help-23

[fathom jewel]

IVT?

[primal sluice]

by solution, does it mean x= something and isnt undefined?

[lean otter]

yes? this expression is well-defined for all reals. (Doesn't mean it is true)

[fathom jewel]

I think intermediate value theorem might help

[junior smelt]

As in, there's some number that you can put in as x into sin(x) + x - 1, and then that works out as zero

[primal sluice]

Oh we haven't learned that

[junior smelt]

(you may have learned a variation of it, or something that's equivalent to it, to be fair)

[junior smelt]

This basically just says if you have a continuous function, say g, and take two points, say a and b, then the function has a corresponding input for all numbers between g(a) and g(b)

Also see: "sign change" etc etc

[primal sluice]

Ah ok

[primal sluice]

is sin(x)+x-1=0 continuous

[fathom jewel]

You could try to find for f(x) = sin(x)+x-1 a negative and a positive value

then by IVT it must be 0 some where between

[junior smelt]

(and really, you want to be asking if the sin(x) + x - 1 is continuous [it is - I'm assuming they don't expect a lot of justification for why])

For contrast, functions like 1/x and tan(x) [ = sin(x)/cos(x) ] aren't continuous everywhere, but you can find parts where they are, but that's a future us problem

[primal sluice]

To prove that this is continuous do i just plug in positive and negative numbers then?

[fathom jewel]

huhh

[red delta]

Sinx is continuous

[primal sluice]

sorry we haven't learned anything on the topic of proving anything 😭

[red delta]

X-1 is continuous

Therefore their sum is continuous

[fathom jewel]

q.e.d.

[red delta]

Properties of continuous functions

[eternal carbon]

🔳

[primal sluice]

oh ok that makes sense ty

[red delta]

Oops

[primal sluice]

what does that mean

[fathom jewel]

quad erat demonstrandum

[red delta]

That was fast

Or quite easily done

[primal sluice]

yes

but tbf im lacking in trig rn

[fathom jewel]

since sine is between -1 and 1

you could tell that for small x you get a negative value for f

It's x-1

[primal sluice]

wdym small x

[fathom jewel]

take x = -1000

for example

[primal sluice]

ok

[fathom jewel]

what value would you get for f(x)

positive or negative

[primal sluice]

negative ig

[fathom jewel]

yea

now take x = 1000 for example

what value would f(x) would be

[fathom jewel]

and so

because f is continuous

no matter how and in what way

it will reach 0 at some point between

and thats what we wanted to show

[primal sluice]

ohh

okay i understand

thank u

the first calc midterm is the thrid week of school can u believe that

[junior smelt]

also 0 and pi/2 [or 90 degrees] btw

[fathom jewel]

pi/2?

[junior smelt]

f(pi/2) is +pi/2 > 0

[primal sluice]

would be 0 right for x=0

oh wait

no -1

[junior smelt]

Yea -1, just realised, was even about to say it's -1 and forgot

[primal sluice]

are you guys undergrad

[junior smelt]

I've already done a masters

[primal sluice]

oh wow in math?

[junior smelt]

Yep

[primal sluice]

was it hard

[junior smelt]

In parts, a bit (but honestly it was more stuff around me that made things difficult than the actual content itself really )

[primal sluice]

oh yeah real

they dont shower

[fathom jewel]

ohh i get it because i toook very big values haha

[fathom jewel]

no i am under my grad

[primal sluice]

highschool?

[fathom jewel]

haha noo

luckily thats over

[primal sluice]

oh ok 😭

[primal sluice]

masters?

[fathom jewel]

wbu

[primal sluice]

im a junior

[fathom jewel]

[primal sluice]

college

[fathom jewel]

i am fighting for my bachelor currently

⚔️

struggling is the better term

[primal sluice]

i chose the easiest majors my freshman year and then decided to add a data science minor which is currently giving me a reality check

so thank u guys for ur help ❤️

[lean otter]

hi guys hoe r

[safe radish]

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[tropic island]

Can anyone explain it to me? I dont even know what that thing inside the () is. I thought it was combination

[plucky elk]

it is

https://mathworld.wolfram.com/BinomialCoefficient.html

[tropic island]

Oh thanks!!!

.close

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[tropic island]

.reopen

[safe radish]

✅

[tropic island]

Wait, how do i use it?

Like i dont get how it went from the lhs to that

[plucky elk]

use binomial theorem

https://www.mathsisfun.com/algebra/binomial-theorem.html

[tropic island]

Thanks

.close

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[granite eagle]

Can anyone recognize what C(A) and N(A) stand for?

[granite eagle]

I assume column and null space respectively but im not sure

[peak estuary]

probably, yes

you would have to check your book/course notes to be 100% sure

but its very likely those

[granite eagle]

Got it
Thank you

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[lean otter]

yo

[lean otter]

quick ques5ion

this

is it

memorization

or is there a reason

[wraith swift]

there is always a reason in math tbh

[lean otter]

so

any ideas

[wraith swift]

think of it this way

u are going clockwise for the third quadrant

[lean otter]

yh

[wraith swift]

so u do pi-a where it will be

but clockwise is conventionally negative

so u do negative

outside

eg torque in physics

[lean otter]

why cant we do like

like

3/2 pi

-a

to find the angle

[wraith swift]

i think u can but im not sure

[lean otter]

cos

[wraith swift]

must be something to do with the derivation

[lean otter]

each quadrant is 1/2 pi?

[wraith swift]

yea

[lean otter]

so

[wraith swift]

thats how i remember it

[lean otter]

for first quadrant

we just do

wait

wait

ffs

im confused now

[wraith swift]

wait 1 sec

[lean otter]

k

[wraith swift]

its because

prinicipal value lies between -pi to pi

[lean otter]

wdym

[shadow sparrow]

those are some blur images

[lean otter]

mb gang

[wraith swift]

https://math.stackexchange.com/questions/174154/how-to-get-principal-argument-of-complex-number-from-complex-plane

[lean otter]

why does a have to be acute?

[wraith swift]

convention i guess

we have other

argumetns and values to have a general value

for eg

general value of argument: 2npi + theta

and least positive value of argument: (0,2pi]

[lean otter]

wait

arg z

isnt that not a

arg z is the angle no

[wraith swift]

yea

[lean otter]

wait wait

arg z = pi-a

mate

how does that work

if arg z and a

are the same thing

[wraith swift]

only in the second quadrant

[lean otter]

this is so confusing

i must be special

[wraith swift]

not rly

im not very good at complex numbers just have a basic understanding

[lean otter]

HOW AM I NOT GETTING THIS

[wraith swift]

maybe someone better can help

[lean otter]

nah appareciate the effort man

[safe radish]

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[safe radish]

@spiral saddle Has your question been resolved?

[obsidian oracle]

I've seen you find SnT in a previous channel

since SnT is a subset of H, you should be able to find k

[spiral saddle]

we take a linear combination of the basis vectors of T and put it in vector form and then made it satisfy the equation for S

[spiral saddle]

how is that related

being a subset

with k

hmm

<@&286206848099549185>

someone give me q hint

with some handhold would be good

why do you say S n T is a subset of H

is that true? if so ,why?

[covert yoke]

@spiral saddle consider what would happen if S cap T was not a subset of H. That would imply that it contains values outside of H.

If we union this with any other set, then it can only add members to the set.

We union it with W and find that the resulting set is H. But this set (W + S cap T) contains members outside of H. This is a contradiction, as H cannot contain members outside of H.

[safe radish]

@spiral saddle Has your question been resolved?

[spiral saddle]

mmm

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@spiral saddle Has your question been resolved?

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[spiral saddle]

.close

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[lean otter]

How do we do this via restricted stars and bars method without using inclusion exclusion

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[lean otter]

how many distinct ways can the sum of one or more consecutive integers be written as 2024?

[lean otter]

how to solve this?

different orders are not distinct

pls help

[buoyant shadow]

i think you need a computer?

oh i missed "consecutive"

2024 = 2×2×2×11×23
so e.g. 11 numbers that are on average 184 gives 2024

any odd amount would work, so there's 4 ways with odd amount i think

11 numbers, 23 numbers, 253 numbers or one number

[grim kraken]

So like the first thing you can notice is that you have multiple cases.
1- A solution that works is 2024 (one consecutive number)
2- If you start by adding an even number, for the result to be even (2024 is) you need to add a multiplication of 4 consecutive numbers (so 4, 8, 12 and etc)
3- If you start by adding an uneven number you need to add 3 numbers and a multiplication of 4 (3, 7, 11 and etc)

[grim kraken]

This is useful to have a condition on how many numbers you need to ask

1- 2024
2- If $n=2k$, you must add $4p$ numbers
3- If $n=2k+1$, you must add $3+4p$ consecutive numbers

[flat frigate]

Lilly (NasaExploration)

[buoyant shadow]

then we can do 368 numbers that are 5.5 on average and 167 numbers avg 11.5 and 16 numbers avg 126.5

so 7?

that's pretty cool

oh i see negatives would be fine

so 8

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[sinful wyvern]

Guys, am i wrong?

[arctic raven]

what happened here?

[marble tinsel]

here, you carried over the -x wrong

it should've been positive 4x not negative

[arctic raven]

just before that step as well, it should be -x-2 = 3x-(something that isn't 15 due to the prior step)

[sinful wyvern]

5-2

[arctic raven]

because you hadn't distributed the negative across the entirety of (x+2)

[arctic raven]

you had changed the denominator from x-3 to x-5

[sinful wyvern]

Ohhh ure right

[marble tinsel]

the right side looks good to me

[sinful wyvern]

Okk holdom imma check it againn

[arctic raven]

right side has the same negative-distributing problem

in this step

right-hand side is fine, left should be -(x+2) = -x-2

[sinful wyvern]

Why positive tho?

[sinful wyvern]

Ohhh

[arctic raven]

he was wrong, you correctly subtracted 3x+2 from both sides, so that -4x = (...)

[sinful wyvern]

I just changed this

[arctic raven]

now adding 2 to the right side gives -9+2 = -7

[arctic raven]

the entirely of the right portion in this image looks better

and only this sign problem on the left portion remains

[safe radish]

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[lean otter]

how did the top become 4 and then 1

[lean otter]

lim --> 4

[shadow sparrow]

2^2=4

then we divide the numerator and denominator by 4-x or as they said, "cancel (4-x)", to get 1

hi hit

bit

[arctic raven]

$\frac{4-x}{(4-x){\rm junk}(x)}=\left.\frac{a}{a{\rm junk}(x)}\right|_{a=4-x}=\frac{1}{{\rm junk}(x)}$

[flat frigate]

Flip

[lean otter]

what abt sqrt x^2

[shadow sparrow]

(sqrt x)^2=x

[lean otter]

i see

is that always the case?

[arctic raven]

it's actually |x| but it's approaching 4, a positive number, anyway

so around 4, |x| just behaves like x lol

[lean otter]

so cancelling out gives us 1?

[shadow sparrow]

with how sqrt is defined yes

[shadow sparrow]

dividing 4-x by 4-x gives us 1

[lean otter]

is this always the case

even with different numbers

[shadow sparrow]

unless u are dividing by 0

[dreamy plover]

2^2 = 4 and under root x^2 = x

[shadow sparrow]

in this case 4-x is approaching zero but isnt equal to 0

[lean otter]

ok i understand now

.close

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[shadow sparrow]

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[light junco]

I wanna know what is considered a solution regarding rational equations.

Is it because it isnt considered a restricted value or not?

Can a solution be zero?

[lean otter]

yes a solution can be 0

[real rock]

That’s wild

[craggy sedge]

yes x is 0

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[light junco]

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[stone barn]

Idk what the frick this is but pls help with functions

[snow robin]

f(g(x)) means that instead of x being the input of f you take g(x) to be the input

[stone barn]

Oh

But like idk how to show it’s h(x)

[snow robin]

try to write out f(g(x))

and see what it is

[stone barn]

Okay

Idk I got this, I don’t think I did it right

[lean otter]

ok open brsckets

[fathom jewel]

open them brsckets

[lean otter]

(a-b)² = a²+b² -2ab

[stone barn]

You mean multiply out the brackets

?

[lean otter]

both

[stone barn]

Uh

Okay

Idk what u mean by both

Sorry I’m rlly slow

[lean otter]

.

[stone barn]

Idk man this is what I’m getting

.close

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[sacred crane]

is there someone speak german

[lean otter]

nein

[fathom jewel]

ja

[sacred crane]

can you translate this problem for me

[fathom jewel]

wo hast du probleme

[sacred crane]

on the picture

where should i draw the tangent line

[fathom jewel]

am Punkt P(5,y_p)

y_p kriegst du mit f(5)

[fathom jewel]

sqrt(x-1) stellt nur die obere hälfte da

"halb parabel"

,w plot sqrt(x-1) between 0 and 10

[flat frigate]

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[sacred crane]

that looks cool

[fathom jewel]

sqrt(x-1) kann nie negativ sein

[sacred crane]

du hast recht

[fathom jewel]

jetzt rechne mal f(5) aus

sqrt(5-1)

?

[sacred crane]

aber wo ist tangente

[fathom jewel]

um die kümmern wir uns gleich

[sacred crane]

2

[fathom jewel]

ja

also am Punkt P(5,2) ist die Tangente

Wir müssen die bestimmen

dafür brauchen wir die steigung

wir können dafür die 1. ableitung ausrechnen

weißt du wie ableiten geht

[sacred crane]

1/(2(x-1)^(1/2))

[fathom jewel]

alter schwede

richtig

jetzt an der stelle x = 5

f'(5) ausrechnen

[sacred crane]

1/4

[fathom jewel]

richtig

Die tangenten gleichung ist eine geradengleichung der form y = mx +c

wir haben m

[sacred crane]

m=1/4

[fathom jewel]

,, y = \frac{1}{4}x + c

[flat frigate]

bacc

[fathom jewel]

Um zu c zu bestimmen

können wir den Punkt P einsetzen

weil wir wissen das dieser Punkt auf der tangente sein muss

P(5,2)

[sacred crane]

c=3/4

[fathom jewel]

alter

krass

,, y = \frac{1}{4}x + \frac{3}{4}

[flat frigate]

bacc

[sacred crane]

danke

[fathom jewel]

,w plot sqrt(x-1) and x/4+3/4 between -3 and 10

[flat frigate]

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[fathom jewel]

stabil

[sacred crane]

wie kann ich das Flachenstuck skizzieren

[fathom jewel]

das flächen stück ist sozusagen die region, die beide funktionen einschließen

mit der x-achse

also einfach mal die beiden funktionen zeichnen im koordinatensystem

dann siehst du schon

[sacred crane]

okay

was sollte ich jetzt machen

[sacred crane]

i see

[fathom jewel]

Jetzt will man dieses Flächenstück rotieren um die x-Achse

Da gabs eine Formel für

Wir brauchen übrings 2 Integrale

[sacred crane]

integral

[fathom jewel]

,, V = \pi \int_a^b f^2(x) : \dd x

[flat frigate]

bacc

[sacred crane]

warum ist es f^2

[fathom jewel]

so ist die Formel

die Funktion f wird quadriert

Im Prinzip addieren wir Kreissegmente auf die zu einem Volumen führen

[sacred crane]

ach so

[fathom jewel]

dx ist die Dicke des Kreises (infinitesimal klein)

[sacred crane]

wie vinden wir a und b

finden

[fathom jewel]

Und die Kreisflächen Formel war pi * r² und f(x) ist sowas wie der Radius des Kreises wenn du die das Bild anschaust

[fathom jewel]

Das kann man hier ablesen

unser erstes Integral geht von -3 bis 1

wie ist es beim zweiten?

[sacred crane]

1; 4

oder 3

[fathom jewel]

nicht 4

der Punkt P war bei x = 5

da endet es

[sacred crane]

5

[fathom jewel]

ja

So beim ersten Integral welche Funktion benutzen wir?

[sacred crane]

tangent?

[fathom jewel]

yes

[sacred crane]

ich habe eine Frage

[fathom jewel]

,, V_1 = \pi \int_{-3}^1 \left ( \frac{1}{4}x + \frac{3}{4} \right )^2 : \dd x

[flat frigate]

bacc

[fathom jewel]

yes

[sacred crane]

könnten wir zuerst tangent -3; 5 finden und von dem Volumen von Parabola subtrieren?

[fathom jewel]

das wäre so grob gesagt

das zweite intgeral

super 😄

,, V_2 = \pi \int_{1}^5 \left ( \frac{1}{4}x + \frac{3}{4} \right )^2 - \left ( \sqrt{x-1} \right )^2 : \dd x

[flat frigate]

bacc

[fathom jewel]

,, V = V_1 + V_2

[flat frigate]

bacc

[sacred crane]

mein deutch sehr schlecht😂

[fathom jewel]

bist du deutsch?

[sacred crane]

nein

[fathom jewel]

lol

but did you understand

[sacred crane]

yeahh but i cried bit inside

[fathom jewel]

the sqrt(x-1) is a Wurzelfunktion

but I agree that it resembles a half of a parabola

[sacred crane]

i think i can finish this one by myself

thank you very much

[fathom jewel]

bitte sehr!

[sacred crane]

do you mind if i ask you sometimes some math problems

[fathom jewel]

you can ask them here on the server

[sacred crane]

okay thank you

,close

[fathom jewel]

.solved oder .close

[sacred crane]

.solved

[safe radish]

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[lilac sun]

Why aren't these equal

[plucky elk]

,tex .freshman

[flat frigate]

pizzanator

[lilac sun]

But I thought int[f(x) - g(x)] = int[f(x)] - int[g(x)]

[plucky elk]

This is correct

[plucky elk]

This is what you did next with x=f(x) and y=-g(x)

[lilac sun]

For context I'm trying to do the slicing method using area of square

Where f(x) = 2-x^2 and g(x) = x^2

[lilac sun]

So what would be the correct approach on separating the 2 integrals

[plucky elk]

Expand and simplify once you do this

[lilac sun]

Can you give me example

[plucky elk]

(x+y)^2 = x^2 + 2xy + y^2

[plucky elk]

Use that to expand the left integral first

[lilac sun]

f(x) = x^4 - 4x^2 +4

[plucky elk]

Now plug that in here for f(x)

g(x) = x^4 as you correctly identified

Oh wait no you didn't

Then do the subtraction

[lilac sun]

Think it's wrong

[plucky elk]

Why do you think that

Show the original problem

[lilac sun]

Find the volume of the solid whose base is bounded by the curves y = x^2 and y = 2-x^2 and whose cross sections through the solid perpendicular to the x axis are squares

[plucky elk]

Yea the first one is correct then

[lilac sun]

I want to know if it's possible to solve it by this without combining them both

[plucky elk]

Why do you think this is wrong

You don't have to combine them both

[lilac sun]

Wait really first one is correct?

[lilac sun]

https://youtu.be/S4vu0317bS4?si=Ee6ZQswT4klrsxfL

I got it from this

[plucky elk]

Okay?

[lilac sun]

The 2nd one is the one that is correct based on this video

Sooo which one is the correct one

[plucky elk]

Trust the video

[fathom jewel]

Trust the process

[lilac sun]

So then which one is correct here

[fathom jewel]

Thats like asking a² - b² = (a-b)²

Oh it was already mentioned mb

[lilac sun]

I'm confused which one is the correct answer then

[lilac sun]

From this problem

[fathom jewel]

I agree the first one is correct

You calculate both volumes and then you subtract the smaller one from the bigger one

The second one would be like the volume of 2-2x² and that would also be a different shape

[lilac sun]

There's 2 attempts on that image which one? The bottom or top

[fathom jewel]

In order to get

You do these two volume integrals

[fathom jewel]

We subtract yellow volume part from purple to get to this

So the top option is correct

The bottom option calculates the volume of this

[lilac sun]

Then that means the video is incorrect? Cuz the video one has the bottom one as it's answer

[fathom jewel]

Also you forgot π

In both options too

[lilac sun]

?

[fathom jewel]

I understood their method

Are you supposed to revolve it around the x axis?

Cause if so then you cannot use this square/slicing method, it seems to me you get a different volume shape than from rotating around an axis

[fathom jewel]

You cannot use the washer method for this then

[lilac sun]

Idk I'm just trying to follow the tutorial on video

[lilac sun]

I'm pretty sure the 2nd one is the correct answer here (that's his answer and that's chatgpt answer)

[fathom jewel]

yes it is

[lilac sun]

So the 2nd one is the correct one? And not the first after all?

[fathom jewel]

the first one is the washer method, used to calculate the volume of function that you rotate around an axis

[lilac sun]

Wait so I did the "washer" method by accident ig

Lmao

[lilac sun]

Idk washer method yet

I'm in slicing method rn

[fathom jewel]

i'll show you some wait

lmao i cant draw it properly

[lilac sun]

What's that

[fathom jewel]

This like your region and now imagine you rotate it around the axis 360°

You will get a way different shape than from your slicing method

[lilac sun]

So that's the washer method?

[fathom jewel]

yes if you rotate stuff around an axis

however you calculate squares

[lilac sun]

For the slicing method using squares ~4.26 should be the answer right?

@fathom jewel

[fathom jewel]

yea

,calc 64/15

[flat frigate]

Result:

4.2666666666667

[lilac sun]

OK thanks

.close

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[fathom jewel]

. this would be like your shape if you rotated your region

. so it's like very different

[lean otter]

Part b

How to do

[lean otter]

Oh then how do u exactly solve this further

I get how u made this into disguised form

But

What to do next

[tight meteor]

Use your answer from part a

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[lean otter]

Wait my part a is wrong

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[crude star]

how can the boundary points of S be {1, 2}?

shouldnt it be {1, 3}?

[lean otter]

it should be

[crude star]

alr ty

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[little spindle]

I've used a website like pubmed or pubchem to look up chemistry or medical related research papers. What website do you often use to look up mathematical papers?

[plucky elk]

This is better for #math-discussion

Help channels are for math questions, typically homework

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[azure vigil]

Hola. Mi problema tiene que ver con logica proposicional.
No comprendo cual es la diferencia entre la implicación logica y la condicional material. Tengo entendido que tienen el mismo valor de verdad en la tabla de verdades, y que en tales proposiciones tal que P y Q, la implicación se escrible "P => Q" y la condicional material se escribe "P->Q".

[lean otter]

in this ode, using existence theorem we get that there exist no solution in neighbour of (0,0) as dy/dx =f is discontinous in rectangle around (0,0) but then they say there's two solution at (0,0)...how is this possible...isn't it against the existence theorem? I'm so confused, help! 🙏

[safe radish]

@azure vigil Has your question been resolved?

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

[lean otter]

#❓how-to-get-help

[safe radish]

@azure vigil Has your question been resolved?

[fast gazelle]

(the same question, or at least i think probably the same question, was asked here #proofs-and-logic message and i answered there)

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[vapid hull]

The deck of a bridge is suspended 275 feet above a river. If a
pebble falls off the side of the bridge, the height, in feet, of
the pebble above the water surface after t seconds is given
by y − 275 2 16t
2
.

(a) Find the average velocity of the pebble for the time
period beginning when t − 4 and lasting
(i) 0.1 seconds (ii) 0.05 seconds (iii) 0.01 seconds
(b) Estimate the instantaneous velocity of the pebble after
4 seconds.

[vapid hull]

My work so far

<@&286206848099549185>

[modern tree]

Sup

[vapid hull]

Hey

[modern tree]

What is it

[vapid hull]

I realized I did it wrong and I am completely thrown off on how to do this or what the question is even asking me

Calculus

Calcus 1

calculus 1*

[modern tree]

What is the question

[vapid hull]

This

[modern tree]

Say it in text

[vapid hull]

The deck of a bridge is suspended 275 feet above a river. If a
pebble falls off the side of the bridge, the height, in feet, of
the pebble above the water surface after t seconds is given
by y − 275 2 16t^2
.

(a) Find the average velocity of the pebble for the time
period beginning when t − 4 and lasting
(i) 0.1 seconds (ii) 0.05 seconds (iii) 0.01 seconds

[modern tree]

(a)

The given function is
y

275

16
t
2
.

The average velocity can be calculated by using the formula,
v
avg

y
t
2

y
t
1
t
2

t
1
.

(i)

Here
t
1

4
and
t
2

4
+
0
.
1

4
.
1
.

So,

v
avg

y
4
.
1

y
4
4
.
1

4

=
275

16
4
.
1
2

275

16
4
2
0
.
1

=
275

268
.
96

275

256
0
.
1

=
6
.
04

19
0
.
1

=

129
.
6

(ii)

Here
t
1

4
and
t
2

4
+
0
.
05

4
.
05
.

So,

v
avg

y
4
.
05

y
4
4
.
05

4

=
275

16
4
.
05
2

275

16
4
2
0
.
05

=
275

262
.
40

275

256
0
.
05

=
12
.
6

19
0
.
05

=

128

(ii)

Here
t
1

4
and
t
2

4
+
0
.
01

4
.
01
.

So,

v
avg

y
4
.
01

y
4
4
.
01

4

=
275

16
4
.
01
2

275

16
4
2
0
.
01

=
275

257
.
28

275

256
0
.
01

=
17
.
72

19
0
.
01

=

128

--

(b)

The instantaneous velocity of the pebble after 4 seconds can be calculated by finding the derivative of the function
y

275

16
t
2
.

d
y
d
t

d
d
t
275

16
t
2

=
d
d
t
275

d
d
t
16
t
2

=
0

16
2
t
2

1

=

32
t

Substitute t=4 into the obtained derivative.

d
y
d
t
t

4

32
4

=

128

[vapid hull]

Why is it all vertical?

Nvm I copied and pasted into docs, ima read it rq

[vapid hull]

Dude its kind of confusing me would you mind sending what you copied and pasted but with a screenshot?

[modern tree]

Ok sure

These will work

[vapid hull]

It does thank you

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[cold jasper]

hihi. im having kind of a debate with my gf because she says there's something wrong with this process because "the -5 is multiplying", however when i expand it (second pic), it works. it says "factor by common factor"
did i do something wrong? is this process incorrect? 👀

[mellow cypress]

First one is correct

Second one doesn’t follow the method requested

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[sour haven]

MY I get help please?

[violet shore]

For which questions ?

[sour haven]

Oop. Basically all except one. I’m not confident at all

If that takes up too much of ur time could you help me with 3-4?

[violet shore]

Alr

[sour haven]

tysm

[violet shore]

Just to check have u learnt binomial expansion?

Cuz doing that expansion by hand is brutal

[sour haven]

yep. Taught in class I just don’t get it at all

[violet shore]

So all binomial expansion is. Is a way to expand brackets like (x-1)^of a big number fast

So think of it as a formula to speed up expanding brackets. Thats all it actually does

There is 2 main methods to do it when the power is a whole number. Both are fundamentally the same. Just preference. You can use the choose function. Or you can use pascals triangle

[sour haven]

Tysm! Yeah I think I just clicked. It’s really 8 I’m confused about now

[violet shore]

Are you familiar with pascals triangle

Dw if not

[sour haven]

Yeah. I had a friend explain it to me

[violet shore]

So basically our coefficients of our expansion for q3. Are just the 8th row of pascals triangle. However our number term (-1) makes some of our terms negative. Specifically every 2nd term. So all the coefficients are. Is pascals triangles 8th row. Add minus add minus. The whole row

I make the answer 0

I recommend laying out ur binomials like this

It makes it really easy to see whats going on

For just coefficients however you can use a much quicker method.

For q4 thats just some annoying expansion. The best trick is to learn to only use terms that are relevant to ur unknown. But beyond that not much to say

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[tardy copper]

does normalized decimal floating-point form mean 0.321 or 3.213

[tardy copper]

I am lost by the definition my book provided

[torpid fable]

Send

[tardy copper]

Am I missing something? is it only 1. if the first int is a 1?

[torpid fable]

It would be 0.321 according to what your book says. It doesn't really matter which one you use, but I guess in binary it may make more sense since we always know the first digit that will be before the "binary" point, since it will always be 1, and the computer can implicitly know that without storing a bit saying its 1, while in decimal it could be 1 through 9

[tardy copper]

Ok, so all its saying is that, for this case, we are considering the form to be 0.12345 where the first number to the right of the decimal is not 0

[torpid fable]

yeah

[tardy copper]

ok thank you

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[floral ermine]

so for this function:
f(x) = -x+b if x<-4,
2 if x = -4,
-4/(x-b) + 0 if x>-4 and x≠b

for what values of b does f have a jump discontinuity at -4?

[sly needle]

wait, if x < -4, why is x set to -4 if -4 < -4 is a false statement

[floral ermine]

its a piecewise function, limits

[sly needle]

oh

so it starts at the origin, goes down 1 and then right 1 infinitly to the right, but to the left, it stops when x=-3?

[floral ermine]

thats what the graph looks like

[manic radish]

Your function f has a jump discontinuity at x=-4 if f just to the left of x doesn't equal f just to the right of x. How can you write this algebraically?

[floral ermine]

lim 4 to the left -> 4+b=2 -> b=-2
lim 4 to the right -> -4/(-4-b)=2 -> b=-2

[floral ermine]

is that what you meant?

coz like i know that the function is continuous at b=-2 so would the jump discontinuity be (-♾️, -2)U(-2, ♾️ )

[manic radish]

I mean $f(x)$ has a jump discontinuity at $a$ if
$$\lim_{x \rightarrow a^-} f(x) \ne f(a) \quad\mbox{or}\quad \lim_{x \rightarrow a^+} f(x) \ne f(a)$$

[flat frigate]

Shuba

[floral ermine]

lim -4 to the left would be -2 ≠ 2 and
lim -4 to the right would be -2 ≠ 2 so both would have jump discontinuities?

[manic radish]

\begin{itemize}\item what is $f(-4)$ \item what is $\lim_{x \rightarrow -4^-} f(x)$ \item what is $\lim_{x \rightarrow -4^+} f(x)$\end{itemize}

[flat frigate]

Shuba

[safe radish]

@floral ermine Has your question been resolved?

[floral ermine]

1. 2
2. -2
3. -2

[floral ermine]

.close

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[ivory elm]

Hey

the answer i got is 3sqrt(17)

and idk where i went wrong

i solved a =12

and then got PF = 3i + 12j

[untold scaffold]

theres a calculation mistake in this

[ivory elm]

Where is it

?

oh wait

should it be -OP

yeah ok lol

ty

[untold scaffold]

alrr

[ivory elm]

.close

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[pure bough]

can someone help me prove this with mathematical induction please

[pure bough]

proving the base case and getting the IH were easy

i tried using the IH but i somehow got my answer as 2^k

for n = k + 1 the whole LHS becomes 2^k - 1 + [(k+1)!/(k+1)!(k+1-[k+1])!]

[peak estuary]

not sure how you got that

[pure bough]

ill send my working

this is how i manipulated the IH

i dont know how to go further than this

[peak estuary]

why do all the binomial coefficients on the left still have k at the top

[pure bough]

i tried using the combination formula and got the LHS as 2^k

OH RIGHT

i get it

but then im even more confused, how on earth would i use induction here

[peak estuary]

do you know pascals identity?

[pure bough]

i know that NC0 + ..... + NCN = 2^N if thats what youre talking about

but i dont know how to prove that either

[peak estuary]

https://wikimedia.org/api/rest_v1/media/math/render/svg/7fe0036d81a5b9b34dff56f2853676fc3e1c5f29

[pure bough]

Oh, no i didnt know this

[peak estuary]

ehh why no preview

[peak estuary]

well thats essentially the same thing as what you are trying to prove

[pure bough]

haha, here it is for anybody else who may control f this in the future

[peak estuary]

do you know the relationship between pascals triangle and binomial coefficients?

[pure bough]

theyre the same right

[peak estuary]

basically, yes

and you know how in pascals triangle one entry is the sum of the two above

[pure bough]

yes

[peak estuary]

this is exactly that fact

expressed with the relevant binomial coefficients

note that (intuitively) it lets you get from n choose stuff to n+1 choose stuff

which is what you need in your proof

[pure bough]

i see

would inducbtion be the best method here?

[peak estuary]

yes

well, there is also a combinatorial proof which is also nice

but the induction proof also works nicely

[pure bough]

alright, thanks. im new to both binomial expansions and proofs so im trying to practice as much as i can

[peak estuary]

using this identity

[pure bough]

yea we havent learnt combo in school yet haha

ive had to prove divisibility with induction as well, not modular arithmethic

[peak estuary]

as a tip, write out the full sum for like n=5 or something and then use the identity on every term

that will help you understand what will happen in the general case

[pure bough]

piggybacking on that, is there a way of applying pascals identity here?

I can perhaps factor out the binomial coefficients or something

cause the sum of the coefficients will be 2^6

[peak estuary]

no, thats classic binomial theorem

[pure bough]

How do I approach a problem like this?

[peak estuary]

look at the binomial theorem and compare the two expressions

[pure bough]

rewrite it in its original form?

(1/3 + 2/3)^6

OH WAIT

yea thats actually really nice

thanks mate

[peak estuary]

yw

[pure bough]

.close

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[digital bay]

I am confused if f o g and g o f even invertible?

[digital bay]

thank you

I know how to do if neglect this

but I just wonder if the question should be asked like this

[stoic dune]

Not an inverse. -1 stands for "inverse image" here. The set of all inputs that map to the given output

Indeed, the function is not invertible

[digital bay]

OH ok i understand now

thank you so much

.close

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[keen reef]

how u do thise??

[lime dust]

!original

[safe radish]

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

[thin bridge]

try to simplify what you can

[keen reef]

the brackets???

[thin bridge]

start with what's inside the ()

[keen reef]

ye

it will be 3^1/2

[thin bridge]

no

[keen reef]

ayy....

[thin bridge]

how are you getting htat

[keen reef]

gettting rid of the threes...

by divideing

[thin bridge]

how does that leave you with 3^1/2

[keen reef]

oh i meant e^1/2

rip my bad

[thin bridge]

also put ()

e^(1/2)

[keen reef]

okok

[thin bridge]

now that you have
$$=3 + 2\ln\br{e^{\frac12}}$$
try to simplify that further

[flat frigate]

ℝαμOmeganato5

[keen reef]

the ln and e cross each other out right??

[thin bridge]

yeh

[keen reef]

so then it would be 3 + 2 (1/2) right??

[thin bridge]

yes

[keen reef]

so does the ln and e cross out each other entirely??

or just = 1?

[honest perch]

e is a number

[normal moss]

You know the rules for logarithms right?

[honest perch]

ln is a function and ln(e) = 1

[keen reef]

yo okok gocha

[honest perch]

ln(x) is defined such that e^(lnx) = x

[keen reef]

ohhhhhh i see

[honest perch]

so e^(ln e^1/2) = e^1/2 so obviously ln e^1/2 = 1/2

[keen reef]

ayyy i get it now

yo thanks

[honest perch]

welcome

[safe radish]

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[safe radish]

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

[sleek lava]

1

[frozen marlin]

okay well first off

notice that you can make an equation for the sloped wall

imagine this as a coordinate grid

[sleek lava]

yep 🙂

[frozen marlin]

what equation do you get?

[sleek lava]

1 sec

[frozen marlin]

sure

[sleek lava]

y=1.5x-45

[frozen marlin]

great job

[sleek lava]

more confused abt finding the parametric

[frozen marlin]

now, we know that it intersects the parabola at one point only

[frozen marlin]

what do you mean

[sleek lava]

its supposed to be a parametric equation for my mvc class

cus its tracing through time

[frozen marlin]

but-

you don't need to find a parametric equation here

only the coordinates of the point of impact

and the time of impact

[sleek lava]

it states its defined parametrically

[frozen marlin]

yeah, the projectile's path is defined parametrically

[sleek lava]

wait ok i figured it out

[frozen marlin]

which means both the x and the y are defined with respect to another parameter, "t", which is time

[sleek lava]

ty

[frozen marlin]

np

[sleek lava]

.close

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[ionic zenith]

for proof by induction

[ionic zenith]

im concufsed why it goes from there to there

since if its 2^r

r becomes just k since (k+1-1) = k

i justr dont see what they did with sigma fnc

[frozen marlin]

that's why you add 2^k

you change the upper index to k

and pull out the last term

[ionic zenith]

dont rly get that

[mossy lotus]

they split the summation into two parts, one where they know the answer from previous info and other where they dont

[deep briar]

the summation is
2^1 + 2^2 + 2^3 + 2^4 .... 2^k-1 + 2^k, so they took out the last 2^k and changed the index of the summation

[mossy lotus]

from the second step of induction, we know the sum till k terms, so we take the additional term out of the summation

[ionic zenith]

is it this for when n = k+1?

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[ionic zenith]

.close

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[night meteor]

Hi, I am working on a graph shwoing relationship between volume of fluid in coca cola bottle and the wet surface area, and I am having some trouble with the bottom part right now. the bottom part is a weird pentagon shape (i dont know how to describe it but i have attached some photos)
Help is appreciated tyty

[night meteor]

How do i find the surface area of the bottom part?

[hearty egret]

it seems star-shaped

[night meteor]

yes it does

is there any way i can use high school calculus for this?

[hearty egret]

maybe you could contact some cocacola factory to give you some information about the model of the basis

[night meteor]

but is there a way lets say to know the wet surface area when the coke is filled to a speicfic water level (red lines)

cus my hw requires me to do it with math 😭

[hearty egret]

try to contact them https://www.worldofcoca-cola.com/

they should know what form the basis has

[night meteor]

so asking for the blueprint of the coca bottle and obtaining functions then doing surface area of revolution am i right?

[hearty egret]

yes, it might work

ohh but it is not a revolution surface

[night meteor]

ok ill ask, but cus my work is due like tomorrow i dont know if they'll reply in time

[night meteor]

huh?

[hearty egret]

there is a star in between ..so it is not a revolution surface due to the star at the bottom

[tropic mirage]

heyy i am new at this app

or site

[night meteor]

yo wsp

[night meteor]

oh then is it possible to use calculus for surface area still?

[tropic mirage]

i wannabe good in maths i also have an exam next year around this time can you guys please guide me a bit

[night meteor]

you guys can go to #❓how-to-get-help

[tropic mirage]

okk

[granite eagle]

Would it be possible to do the reverse?
Like pretend the bottle DOESNT have the grooves, subtract off the extra space and then you’re done?

[night meteor]

ye ive thought of this, but my method is to

1. plot graph of volume against water level
2. plot graph of wet surface area against water level
3. combine both

by doing this i wouldn't be able to get the volume values if water level is at a level shown in my diagram

[bronze gust]

What class are you taking 😭

[night meteor]

im j ust doing my ib math internal assessment

[bronze gust]

Oh

Project work moment

[night meteor]

ye sad

i think imma just use a 3d model online

.close

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[edgy canopy]

I simply do not understand how it went from the plot of -log() being regressive to it being growing simply by subtracting f from its input of 1, keep in mind f is a value between 0 and 1

[lean otter]

Maybe look at the graphs of F(x) = x and G(x) = 1-x for values of x between 0 and 1 to understand at least partially what is going on.

I don’t understand the context of what you posted in full detail, but if f is a value between 0 and 1, then 1-f is also a value between 0 and 1 but in reverse: When f approaches 0, 1-f approaches 1, and vice versa

[edgy canopy]

yea that makes sense to me, but the fact that he simply flipped the -log() plot is what doesnt make sense to me.

example

-log(0.9)=0.04575749056

but if you look at the right screenshot then it would be far higher

doesnt make sense, maybe he did it to simplify the concept?

still very annoying

i hope ive overseen somethng

its andrew ng's machine learning specialization for some slight context

:/

he shouldve just stuck with the first graph, the one on the left and it wouldve made perfect sense

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@edgy canopy Has your question been resolved?

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[icy ferry]

<@&286206848099549185>

Can anyone help

[delicate musk]

remember to ping the helpers 15 minutes after you asked the question, not instantly

[icy ferry]

oh ok i will wait 15 min

[delicate musk]

but ill see what i can do

[icy ferry]

ok

[delicate musk]

i think the answer is 674

[icy ferry]

um how

[lean otter]

,calc ceil(2020/3)

[flat frigate]

Result:

674

[lean otter]

it's asking for the best case so for each "button click" you want new lights to be turned on

[icy ferry]

for the three changes they have to be used on differnet lightbulbs

[delicate musk]

what i did was divide 2019 (closest number that can be divided by 3 without decimal numbers), and i got 673. i actually added one more because another press is needed

[delicate musk]

exactly what i did

[icy ferry]

you have to use all three changes

[vast cloud]

When there are 4 lights, you need 4 presses

[delicate musk]

exactly- wait im confused

[vast cloud]

so the ceil(N/3) strat doesnt work

[icy ferry]

yes 4 lights need 4 press

[vast cloud]

But with 5 lights, u can get away with only 3 presses

[icy ferry]

yeah

[vast cloud]

and 6 lights ofc ubcan do with just 2 presses

[delicate musk]

im pretty sure it says when there are 3 lights, one press is needed. my brain is not working

[icy ferry]

my teacher said to find a pattern

and then 7 needs 3 press

[vast cloud]

it takes 673 presses to toggle the first 2019 lights (so that the last six lights look like XXXXXO) then XXXOOX, XXOXOO, XXXXXX. 676 in total

[icy ferry]

oh

ok

thx

[delicate musk]

oohhh

[vast cloud]

it takes 672 presses to toggle the first 2016 lights, so that the last 5 look like XOOOO, then OXXOO, then XXXXX, so u can acc do it with 674, so they were right after all but only by accident

[delicate musk]

now i realized my mistake

thanks

[icy ferry]

oh so its 674

[lean otter]

ceil(674/3) works because for the 4 remaining lights, i need exactly 2 clicks which is basically ceil(4/3) and that's the sole edge case

maybe semantically you could argue that ceil(674/3) would need more context

[vast cloud]

i mean that in general for N lights its not ceil(N/3)

[lean otter]

Oh, probably would need to try it out for all N = 2 (mod 3)

but yeah fair enough

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@icy ferry Has your question been resolved?

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[whole minnow]

try derive

[outer vector]

im not really good at deriving with these questions

[whole minnow]

Radius

[whole minnow]

Should be equal

Area of sectors can be written as 1/2r^2 theta

Half of radius and arc length gives you the area

[whole minnow]

$S=πR²×L/2πR=LR/2$

[flat frigate]

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

[whole minnow]

Assume the degree of AOB is n

You have $\pi r^{2} \frac{n}{360}$

[flat frigate]

Aestusy

[outer vector]

Ohh

[whole minnow]

n can cancel out

Because the arc length$l=2 \pi r n/360$

[flat frigate]

Aestusy

[whole minnow]

so they cancel

You associate n and substitute in the first equation

Write n in terms of l you can associate area with l and r only

[whole minnow]

From here you associate n in terms of l because you only have l you don’t have n

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[wise hill]

Could someone please help me?

[sleek plank]

have you tried [-1, 0, -5] for the last row?

[wise hill]

How did you get that? No. I did not.

[dapper star]

-x = -5

[wise hill]

I wonder why its not ok to divide both sides by -1

Or why it shouldn't be

doing that gives x = 5

There is no y so you'd need to do x + 0y = 5