle]

Actually too much mastur

[green crypt]

uh YEAH

[orchid shuttle]

So hand shake when I write

[green crypt]

oh

i just understood what you meant

[devout snow]

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[ripe grove]

Let a = (1, −3, 2), b = (2, −1, 1) and c = (2, −5, 4) be vectors in R3
.
Write the vector c as a linear combination of the vectors a and b.

My peers are being so collectively strange about this. Just to confirm I'm not a psycho; we can't solve this yep?

[ripe grove]

I.e. there exists no linear combination

[wicked turtle]

they're linearly independent

so no, it's not possible

[uncut crow]

[ripe grove]

Sane people still exist 😭

.close

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[bitter garnet]

Hello, I would like to learn geometry. I am a beginner at it.
I am planning to learn it alongside number theory and abstract algebra

[bitter garnet]

I am seeking for recommendations thereof.

[rare kernel]

i think #study-discussion

[bitter garnet]

.close

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[sharp quartz]

Given a block ABCD.EFGH, with AB= 6; BC=4; CG=3. Find AD to BE

[sharp quartz]

so I got the answer (6√5)/5

But I'm not sure if it's correct

I used the comparison of area of ABE

[devout snow]

@sharp quartz Has your question been resolved?

Show your work, and if possible, explain where you are stuck.

[sharp quartz]

<@&286206848099549185>

my work:

I'm stuck cus the answer I found isn't in the options

[devout snow]

@sharp quartz Has your question been resolved?

[sharp quartz]

<@&286206848099549185>

[fossil anvil]

bro

[sharp quartz]

dude I feel so gaslit

by these questions

chatgpt for some reason answered 6

[sharp quartz]

(my friend said)

[devout snow]

@sharp quartz Has your question been resolved?

[true pollen]

You found the shortest distance between A and BE, but the question is to find the distance between AD and BE. You need a distance that is represented by a line that is perpendicular to both lines. Your lenght is only perpendicular to BE. But this link explains the whole thing: https://youtu.be/t7vDAe3-PV8?si=PBYAIqJZuvOlNI4U

[true pollen]

Hope my answer helps

[devout snow]

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[karmic sun]

Hey,
Can anyone please explain why 0^0 is undefined even though any other number raised to 0 is 1?
Thanks!

[polar chasm]

Can you explain why 0^0 is undefined even though raising 0 to any positive number gives 0? 0^x = 0? (for positive x)

if we look at it my way, it should be 0. Your way it should be 1. So which one should it be?

the simplest resolution is just leaving it undefined. For specific context (e.g. in combinatorics), its sometimes helpful to define it, usually to be 1.

[karmic sun]

Ohh, I see. Thank you!

This helped

.close

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[vast drift]

For context, Im taking a computer vision and image processing course and on their prerequisites, it hardly mentions anything about math background

Then I found out after starting the course that most of it is discussing applications of fourier transforms on processing of images for refining training data set (seems to occupy quite a notable portion of the course)

ATM Im self studying fourier transforms because I have never learnt it, my question is how important the more 'theortical' part is, i.e., do I need to understand some measure theory etc, will it significantly hinder my understanding of how fourier transform works if I dont understand the more 'pure-math ish part'

I do want to gain a holistic understanding but I don't think my time permits me to do that atm, probably will be revisiting fourier transforms quite formally in a math course next term.

[supple knot]

you probably don't need to know measure theory to do fourier transform problems

e.g. this hw set has 0 measure theory
https://eceweb1.rutgers.edu/~gajic/psfiles/HW_4.pdf

the minimal measure theory you'd need is enough to understand the delta functional. it can come up if you ask yourself "how can something have area 1 but also 0 at all points except 0"

[stone stump]

lots of people understand fourier transforms well enough for their purposes without having a single clue about how lebesgue integrals or distributions work

[vast drift]

okay I will look into that then.
Just not exactly sure how much detail Im missing out because some resources and vid I watched kept mentioning L1 spaces and Lebesgue integrals and they currently mean nothing to me

Thx 🙏

.close

[devout snow]

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[scenic bridge]

@devout snow how to smart in math:v

[supple knot]

#discussion

[scenic bridge]

Okie:v

[devout snow]

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[heavy current]

my equations for part b) are
\begin{align*}
\ddot{r} - r \dot{\theta}^2 - r^2 \sin^2 \theta \dot{\phi}^2 & = 0 \
\ddot{\theta} + \frac{2}{r} \dot{r} \dot{\theta} - \sin \theta \cos \theta \dot{\phi}^2 & = 0 \
\ddot{\phi}^2 + \frac{2}{r} \dot{r} \dot{\phi} + 2 \cot \theta \dot{\theta} \dot{\phi} & = 0
\end{align*}

[woven radish]

higher!

[heavy current]

I'm confused on the last part though

how do I verify that solutions of these correspond to straight lines?

[faint zinc]

Not an expert in diff geo, but my first instinct would be "simply" convert from spherical coords to cartesian.

[heavy current]

that sounds like an absolute horrid mess

[faint zinc]

Yeah

[heavy current]

I had that idea but really wanted to avoid it, since this problem's been enough work as is

I suppose if I convert these eqs into Cartesian, I'd end up concluding smth like \ddot{x} = \ddot{y} = \ddot{z} = 0?

[faint zinc]

Well, maybe the inverse problem would be easier: take a general line in Cartesian coordinates, convert to spherical and then show that you get what you wrote

[woven vale]

is it enough to verify that the curve maintains a constant angle with a known line for all t?

[heavy current]

hmm

[faint zinc]

Most line pairs in 3d don't intersect and aren't parallel. (Skew)

[heavy current]

I'm not sure how you'd get from a line to these equations though

I can see how you might get to a line from these eqs

[heavy current]

like the hope would be to get this I think, and then conclude that the solution to the eqns is a line

...it's true that x.. = y.. = z.. = 0 implies that the solution is a straight line, right?

[faint zinc]

Or a point

[heavy current]

no curvature in the x, y, or z directions due to 0 2nd derivative

[heavy current]

ehh, I'll ignore that case

[faint zinc]

But that being said, there are other situations where you get a straight line with non-zero second derivatives. For instance, accelerating motion along a line

[heavy current]

hm, that's true..

I guess the only way to figure out what I'll get is to actually do it, but I really kinda well... don't want to do it right now

[faint zinc]

That's fair, it looks heinous

Best of luck!

[heavy current]

then I'll put it off till the 'morrow

thank you for the help OE

I'll close this for now c:

[faint zinc]

Maybe shoot your prof an email

[heavy current]

it's not for a course actually

[faint zinc]

Ask if that's the path forward, or if there's a clever solution

[heavy current]

I'm doing a DRP with a grad student, and she's given me this problem to work on, but idt she's yet solved it herself

[faint zinc]

Oh nice

[heavy current]

alright, I'll go and do another problem for now then

[faint zinc]

[heavy current]

and return to this tomorrow

thank you OE!!!

.close

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[devout snow]

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[fickle moon]

do these look good?

[supple knot]

yea i think so

[fickle moon]

this might be a stupid question but idk how to do questions without variables. like how would i do this one tho

[supple knot]

well do you know how to plot y = 6?

[fickle moon]

yuh

[supple knot]

do you know how to plot a single point (-2, -6)

[fickle moon]

yuh

[supple knot]

f(x) is the same as this plot except at one point

do you know where?

[fickle moon]

but like why would i plot one at -2, -6

isnt -2 restricted

[supple knot]

oh sorry i have those swapped

f(x) = -6 everywhere except one point, and that other point is ?

[fickle moon]

-2

[supple knot]

x=-2, what's the y value?

[fickle moon]

6

[supple knot]

right. draw that graph and try to answer the limit question

[fickle moon]

okok thank you sm

[fickle moon]

would i just write this

[supple knot]

The answer to the limit is correct but the plot is not

There's only one value of x that f(x) equals 6

[devout snow]

@fickle moon Has your question been resolved?

[supple knot]

Perfect

[devout snow]

@fickle moon Has your question been resolved?

[fickle moon]

would i label the lines like this or with the limit

[fickle moon]

.close

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[rich schooner]

is there a way to figure out each value without looking at the unit circle? i don’t have it memorized so i just looked at it

also, could you help me on the 2 that I have wrong

[royal laurel]

No

You should have all of them memorized and be familiar with the rationale behind it

[bright phoenix]

Kind of... but not really there are some tricks to learn common ones without having to brute memorize the circle . it'll serve you better in the long run to get comfortable traversing the unit circle though

you can quickly convert radians to degrees as every pi radians is 180 degrees. That may help you visualize the angles better if thats the problem

[devout snow]

@rich schooner Has your question been resolved?

[devout snow]

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[keen sundial]

what is this slide saying?

[supple knot]

Do you know what correlation is

[keen sundial]

yes

[supple knot]

What examples did you learn that have a measurement that depend on two correlated variables

[keen sundial]

an incallibrated scale

[devout snow]

@keen sundial Has your question been resolved?

[keen sundial]

not sure what a quadrature is

[devout snow]

@keen sundial Has your question been resolved?

[devout snow]

@keen sundial Has your question been resolved?

[devout snow]

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[devout snow]

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[last parrot]

Multiple choice?

[devout snow]

@small surge Has your question been resolved?

[small surge]

Nope

[mint cairn]

This is a depressed cubic.

The discriminant of a depressed cubic in the form of $x^3 + cx + d = 0$ can be calculated using the formula below:
$\Delta = -4c^3-27d^2$.

To have two equal roots, $\Delta = 0$.

[woven radish]

Restarter

[devout snow]

@small surge Has your question been resolved?

[small surge]

What is depressed cubic?

[mint cairn]

Basically a cubic without the x² term.

[small surge]

I see

[twilit comet]

!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.

[small surge]

I don't understand the discriminant thing

.close

[devout snow]

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[trim pewter]

it doesn't appear to be a solution?

[twilit comet]

i mean it was just the straight up answer

[river rune]

it looks like a solution, but ren should have targeted Restarter's message instead of the LaTeX rendering.

[river rune]

!nosols, just for your information.

[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.

[twilit comet]

yeah accident, mb

[mint cairn]

So step by step guide aren't allowed.

Noted.

I'll just observe next time.

[small surge]

.close

[upbeat forge]

to be fair you kinda left the easiest part for the OP without even giving them a chance for a challenge and response.

[upbeat forge]

it's closed already.

[mint cairn]

You have closed it.

[mint cairn]

Well as you can see.

The OP didn't understands it.

So it's not a solution for him. 😁 👍

[upbeat forge]

I don't think posting a solution that isn't understood is still a justification for (pretty much) direct solsposting, but sure.

[mint cairn]

If that's so, alright then.

I'll refrain myself from posting anything in the future. 😁 👍

The last time I did that.

Nobody mentioned.

So was it a biased opinion then?

[upbeat forge]

I don't know where else you did that, so I only mentioned it on the one time I saw it.

of course, I'm not the absolute authority here, so if the mods say you're fine, then I apologize and you may continue helping your way.

I'm not going to dig through your history and reprimand every single occurrence.

[mint cairn]

Sorry dear for the misunderstandings.

[mint cairn]

Not sure if he's the mod or not.

[devout snow]

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[small surge]

.close

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[karmic basalt]

how to do 3)?

[devout snow]

Please don't occupy multiple help channels.

[karmic basalt]

@quaint hearth ici svp

[woven radish]

woody6978

[karmic basalt]

par quelle propriete?

PGCD(a;b)=PGCD(a;a-kb)?

[quaint hearth]

oui

je sais pas si vous avez le droti de l'utiliser

[karmic basalt]

cest loppose je pense?

[karmic basalt]

moi aussi

jai trouvé un resume

[quaint hearth]

l'opposé c'est à dire ?

[karmic basalt]

cela est la partie des PGCD

[karmic basalt]

j'ai deja modifié le msg

[quaint hearth]

ah oui +1

oui je pense que ça peut s'utiliser c'est un peu implicite ici

[karmic basalt]

c'est dure pour moi

honnetement

ce casse ma tete

[quaint hearth]

tkt au debut c'est difficile puis en faisant pleins d'exercices tu trouveras surement celui ca super facile

[devout snow]

@karmic basalt Has your question been resolved?

[karmic basalt]

je n'ai trouvé pas une solution

je penser que je va passer a l'anlyse

merci!

.close

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[devout snow]

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[coral zephyr]

Can I get help on this one

[dull parrot]

how is v(t) related to a(t)?

[coral zephyr]

I need a lot of help with intuitive thought

[coral zephyr]

Velocity is the integral of acceleration?

[dull parrot]

yep, or you can say accleration is the rate of change of velocity

[coral zephyr]

Yes

[dull parrot]

so how can you get v(t) then?

from the first question

[coral zephyr]

Integrate acceleration?

[dull parrot]

yep

now you're given that a(t) = -5cos(t)i - 5sin(t)j

[coral zephyr]

Yea

[dull parrot]

since you said to get v(t), you integrate a(t), you just need to integrate -5cos(t)i - 5sin(t)j

[coral zephyr]

But what about the initial stuff

[shy osprey]

but remember after integrating, the limits are from time t to 0

hence you will have to subtract the given initial value of v, v(0) = ....

[coral zephyr]

Okay I don’t understand

[shy osprey]

when you integrate a(t), you gotta keep some limits right?

[coral zephyr]

I don’t understand where limits came from

[shy osprey]

okay so when we intregrate acceleration, we cant use indef integ

as we want velocity at a given time right?

v(t) basically means velocity at definite time 't'

hence why the def integration and limits need to be applied

[devout snow]

@coral zephyr Has your question been resolved?

[devout snow]

@coral zephyr Has your question been resolved?

[coral zephyr]

<@&286206848099549185>

Can I get help on this

[shy osprey]

okay how about a different approach

[coral zephyr]

Okay

I integrated a(t)

[shy osprey]

and whatd u get?

[coral zephyr]

I got v(t) = 5sinti - 5costj

[shy osprey]

erm

[coral zephyr]

Oops

What happened

[shy osprey]

yeah integral of cos x isnt -sinx

its just sinx

[coral zephyr]

Oh okay

Alright so

[shy osprey]

you differentiated it i think

[coral zephyr]

Oh

So switch signs

[shy osprey]

similarily integral of sinx is -cosx

[shy osprey]

yup

[coral zephyr]

v(t) = -5sinti + 5costj

[shy osprey]

thats correct!

okay good

but the problem here is that it initally too had some velocity

so inoder to find the velocity at a given time, you need to add its intially velocity

this is one of those basic concpets of physics

[coral zephyr]

Okay now I’m confused

I never took physics

[magic wind]

v(t) = -5 sin(t) i + 5 cos(t) j + C, determine C with v(0)

[coral zephyr]

Wait what

[shy osprey]

??

[coral zephyr]

<@&268886789983436800>

Alright

[coral zephyr]

What’s the objective here from this problem

[magic wind]

v(t) is the antiderivative of a(t), but you can always add a constant value to it. This constant can be calculated with the initial consitions.

Same procedure from v(t) to r(t)

[coral zephyr]

I don’t get the initial condition part and it’s relation to the constant C and why we are considering this to begin with

[shy osprey]

hey i think you gotta explain in more basic terms i tried the same thing, he got a lil confused

[magic wind]

Example: if a(t) = 2 -> v(t) = 2*t + C

[coral zephyr]

Yes

[magic wind]

If v(0) = 16, then 2 * 0+C = 16 => C = 16, thus v(t) = 2*t + 16

sorry, fixed that typo.

[coral zephyr]

Okay so

v(t) = 2t + c is a family of functions that are valid anti derivatives of a(t) correct

[magic wind]

yes

[coral zephyr]

Okay and an initial velocity will tell you the instantaneous rate of change of the object at an initial time correct

Instantaneous rate of change = velocity

Right

[shy osprey]

rate of change of displacement

[magic wind]

It does not have to be the initial time, you have to know the velocity for some time

[coral zephyr]

I’m tryna understand the logic behind the initial values

[magic wind]

You could solve for c with v(5) = 20 for example.

[coral zephyr]

Wait so essentially we want the correct function of the family of functions so that’s why we need to solve for C?

[icy sail]

To calculate the constants of integration

[magic wind]

correct

[coral zephyr]

And the logic behind the initial conditions is that you end up with that C value

Because it actually narrows down all the possible anti derivatives into the scenario we need to consider

Right

[magic wind]

Yes, it "narrows it down" to one possibility.

[coral zephyr]

Alright, can I ask another question regarding derivatives like

Position-> velocity -> acceleration?

[magic wind]

Sure

[coral zephyr]

Okay so I just wanna make sure I have this right

When we talk about derivatives for vector functions, we are talking about the instantaneous rate of change at a point on the curve represented by the vector function

What that means is the displacement from some point a to another tiny point / a tiny amount of time ?

[magic wind]

That sounds correct.

[coral zephyr]

So I can say here that the derivative is the instantaneous rate of change of position right

[shy osprey]

yes

[coral zephyr]

And subsequently I can say that acceleration is the instantaneous rate of change of the instantaneous rate of change of displacement?

Which is just the instantaneous rate of change of velocity?

[shy osprey]

yup spot on

[coral zephyr]

And the rate of change would be the displacement vector / some time right

[shy osprey]

more accurate would be to right in derivative format

aka

dr/dt

[coral zephyr]

But like geometrically what would that be

@shy osprey

As the green vector gets closer and closer to the blue one you get their difference to be the displacement vector at a very very small disposition?

[shy osprey]

thats just the explanation of the derivative format but yeah thats the geometrical one

[shy osprey]

yes

when it becomes infinitesmally small, you call it instantaneous

[coral zephyr]

Why do we express that very very small displacement (red vector) as a vector necessarily?

Because when I think of displacement I think of changing from one point to another right

So why do we wanna express that as a vector necessarily here

[shy osprey]

as you are expressing other positions, blue and red as vectors too

[coral zephyr]

@shy osprey I have another question, why do we end up with another function for velocity when deriving the position function?

Like why do you get a desperate curve

[shy osprey]

as in ?

[coral zephyr]

Okay so let’s look at it this way If you differentiate a position function you are basically finding the instantaneous rate of change of position on all points on the curve right ?

[shy osprey]

correct

[coral zephyr]

So why do you get a new function then

Like a new graph for velocity

[shy osprey]

because differentiation and integration is basically changing the fucntion right?

[coral zephyr]

Is it because all those instantaneous rates of changes of positions give you a new position or something and all those new positions connect to a new path or something ?

[shy osprey]

if you look at it geometrically, at each point, the chnage is different than the actaul values

[coral zephyr]

No but I mean like intuitively

[coral zephyr]

Ooooooh

And all those changes are changes in position

And then it’s a new position

[shy osprey]

yupp

yup

[coral zephyr]

Connect all new positions together

[shy osprey]

there we go

[coral zephyr]

= whole new path

Oh my God finally

Now I know what derivatives are after 2 and something years of first learning them

🌝

[shy osprey]

aw thats nice

[coral zephyr]

Wait new question 😂

[shy osprey]

sure go on

[coral zephyr]

So like what about integrals

I know they undo everything

But how does that work geometrically

Also we know integrals to also be for calculating areas and stuff so that’s a lil confusing

For why areas and volumes are related to rates of changes

[shy osprey]

explaining integrals geometrically might be a little challenging at your level

[coral zephyr]

Oh

[shy osprey]

but for now just go with this

this should be sufficient

[shy osprey]

btw if udm me asking, which grade?

[coral zephyr]

undergrad

[shy osprey]

ah so ur 19-20?

[coral zephyr]

sure

i could be

i could be not

i dont wanna specify tbh

[shy osprey]

mr aristotle over here🥀

[coral zephyr]

😂

[shy osprey]

oh sorry

[coral zephyr]

youre good

[shy osprey]

.close?

[coral zephyr]

can we keep it open for a little I might have more questions

[shy osprey]

sure why not

but i gotta head to sleep bro

4am here forme

[coral zephyr]

no worries man have a great night and thanks for your help

[coral zephyr]

<@&286206848099549185> I have a question, so I understand what derivative is graphically but what does the rate of change really give us

Like what does it really mean when we say velocity aside from derivatives and stuff

What does it mean to say a rate of change of something

[weak rapids]

Yeah nice question...I would also like a explanation to this

[sage ibex]

how much it changes given a rate=$

or speed of change

how fast or slow it changes from a to b

[coral zephyr]

okay why is point b - point a / time b - time a that thing you just described

[sage ibex]

say before 10 years my height was 100 cm now 160 cm

the difference in time is 10 years and difference is height is 60

so the speed is 6cm/year

this is average rate of change

[coral zephyr]

can you call that velocity in this instance

[sage ibex]

ofc because i wont get my height decreased

+6cm/year

[coral zephyr]

So like velocity isn’t something tangible in the same way position is right

like velocity is just how much something changes / time correct?

like thats just what it is you cant like see it or something

[weak rapids]

Yeah we can say that

But you know I'm coming from the psychology to understand mine cause from the question you asked...it kinda makes me feel like you want your answers beyond maths

[weak rapids]

So let me tell you what I know

[sage ibex]

yes but with direction which says it grow or degrow

[coral zephyr]

go ahead

[coral zephyr]

so velocity is just a scale that tells us how something changes per a specific amount of time

[sage ibex]

yeah right

[weak rapids]

So for me this is wat it is...
It measures how one thing responds when another thing changes.
If time moves, how does position respond?
If price changes, how do sales respond?
If effort increases, how does performance respond?
That’s rate of change.
It tells us not just what is happening, but how the situation is evolving.
In real life, your brain uses rate of change thinking all the time... especially in decision making.

If someone is walking and you see water on one side and nails on another end, you can predict their next step, that it will definitely be a clear spot where it is comfortable (the result). Why? Because you observe the current movement and the surrounding factors. From that, you can tell the likely direction of the next move.
That is the same idea mathematically.
When we look at a curve on a graph, maybe I have an irregular curve like that of a trade...now let me say that at a maximum point it start sloping downwards and I know that it won't eventually reach the bottom and out...it might change at another certain time, I can pick a point and ask:
“At this exact moment, which direction is this moving, and how strongly?”
If the rate of change is positive, it’s moving upward.
If it’s negative, it’s moving downward.
If it’s zero, it’s about to change direction or pause.
So rate of change is like a prediction tool.
It tells you the immediate next tendency of a system based on how it is moving right now.
So in simple words let me say:
A function tells you where you are.
Rate of change tells you where you’re heading next.
That’s the real-life meaning behind it...

[ocean quest]

Velocity is just speed

Although changed from a scalar to a vector form

Speed is the absolute value of velocity

[coral zephyr]

Speed?

Interesting

[coral zephyr]

When you say “where you’re heading next” that almost sounds like the position function itself

[coral zephyr]

You mean in the context of vector valued functions right

[willow path]

velocity is just speed in a certain direction

you can interpret this a few ways

Velocity = speed • unit vector (gives direction)

Speed is the magnitude of velocity

etc

[coral zephyr]

I’m still trying to understand this

@willow path

Like what is “how fast it changes”

[willow path]

velocity is the rate at which position changes

[coral zephyr]

Which is??

[willow path]

in other words, velocity is the derivative of position

[coral zephyr]

Yes but

[willow path]

Say i’m moving at 10ms in the x direction

every second, i’m covering 10 meters

my position at t=0 is 0, t=1 is 10m, t=2 is 20m

etc

the rate at which this position changes, is my velocity

if you plot those positions as a graph, you get a linear function

the slope of that function is the velocity

[coral zephyr]

But like when you’re taking the derivative of a function at a point, you get the rate of change from a point to another very close point almost reaching your original point, that is gonna spit out some value, but what IS that value that’s spat out

[willow path]

The value that’s given is just the derivative at that point, it’s your instantaneous rate of change

If i take the derivative of my position and get a velocity of 10m/s

at that EXACT moment

i’m moving 10m/s

next moment could be different or the same

all that a derivative says is the value of that point

[coral zephyr]

But that’s just a description

Not a value

Right

[willow path]

I’m sorry wdym?

[coral zephyr]

Like

Take this

Or this

I’m dividing the change in one position to another by the time it took for that to happen

Right

@willow path

[willow path]

this is an approximation

[willow path]

the formula is an exact value

for the former, you’re picking 2 points (hopefully close) and getting a value of the average slope

not the slope of an exact point

the second formula is the limit definition of a derivative

[coral zephyr]

Yeah I forgot to add the limit on the first photo

But I meant to represent velocity in two different applications

Normal functions and vectors

[willow path]

Velocity can be represented both ways yes

[coral zephyr]

But I’m having trouble grasping the idea of the formula which is the rate of change

Like what are you really getting it’s not clicking for me

Like

You’re dividing distance by the time it took for the displacement to happen

Right

[willow path]

so you aren’t understanding what rate of change means specifically ?

yes in an approximation

[coral zephyr]

I get the math but not the abstract concept

Like I don’t get the essence of it

You know?

<@&286206848099549185>

[willow path]

Essentially

you’re taking the value at one point

because you’re making the difference between the two point infinitely small

that’s what the limit definition does

you take a delta that approaches infinitely close to 0

giving you the answer for the point

does that make sense?

[coral zephyr]

I get the operation mathematically but there’s two things I’m still not fully understanding:

1- once you have the velocity function that maps to all the infinitely small rates of change between every single point, when you plug in time t , you get it’s instantaneous rate of change of that point t and the infinitely small point right after it? Right before it?

Let’s start with 1

[willow path]

there is not “infinitely small point right after it

it’s just that pint

if hou take a measurement that’s infinitely small, it just gives you the point

think about it this way

i’m getting infinitely close to a value, Delta= 0000…..

can there be anything after an infinite string of zeros ?

no

there isn’t an end to infinity

so there’s no “infinitely small point” next to it, it’s just the individual point

[coral zephyr]

But then that point at that time would just be still

Right

There wouldn’t be a rate of change because that point being there isn’t changing

[willow path]

Yes you’re taking a snapshot of a moment

Like a picture of time at the moment measured

If i take a picture of someone running, were they not still running?

[coral zephyr]

Ohhh

Okay so

[coral zephyr]

What’s the point of the limit and the definition we have then

[ionic heron]

what grade

[coral zephyr]

Undergrad

I know what a limit is I’m just trying to understand the abstraction behind derivatives

[willow path]

proves derivatives work

it’s exactly as it sounds, it’s a definition

it tells you that a derivative is when you take the change in value of a function, and shrink it down until you’re measuring just one point

and what it’s instantaneous change is

same way we can define an integral with a summation

[coral zephyr]

Bro who’s reacting like that

The hell

Weirdos

[willow path]

Idk 🙁

[ionic heron]

Is it undergrad algebra?

[coral zephyr]

So

[coral zephyr]

Calc 3

Yeah I know

[light storm]

what do yyou need help with

[coral zephyr]

Understanding derivatives fully

So so far I understand that derivatives are the instantaneous rate of change of some point

But you wanna measure it’s change from point to another tho

This isn’t making sense

[willow path]

If you use point slope formula

you’re measuring the change between 2 identified points

we want to find the derivative of the initial point right

P1

[coral zephyr]

Yes

So if you wanna take a rate of change between it and another point

[willow path]

so half the distance between the next point

[coral zephyr]

How can it be that point only

[willow path]

our answer is more accurate

half it again

and again, and again

[coral zephyr]

But it’ll never be it right

[willow path]

if you half it infinitely, that difference shrinks to zero

not just close to zero

it is zero

[coral zephyr]

It’ll always be so close to the point where it’s almost that point

[willow path]

i think you’re having trouble grappling with the concept of infinity

it’s not just close

infinitely close is the same as being at that point

[coral zephyr]

So what does limit as Δt approach 0 mean

[willow path]

infinity never ends, if i’m infinitely close im standing right on that point

[coral zephyr]

It’s approaching right

[willow path]

it means it is getting infinitely close to 0

which fundamentally is 0

[ocean quest]

Notably, in this case it only works because what you are looking at is differentiable in the first place

[coral zephyr]

So it’s literally that point

[willow path]

Yes that is one caveat

[ocean quest]

technically no

[coral zephyr]

Okay so then we are dividing by 0 or what 🙂

[willow path]

essentially it is, the notation helps us define how we approach that point without running into the issue of dividing by 0

technically not the point, but for purposes of our calculation it is infinitely close to said point

fun nature of infinities

[ocean quest]

like, if you have the function $\frac{(x+1)(x-1)}{(x-1)}$, the function is discontinuous at x=1, but the limit of the function still exists at x=1

[woven radish]

Mr. BananaHead

[coral zephyr]

Yes but in terms of derivatives

There’s rate of change that we wanna calculate

Shouldn’t it be between TWO points???

[willow path]

instantaneous rate of change

[coral zephyr]

Because that’s the only way you can take an average to begin with

Like

[willow path]

no there doesn’t need to be 2 points as we make an infinitely small gap

there is no average

a derivative isn’t an average

it is an exact value at the point

[ocean quest]

the essence of the limit process is that you are removing that second point while still getting a useful result

[willow path]

(assuming no discontinuity

[coral zephyr]

Oh

Wait so

This is confusing ngl

I’m tryna wrap my head around how we are considering the change in displacement/ time of just a single point

[willow path]

i’d recommend finding extra content to watch on how to better understand the nature of infinities

[ocean quest]

yeah, calculus is difficult to learn because when you learn it, it is very different from any other mathematics you've taken beforehand

[willow path]

incredibly helpful for calculus, they can be weird so it’s about coming to terms with their implications

[willow path]

because velocity isn’t displacement/time

average velocity is

[coral zephyr]

Velocity is instantaneous rate of change from a point to another ?

[willow path]

what you have there is the formula for average velocity, velocity itself is defined as the derivative of position

dp/dt

it is the rate at which your position changes at your exact point, not from one point to another

[coral zephyr]

Oh

Okay how does your position change at its exact point

[willow path]

position itself doesn’t change at the exact point, it just defines the rate of change at that point

position changes relating to whatever position function you’re using

in the most simple case, a linear one

Take p(t) = 2t

each t value is your position, that derivative just tells you how fast your position will be changing

in this case v(t) = 2 so constant change

[coral zephyr]

Yeah but like

[ocean quest]

maybe it would actually be easiest to explain in terms of integration

[coral zephyr]

How does the limit as h approaches 0 of this give you what you just said

[ocean quest]

let's say you know someone's current position and speed

or, velocity in this case

[willow path]

that h becomes smaller and smaller

until it is essentially zero

so imagine that second point being dragged closer and closer until it’s on top of the other point

[coral zephyr]

Yes

Then you have a difference of what divided by what?

When that happens

When it gets smaller and smaller infinitely

[ocean quest]

if the speed is constant, it's very easy to calculate their future position for some time t. it's just the constant velocity c multiplied by the change in time t, and you get ct

[coral zephyr]

Then it’s what - f(a)?

[ocean quest]

but if the velocity is not constant, then it will follow some derivative function f'(t), and you will need to integrate along that function from the current point in time to some future time t to find their future position

[willow path]

try it out

using the limit definition to calculate dy/dx (x^2)

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

[coral zephyr]

(x+h)^2 - x^2 / h

[willow path]

x^2+2xh+h^2 then -f(x) = -x^2

simplify from there by dividing h out

[coral zephyr]

That’s so weird

Okay yes

But like

[ocean quest]

the main idea here is that $h \neq 0$

[woven radish]

Mr. BananaHead

[coral zephyr]

But @willow path said it’s eventually 0

[ocean quest]

it's not

[willow path]

for all practical purposes it is

it’s not exactly 0

[ocean quest]

the limit is a separate value from the value of h itself

[willow path]

yes the calculation is performed first

anyways, i have to go to a club meeting so ill let Banana take it from here goodluck man

[coral zephyr]

Thanks for the help

[ocean quest]

it's like how $\lim_{x \to \infty} \frac{1}{x} = 0$ but $\frac{1}{x} \neq 0$ for all x

[woven radish]

Mr. BananaHead

[coral zephyr]

Hm

Okay so let’s just get this clear

Velocity at a point is what

[ocean quest]

it's an object's current speed in a certain direction

[coral zephyr]

Okay, so

[coral zephyr]

Are we considering any displacement

[ocean quest]

for our purposes, we are only looking at motion along a single line

so direction is either positive or negative

[coral zephyr]

But I thought we established velocity = instantaneous rate of change = change in position/ certain time

Is this correct

[ocean quest]

if you are looking at 2D or 3D motion, you can have a separate derivative for each direction of motion

then you start getting into partial derivatives and stuff like that so to keep it simple, it's best to only look at motion along a line for now

[coral zephyr]

Yeah but did I have the definition right ?

[ocean quest]

yes

[coral zephyr]

Okay so when we say change in position / certain time

What’s the change in position at some point a when considering it’s derivative

[ocean quest]

basically it gives you an approximation of where the function will be close to the point you take the derivative at. This takes the form of a tangent line at that point

the closer you are to the point, the better the approximation the tangent line yields

[coral zephyr]

But we are talking about “change in position”

So from point a to another point?

So another point - point a

/ time it took to go from point at to another point

See rhinos what’s confusing me because it’s from point a to where

If it’s at that point