help-26

[craggy haven]

and mapped it into Q

[waxen flame]

oh yeah that works too, but you have to worry about multiple decreases

[craggy haven]

you would encode those as a run length of 0

[sturdy jasper]

what does the tuple represent

[waxen flame]

basically you encode the value of the function at each point with a number of 1s and use 0s to separate the points if they're different

or something like that

honestly don't even remember anymore LOL

[craggy haven]

look at the functions of {0,1} -> N that we wrote out

100000000000
110000000000
111000000000
111100000000
111110000000
111111000000
... (infinitely many)
111111111111... (forever)

[sturdy jasper]

mhmm

[craggy haven]

these each map to a natural number describing how many 1s are at the front

[sturdy jasper]

what if there’s no 1s

like 20000000000000

[waxen flame]

then you just write (2, 1) (0, infinity) or whatever

[craggy haven]

no it would be (1, 0, infty) because there's one 2, zero 1s, and infinity 0s

(your way would work as well but this is what snow was describing)

[sturdy jasper]

so 10000000000 would be (1, infinity)? and it maps to the natural number 1

[sturdy jasper]

but what would this map to

[craggy haven]

eh, don't worry about mapping it to natural numbers

lean heavily on the fact that |N x N| = |N|

[sturdy jasper]

so is it basically a set of finite length strings which is countable?

[craggy haven]

as long as those strings have finite length and each slot has countably many symbols to choose from, yeah that's countble

[sturdy jasper]

hmmm but they aren’t finite tho right since it can go on forever

[craggy haven]

can they?

[sturdy jasper]

isn’t there an infinite number of natural numbers for the input

well at some point it’ll repeat tho

but if we just take the non repeating portion yeah ig it’s finite

[craggy haven]

remember what these strings are, each element describes how many of a certain number we get in a row

so we reduce 553333333333332222111111111111111... down to (2, 0, 8, 4, infty, 0)

[sturdy jasper]

how do we know that there’s 0 0s

[craggy haven]

we could kinda put whatever we wanted there

but the function i described repeats 1s infinitely

[topaz sinew]

@sturdy jasper Has your question been resolved?

[sturdy jasper]

ok i think i can solve it from here thanks

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[neon iron]

why is the graph of {e^x} like this? {.} represents fractional part function.

[cinder light]

fellow jee 2025 aspirant 🤝

[noble laurel]

the function increases in intervals until every time it hits a new whole number

the intervals decrease in size as x gets larger

because e^x grows faster

and hence it hits a new whole number more often

e^1 = 2.718...

[waxen flame]

those are just pieces of the thing shifted downward

you could shift them up and reassemble the curve

[noble laurel]

somewhere in between 1 and 2, e^x becomes equal to 3.0..... , and at this point the fractional part is 0 and a new interval begins

ad x becomes larger than say 1, or 2, this happens more often

and the intervals become smaller and smaller

,w e^100

[thorny flame]

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[noble laurel]

,w e^101

[thorny flame]

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[noble laurel]

in between one x-value here, between 100 and 101, see how the fractional part must've hit 0 a lot of times

[neon iron]

oh

[neon iron]

i didn't understand what you said

[waxen flame]

imagine you cut up the curve of e^x at each x value where e^x is an integer

and then you let all the pieces fall to the x-axis

that is what you get

[neon iron]

but why did this happen?

[waxen flame]

that's basically what taking the fractional part is doing

[noble laurel]

You should read my responses for some reasoning behind why this happens

[neon iron]

ok

got it thanks

.close

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[nimble crown]

using the remainder theorem, show that x^2+x-6 is a factor of 3x^4+4x^3-17x^2-13x+6

[nimble crown]

so far I have just broken up x^2+x-6 into (x+3)(x-2)

wait nvm

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[rigid epoch]

i have a question regarding modulo
What would be the value of |{[x-a)^2+(y-0)^2 ]-[(x+a)^2+(y-0)^2]|

[rigid epoch]

its okay if its in equation im just confused about

the signs

like how exactly will the signs be affected

And can you transpose modulo

[glossy basin]

What do you mean modulo @rigid epoch

[rigid epoch]

these things | |

around an equation

or value

[glossy basin]

See

(x-a)^2 = x-a
(y-0)^2 = y-O
(y-0)^2 = x+a
(y-0)^2 = y^2

|[(x-a)^2 + y^2] - [(x+a)^2 + y^2]|
can be re-written as
|[(x^2 - 2ax + a^2) + y^2] - [(x^2 + 2ax + a^2) + y^2]|

The signs will change because we r distributing negative sign outside the brackers.
so, now it is,

| (x^2 - 2ax + a^2 + y^2) - (x^2 + 2ax + a^2 + y^2) |

After all they cancel out, we are left with

| - 4ax |

|-4ax| = 4|a||x|

vertical bars or absolute value notation

are (| |)

Eg - f x - 5 is positive, then |x - 5| = (x - 5).
If x - 5 is negative, then |x - 5| = -(x - 5) = -x + 5.

In both cases, the result is a non-negative value.

In both cases, the result is a non-negative value..

Any other question?

Is your question solved?

[rigid epoch]

im still a little confused

[glossy basin]

Ok can you explain in what part?

[rigid epoch]

so lets take an example

|5x-2y-19|=18

then it would become

5x-2y-19=18

?

[glossy basin]

Ok.

Consider both the positive and negative cases.

Positive Case:
We set the expression inside the absolute value bars equal to the positive value on the other side:

Negative Case:
We set the expression inside the absolute value bars equal to the negative value of the other side:

Positive case : 5x - 2y - 19 = 18

[rigid epoch]

so it would be +/- 18?

[glossy basin]

Nevative case : 5x - 2y - 19 = -18

Wait.

I explain better.

[rigid epoch]

oke

[glossy basin]

Positive Case:
We set the expression inside the absolute value bars equal to the positive value on the other side:

5x - 2y - 19 = 18

---

Negative Case:
We set the negation of the expression inside the absolute value bars equal to the positive value on the other side:

-(5x - 2y - 19) = 18

---

Expanding the negation:

-5x + 2y + 19 = 18

---

Now we solve this equation for x and y.

So, in total, we have two equations to solve:

Equation 1: 5x - 2y - 19 = 18
Equation 2: -5x + 2y + 19 = 18

Equation 1:

5x - 2y - 19 = 18

We can simplify it:

5x - 2y = 18 + 19
5x - 2y = 37

---

Equation 2:

-5x + 2y + 19 = 18

Simplifying it:

-5x + 2y = 18 - 19
-5x + 2y = -1

So, we have the system of equations:

5x - 2y = 37
-5x + 2y = -1

Stil confused?

Ping me if confused

[rigid epoch]

oh ok i think i got it

so basically

|5x-2y-19|=18 can be two equations

one being 5x-2y-19=18

other being 5x-2y-19=-18

and then we can simplify

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[undone flicker]

By Binomial expansion (x+a)^n has n+1 terms. How many terms in (x+a)^n -(x-a)^n

[tall wolf]

wouldnt it just be 0?

[undone flicker]

Noo

Ohh wait

Edited

How?

[tall wolf]

nvm

it would prob still be n+1 terms no?

[craggy haven]

try it with small n

like 1

[undone flicker]

N=1
So a
1

@craggy haven

[craggy haven]

intuition comes from multiple examples generally, not just one

[undone flicker]

I see right

[soft rune]

You can prove it using the binomial theorem

The summation formula

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[west pendant]

We want to select an ordered pair (A,B) of proper subsets of [n] so that A ∩ B /= ∅, where [n] = {1, 2, 3, 4, ... , n}. In how many different ways can we do this?

[wary tulip]

for each size A could be, count how many valid B's you could choose

so i mean...
if |A| is 0 how many B's could pair with it?
if |A| is 1 how many B's could pair with it?
if |A| is 2 how many B's could pair with it?
and so on

if you can do that and know, for each [k] in [n] union {0}, how many subsets of [n] of size k there are then you should be good to go

[topaz sinew]

@west pendant Has your question been resolved?

[west pendant]

@wary tulip I know there are 2^n subsets for A and 2^n subsets for B. So, you can make a total of 4^n total pairs correct?

Then, you consider how many pairs are there such that A ∩ B = empty set.

And, I guess you subtract that from 4^n?

[wary tulip]

yes you can do that

[west pendant]

How would I figure out how many pairs there are such that A ∩ B = empty set though?

[wary tulip]

let's say n is 8 and A has 3 elements

how many possibilities for B?

[west pendant]

Not really sure.

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[wary tulip]

the sets that don't intersect A are the ones that have only the 5 elements in [8] that aren't in A

[west pendant]

Agreed. But, still not sure how I'm able to figure out the possibilities for B.

[topaz sinew]

@west pendant Has your question been resolved?

[west pendant]

@wary tulip You still there? I was thinking that there would be 3^n choices since each element could either belong in A, B, or neither since we want the condition such that A intersect B is empty.

Then, the answer would be 4^n - 3^n right?

[topaz sinew]

@west pendant Has your question been resolved?

[keen raptor]

Yeah that works

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[nimble crown]

can someone check if I got this right?

[nimble crown]

the questions asked me to find y^2=f(x) of the function

theres no function, onnly the graph was given, the pencil is mine

[prisma badger]

I must say, great graph!

You captured all of the essence of what a graph of y²=f(x) would look like

[nimble crown]

really?

[prisma badger]

Do you need to justify anything here?

[nimble crown]

don't root graphs need to have an asymptote when there is an x-intercept?

because I was thinking of adding one

[prisma badger]

asymptote? At x=-1?

[nimble crown]

vertical asymptote

[prisma badger]

But why?

[nimble crown]

oh not asymptote my mistake

my book says a graph with an x intercept make a vertical tangent

hold on let me take a picture real quick

of a different question

[prisma badger]

That may be the case if you were graphing y = 1/f(x) or something

[nimble crown]

,rotate

[nimble crown]

If you look in this example, I had to put vertical tangents so the graph to the left from zero would be like a semi circle

[prisma badger]

Hmm, ig you would need chain rule to really pin down if it would have undefined slope there

Take something like f(x) = |x|^3. Then the graph of y = √(f(x)) has no asymptote at 0

Even though y = f(x) has x-intercept at x=0

But if you take say f(x) = |x|^(1.5) it again looks like the same sort of curve but now graph of y = √(f(x)) involves undefined slope at x=0

So ig in your original question, it's not really possible to tell which of the above two behaviours your function falls in (around x = -1) unless you have the actual equation

[nimble crown]

i see...

let me input this in desmos real quick so I can understand what is happening

[prisma badger]

That's a good idea

[nimble crown]

I don't think I am inserting this in it right

[prisma badger]

You need √(f(x))

[nimble crown]

oh my mistake

ok I get this

I know how to graph something like this but what equation would produce that original question I was asking

[prisma badger]

Problem with that is it could be anything even a piecewise defined function

[nimble crown]

indeed...

[prisma badger]

Unlike the clearly cubic looking second question

[nimble crown]

but what function makes a reatively simple cubic

turn into this

because my logic tells me

[prisma badger]

No cubic written in terms of x has horizontal asymptotes

[nimble crown]

this is what my logic tells me

but the latter is right

[prisma badger]

But if the curve near x=-1 behaves like |x+1|^3 then you wouldn't have a vertical tangent

[nimble crown]

can you explain that?

[prisma badger]

,w graph y = |x+1|^3

[thorny flame]

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[prisma badger]

If your function was piecewise defined then maybe near x=-1 it has this behaviour

But,

,w plot y^2 = |x+1|^3

[thorny flame]

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[prisma badger]

But now look what it becomes

[nimble crown]

oh I see what your getting at

[prisma badger]

More absurdly look at

,w plot y^2 = |x+1|^2

[thorny flame]

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[nimble crown]

what in the world

[prisma badger]

Which stems from perfectly normal

,w plot y = (x+1)^2

[thorny flame]

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[nimble crown]

is it because its not actually intercepting the x-axis?

more so its just 'grazing' the x-axis

[prisma badger]

It's partially because at the intercept on x-axis the function has derivative 0

[nimble crown]

oh I see

how does that relate to the root though?

[prisma badger]

For the second problem, your graph intersects x-axis at a non-zero slope

[nimble crown]

non-zero slope is referring to gradient yes?

[prisma badger]

Yeah you could say that

[nimble crown]

alr

[prisma badger]

Try to look at how chain rule works out

[nimble crown]

wait a second

I just found this image

it doesn't have a vertical tangent because assuming R is on the x-axis

it would just become horiztonal and therefore there would be no vertical in the first place

[prisma badger]

Are you referring to tangent to this graph itself or square root of this graph?

[nimble crown]

tangent to the graph itself

[prisma badger]

I mean, you could have vertical tangents at roots

Take say y = cbrt(x)

[nimble crown]

ok

[prisma badger]

Would you say y=x^3 grazes the x-axis at origin?

[nimble crown]

no

ok good point

I think I'm not 100% fully understanding this, I think I will just ask my teacher tomorrow

thank you for the help Euclid though for answering the original question I had

sorry I had you go on a wild goose chase

[prisma badger]

Also, I am not saying that the square root graph cannot have vertical tangent there just that we can't say for sure it does

[prisma badger]

Did kinda enjoy it though

[nimble crown]

no I get that, its just this is being taught differently to what you know, I feel like

thanks for the help though

I really appriciate it

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[white marlin]

Hey 👋 I’m in the 7th grade and I’m a bit confused on how to solve this cus one of my teachers got an answer different form another and idk how to solve it

[orchid gust]

You know how to divide fractions?

[white marlin]

Yeah but the teacher did this tying we’re he like swapped to one on the right

And made it like a multiple

[orchid gust]

Yeah then you multiply

You do the same here

[white marlin]

Oh oki Ty

[orchid gust]

Then you simplify from there and you're welcome

[orchid gust]

🙂

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[sudden sapphire]

Hello @everyone How can u be sure that the speed of light is constant
I mean if the time changes will it be constant then

[grim jacinth]

Bro really tried to ping everyone lol

[full falcon]

lmao

[grim jacinth]

You can look up the speed of light

It's constant everywhere

[sudden sapphire]

even when time slow down

?

ah u didnt said everywhen

[grim jacinth]

everywhen?

[grim jacinth]

yes

This is moreso physics

[drifting swift]

there are experiments that show light speed is indeed constant no matter how quickly your frame of reference is moving

[sudden sapphire]

so it is variable compare to time

[grim jacinth]

Go to the physics server for this

[grim jacinth]

What?

[sudden sapphire]

there is no official physics server

[grim jacinth]

#old-network message

[sudden sapphire]

Ah thanks

Bye

.close

[grim jacinth]

👋

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[tribal marlin]

I tried solving it using expansion of sinx

But it just became too lengthy

[grim jacinth]

Why not start with u = sin x?

[tribal marlin]

Not substitution again :whyyyyyyyyyy:

That become $\frac{U - U^U}{1-U+lnU}$

[thorny flame]

_kapp_

[grim jacinth]

That doesn't help a ton does it lol

It still looks bad XD

[tribal marlin]

Ikr

Once again - L hospital not allowed

and even if it was allowed i don't think i would differentiate such a thing

[grim jacinth]

What if you ask wolfram to do it?

[tribal marlin]

How do i do that?

[grim jacinth]

Maybe try plugging it into wolfram alpha and see if it can give you an answer

[tribal marlin]

The thing is idk how to do that........😶

Like how do i put limits on it from mobile

Latex?

[neon iron]

limit value is wrong

[neon iron]

seems alright to taylor

[tribal marlin]

Wdym ?

[neon iron]

taylor expand u^u and ln(u)

[tribal marlin]

Ohhhhhhh

Ok

Lemme try that

[tribal marlin]

I know expansion of ln(1+u) only

And u^u

Do is just expand ln(1+(u-1))

?

[neon iron]

no

we want to taylor about x = sin(pi/2) = 1

[neon iron]

so just this with no extra calculations

[tribal marlin]

Okie i will try that

I will close for now since it would take time

.close

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[silent mortar]

can someone explain what's happening here

[silent mortar]

haven't seen this in professor leonards videos yet

[eternal wing]

try drawing it

choose a to be some number like 1 or 0

[silent mortar]

i dont even know what those numbers are

I've only solved with just the integral sign

[silent mortar]

.

[eternal wing]

do you know what an integral can represent

[silent mortar]

area under a curve

so apparently you do F(top limit) - F(bottom limit)

[silent mortar]

right?

[eternal wing]

yes

but forget antiderivative now

draw it

draw some random positive function

whatever u want

mark x = 4, 3

and let a be some other x value

that's less than 3

and then I want you to see what areas the integrals represent

[soft spindle]

you could also go this route by turning all the integrals into this form and getting rid of the F(a) terms

[silent mortar]

So the answer is positive 2?

cause it's just F(3) - F(4) and those values are given as 6 - 2

[silent mortar]

.

[eternal wing]

its -2

wait

[silent mortar]

3 is on top

[eternal wing]

they wrote it as the integral from 4 to 3

yeah thats weird

so 2

[silent mortar]

ok thanks

.close

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[silent mortar]

.reopen

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[silent mortar]

can u help me with this one too

[glacial frost]

14

[silent mortar]

explain

[glacial frost]

https://tenor.com/view/run-running-running-away-gif-26050957

[soft spindle]

i dont think so lol

[silent mortar]

can u help

[soft spindle]

y = g'(x), so g(x) = integral of y = area under y + C

[silent mortar]

but i dont have a function

[soft spindle]

you can find the area under y from the picture

and you can find the C from g(0) = 8

[silent mortar]

at 15 wouldn't the area be 0

also how is g(0) = 8

is that supposed to g'(0)?

[soft spindle]

well you have a positive contribution from 20 and 4, and a negative contribution from 10, so the total area

[silent mortar]

so it is 14

+8

[soft spindle]

yea, u would add the area from 0 to 15 (14) plus the value of the function at 0 (8)

[glacial frost]

what did i say

[silent mortar]

u were wrong lol

.close

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[slender stratus]

qn 12 part c is asking for ?

[slender stratus]

i have no idea only know its a degree

[hazy pumice]

AF = 36m*cos(35 deg) - 20m,

Angle EAF = arccos(AF/36)

[slender stratus]

so part iii requires u to find angle EAF?

or EAC?

i know how to do it just dont know what they asking for

or EAB

idk...

[hazy pumice]

Oh, and you need to subtract 35 from that

[slender stratus]

ohh i see

alright thanks!

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[still surge]

how did they get 10A= -70?

[soft spindle]

the determinant of the matrix, using the two 0s in the bottom row

or any way you like

[topaz sinew]

@still surge Has your question been resolved?

[still surge]

so we get 10a = -70 by determinant?

[soft spindle]

yea, thats what the lines on the sides indicate

have you computed them before?

[topaz sinew]

@still surge Has your question been resolved?

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[topaz sinew]

Channel closed

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[urban tusk]

how do I find dy/dx for this

what's the first step, can someone guide me through it

[novel echo]

chain rule

[urban tusk]

oh shit I totally forgot about that

okay brb I'll try

what should the answer be?

should it be 3 ⋅ (1-5x^3)^3 ⋅ (-15x^2) or (-45x^2) ⋅ (1-5x^3)^3

or are they both the same thing

[novel echo]

both are the same thing

[urban tusk]

by same thing I mean

both acceptable as a final answer

?

[novel echo]

depends on your professor but both are acceptable

[urban tusk]

okay so how do I find this for this

[novel echo]

the second derivative

[urban tusk]

I know how to do it for the regular one

[novel echo]

hmm product and chain rule

[urban tusk]

but is it diff for chain rule

[novel echo]

wdym

[urban tusk]

this is the final answer

oh wait let me try something

[topaz sinew]

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[urban tusk]

wait what

[novel echo]

woah

Bot tripping

[urban tusk]

yeah I have no idea

can someone demonstrate how to do the working

[novel echo]

are you familiar with:

constant multiple rule
product rule
chain rule

[urban tusk]

I think so but

not really too familiar with 2nd derivative

[novel echo]

ok so, separate the -45 from the rest of the expression

you’re left with x^2(1-5x)^2

[novel echo]

it’s the same as any other derivative

you’ll use the same exact rules

the second derivative just means that you’re taking the derivative of the derivative

[novel echo]

you can use product rule and chain rule to solve this

don’t forget about your constant at the end: -45

you’re working with this $-45 (x^{2} \cdot (1-5x^{3})^{2})$

[urban tusk]

what

[thorny flame]

__saad

[urban tusk]

hold on let me try and process that

[novel echo]

yes

[urban tusk]

wait just to make things clear

are we working with 3(1-5x^2) ⋅ (-15x^3)^3 or (-45x^2) ⋅ (1-5x^3)^3

wait ignore that

I typed it wrong

[novel echo]

$-45x^{2} \cdot (1-5x^{3})^{2}$

[thorny flame]

__saad

[novel echo]

this

[urban tusk]

oh ok

[novel echo]

now, can you calculate the derivative of this?

[urban tusk]

brb I'll try it

btw I think the original question is ^3 at the end

[novel echo]

yes but you took the derivative

now you’re taking the second derivative

the derivative of the derivative

[urban tusk]

-90x

2(1-5x^3) ⋅ (-15x^2)

yeah this is what I mean when I said I don't know how

[novel echo]

let’s work through it step by step

first step

separate the constant

$-45 (x^{2} \cdot (1-5x^{3})^{2})$

[thorny flame]

__saad

[urban tusk]

okay so just so we're on the same page we're starting with (-45x^2)(1-5x^3)^3

right

[novel echo]

^2 at the end but yes

because of chain rule

[urban tusk]

do you mind if we start over from the origin of the question

[novel echo]

sure

[novel echo]

let’s start with this

[urban tusk]

I'll send for the first dy/dx

[novel echo]

$(1-5x^{3})^{3}$

[thorny flame]

__saad

[novel echo]

what is the first step for chain rule

[urban tusk]

so for the first part of the question which is to find dy/dx

this is right?

[novel echo]

yes, that is correct

[urban tusk]

okay and next to fibd 2nd derivative

what's first step

[novel echo]

well you can simplify your current answer by multiplying 3 with -15x^2

to get to this $-45x^{2} \cdot (1-5x^{3})^{2}$

[thorny flame]

__saad

[novel echo]

next step to make your life easier: separate the constant, this is known as the constant multiple rule.

An example of this rule would be: $\frac{dy}{dx}(2x^2) = 2 \cdot \frac{dy}{dx}(x^2)$

[urban tusk]

ok and

[thorny flame]

__saad

[urban tusk]

OH THE CONSTANT MULTIPLE RULE

[novel echo]

the constant in this case is __?

[urban tusk]

wait I'm trying to remember if I learned this or no

is it 45

or 045

-45*

[novel echo]

-45, correct

now you’re left with: $-45 \cdot \frac{dy}{dx} [(x^{2} \cdot (1-5x^{3})^{2})]$

[thorny flame]

__saad

[novel echo]

which rule can you use here?

[urban tusk]

just to confirm this is right

[novel echo]

yes but you’re multiplying the whole expression by -45, that means both (1-5x^3)^2 and (x^2)

[novel echo]

like in this

[urban tusk]

okay

next step is

[novel echo]

figuring out this

[urban tusk]

product rule?

[novel echo]

note that (1-5x^3)^2 is being multiplied by (x^2)

yes

correct

do you know how to use the product rule

[urban tusk]

I think

let me try

welp no I failed

[topaz sinew]

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

[urban tusk]

I'll try again

I think I'm stuck on the part where

[urban tusk]

I'm turning my initial answer into this

[novel echo]

oh

[novel echo]

that’s the same as this

with additional brackets

[urban tusk]

yeah that's still where I am now

[novel echo]

this*

ok so

you’re calculating the derivative of this using the product rule: now you’re left with: $(x^{2} \cdot (1-5x^{3})^{2})$

[thorny flame]

__saad

[novel echo]

the first step: identify your functions, f and g

in this case

f(x) = x^2

g(x) = (1-5x^3)^2

[urban tusk]

ok

[novel echo]

this is the product rule

try using this

[urban tusk]

how do I do it for this though

I'm not familiar with the ^2 at the end

[novel echo]

ok so, you’re using chain rule again

[urban tusk]

oh okay

wit but doesn't that leave me with two brackets

[novel echo]

yes, that shouldn’t be a problem once you get to that part

just like you did in the first part

[novel echo]

apply this rule and then try simplifying it

[novel echo]

correct

try simplifying it now

[urban tusk]

idk how to get started

[novel echo]

you have this $(x^{2} \cdot (2(1-5x^{3}) \cdot (-15x^{2}))) + (2x \cdot (1-5x^{3})^{2})$

[thorny flame]

__saad

[novel echo]

now, are you familiar with the laws of indices?

[urban tusk]

no

[urban tusk]

oh yes

[novel echo]

exponent rules basically

start with the first half

[urban tusk]

how did my equation turn into this

[novel echo]

same thing but in this format

[urban tusk]

is it possible if you show the working from start to finish so I get a general idea of how the question is solved

[novel echo]

sure

[novel echo]

fine if it’s like this?

[urban tusk]

I mean like until the point where it turns into this

[novel echo]

sure wait

[urban tusk]

okay ty

[topaz sinew]

@urban tusk Has your question been resolved?

[novel echo]

starting with the first half: $-15x^{4} \cdot (-2 + 10x^3)$
$\ - 30x^{4} + 150x^{7}$

[urban tusk]

alright I'll try to understand it and if I have any question I'll ask ty

[novel echo]

second half:
$50x^{7} - 20x^{4} +2x$

[thorny flame]

__saad

[novel echo]

final step:
$-45 ((-30x^{4} + 150x^{7}) + (50x^{7} - 20x^{4} +2x))$

$\ 1350x^{4} - 6750x^{7} -90x + 900x^{4} - 2250x^{7}$

[thorny flame]

__saad

[novel echo]

Final answer: $-9000x^{7} + 2250x^{4} - 90x$

[thorny flame]

__saad

[novel echo]

wow that is a mess

[thorny flame]

__saad

[novel echo]

sure

[topaz sinew]

@urban tusk Has your question been resolved?

[urban tusk]

can I use quotient rule for this question?

[primal marsh]

Yes

But you can also do it in part wich is probably easier

[urban tusk]

so this is what happened

initially I forgot that quotient rule was a thing

so I just elimated to denominator and differenciated it the normal way

and that worked and I got the correct answer

BUT

for some reason when I tried using the quotient rule with it, the answer is always wong

wrong*

I'll send a pic maybe you guys can identify what's wrong

oh nevermind there wasn't a problem

I just did mistakes

I guess my question now is

can both x^2 - 1/x^2 and 1-x^-2 be accepted as final answers

[primal marsh]

Yeah

[novel echo]

better to provide the first one as your answer

[urban tusk]

oh really

alright tysm

.close

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Channel closed

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Use .reopen if this was a mistake.

[urban tusk]

.reopen

[topaz sinew]

✅

[urban tusk]

btw

the list of my help questions are like gone

nevermind I found out why

thanks!

.close

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[inner crater]

Did I do the graph right? And how do answer the questions below

<@&286206848099549185>

[primal marsh]

Hey

Just look at the graph at 3am

[inner crater]

I'm still confused

Can you help

[primal marsh]

So the time line is on the x-axis

And the depth is on the y-axis

[inner crater]

Yeah

[primal marsh]

Look at the graph at 3am

Which is x=3

[inner crater]

I didn't put points at 3

I might have did the graph wrong

[primal marsh]

So i think the graph is wrong

At midnight and 0 am it should have 12

But yours have 2

[inner crater]

I'm not sure how to graph it

[primal marsh]

Also i think is should have 2 at 12am

So i think you should put points at 18 and 6

[inner crater]

I was trying to do it based on this example

[primal marsh]

Based on this i think

At 24am the depth is 12

At 0am the depth is 12

[inner crater]

Yeah

[primal marsh]

At 12am since its cycle is 12 it should be 12 as well

[inner crater]

I'm confused on why they started the graph at 8 in the example

[primal marsh]

8 is the level of the ocean

When high tide its gets bigger by 4 m

At low tide the level gets smaller

[inner crater]

Where does it tell you that

[primal marsh]

It donsent say particularly but thats what it means

[inner crater]

So in the example graph they use the 8 scale for mine what scale would I use

[safe acorn]

"No idea"

[inner crater]

?

[primal marsh]

7

I dont know what that bot was

You should use seven because its in the middle

[inner crater]

Leave

<@&268886789983436800>

[inner crater]

Oh ok

I see that they started at 0 then 4 would I start at 3

@primal marsh

[primal marsh]

Put points at
0am, 12
6am, 2
12am, 12
18am, 2
24am, 12

[topaz sinew]

@inner crater Has your question been resolved?

[inner crater]

Thanks

[topaz sinew]

@inner crater Has your question been resolved?

[inner crater]

.close

[topaz sinew]

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[midnight sinew]

Another exercise 5.5.5 asks me to show an analogue of this proposition but replace rational with irrational. This is my proof sketch:

Note: We have already proven that sqrt(2)>0 is an irrational number and it is easy show that (q)/(sqrt(2)) is also irrational for any rational q.

1. Since y>x, and sqrt(2)>0, we have that ysqrt(2)>xsqrt(2)

2. Thus by proposition 5.4.14, we have that ysqrt(2)>q>xsqrt(2) for some rational q.

3. Finally we can just divide across by sqrt(2) to get y>q/sqrt(2)>x and let q'=q/sqrt(x) and we are done.

I'm wondering if there's another way to do without using sqrt(2)?

Something like creating a Cauchy sequence whose limit lies between x and y or something about the supremum of a set?

Was thinking about it but I don't think it's possible to show that a Cauchy sequence will have irrational limit without first knowing that the limit is irrational, similar for a supremum of a set.

[drifting swift]

and it is easy show that (q)/(sqrt(2)) is also irrational for any rational q.
parentheses unnecessary; q/sqrt(2) is just fine

I'm wondering if there's another way to do without using sqrt(2)?
you can use sqrt(3) instead

[midnight sinew]

I see

[topaz sinew]

@midnight sinew Has your question been resolved?

[topaz sinew]

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[sharp lily]

i am having a hard time with this problem

[sharp lily]

here is my work

[sweet shard]

pain. you should simplify the numerator before taking the derivative

[novel echo]

no chain rule for h(x)?

[sweet shard]

,w expand (3x^2 + 5x + 9)(8x^4 + 9x^2)

[thorny flame]

Results provided by WolframAlpha

Click here to refine your query online
Upgrade to WolframAlpha Pro!

Requested by rie.mann#0

[sweet shard]

put that as your numerator and you won't need to worry about chain rule, just quotient rule for a rational function

[sharp lily]

i see thanks! is it the same for this one as well? just expand it?

[novel echo]

yes

[sharp lily]

okay cool let me try these 2 and ill come back to a channel if i still cant get it. thanks for your help!

.close

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[inner crater]

How to solve 1.a and b

[gusty bane]

do you know what a sinusoidal function looks like

[topaz sinew]

@inner crater Has your question been resolved?

[inner crater]

Yeah

I don't understand what the questions asking me

Is it saying are these sinusoidal functions?

[gusty bane]

it’s asking which situation can be modeled by a sinusoidal function

so which of those graphs could be the graph of a sinusoid

@inner crater

[inner crater]

To me they both look like they could be sinusoidal graphs

They both have a cycle

@gusty bane

[gusty bane]

that’s not the only condition to be a sinusoid

i think you can tell one looks like your typical sinusoid

and the other is not

[inner crater]

Yeah 1.a

[topaz sinew]

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[oak quiver]

Hello. Can someone please verify that my answer is correct?

[restive inlet]

show work

[vital relic]

If you submit, the machine will verify it wont it

[oak quiver]

150g will be 75g in 27days. So 150g will be 100 in 18 days. Every 9 days it decreases by 25g. Is this right ?

[restive inlet]

no

decay isn't linear

[vital relic]

it wont be completely gone in 54 days.

or negative in 63 days.

[oak quiver]

What’s the formula for half life?

[vital relic]

refer to your notes, textbook, etc

you shouldnt be asked this question without having been taught it

[oak quiver]

Ok I figured it out. It’s 16 days to reach 100g

A=150(1/2)^15/27

Thanks for checking

.close

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[neon iron]

why doesn't this approximate the value of the integral of sin(x) Interval = [0,pi]? On my calculator it just says "0.383288", while the actual value of the integral is 2. I don't know what I did wrong with the Riemann sums

[neon iron]

i subdivided the area into 100 rectangles, and since the length is all equal it is pi/100

[gray ridge]

x/100?

oh

i see

[neon iron]

i wanted it to slowly reach to pi

if you know what i mean

I actually have no idea what i did wrong

[onyx bramble]

Your notation of the sums seems wrong

[neon iron]

hold on lemme screenshot of my calculator

[neon iron]

this is the actual calculation

[onyx bramble]

what is the actual function

[neon iron]

f(x) = sin(x)

interval: I = [0, pi]

[onyx bramble]

I see

your sums is wrong

[gray ridge]

pi(100-x)/100?

[neon iron]

no just pi /100-x

i think. I'm not sure

[neon iron]

what part exactly?

[onyx bramble]

it should be $\displaystyle \sum{i=0}^{n}sin(\frac{ix}{n})(\frac{x}{n})$

[neon iron]

what does n represent?

[onyx bramble]

wait there error in latex

[neon iron]

$f(x) = x

[thorny flame]

Bring Back Beatrix

[neon iron]

$\f(x) = x

how do i use that bot

$\displaystyle \ f(x) = x

[onyx bramble]

type with $ latex code here $

[neon iron]

$

[onyx bramble]

yours sums should like this $sin(\frac{ix}{n})\frac{x}{n}$

[thorny flame]

Bring Back Beatrix

[onyx bramble]

from $i=0$ to $n$

[thorny flame]

Bring Back Beatrix

[onyx bramble]

if you want 100 samples then it will be $sin(\frac{ix}{100})(\frac{x}{100})$

[thorny flame]

Bring Back Beatrix

[neon iron]

ah yes I see, thank you very much

any idea why my sum didn't work?

https://cdn.discordapp.com/attachments/903486479064522752/1129406128506294352/IMG_4748.jpg

shouldn't it be rougly the same?

[onyx bramble]

yes

[neon iron]

your sum seems to work just fine in the calc

[onyx bramble]

if you are summing from $0$ to $\pi$ then it will be $sin(\frac{i\pi}{100})(\frac{\pi}{100})$

[thorny flame]

Bring Back Beatrix

[neon iron]

$f(x) = x$

[thorny flame]

encouraged

[onyx bramble]

thats right

[topaz sinew]

@neon iron Has your question been resolved?

[topaz sinew]

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[undone flicker]

Why not rational

[undone flicker]

B^2-4ac =6?

Why not rational

[silk pewter]

It's talking about the roots

Use the quadratic formula, you will find out why they are irrational

[empty sail]

If b^2 - 4ac equals a square number, then it's rational

If it's not a square number then the roots are irrational

Because b^2 - 4ac is under the radical in the quadratic formula

So if you take the square root of not a square number, it's a irrational number

[topaz sinew]

@undone flicker Has your question been resolved?

[undone flicker]

Why b^2-4ac is not giving exact solution

[topaz sinew]

@undone flicker Has your question been resolved?

[undone flicker]

@empty sail @silk pewter

[undone flicker]

<@&286206848099549185>

[sullen gyro]

Find the roots using the quadratic equation and you'll find x= √7 ± 3

b²-4ac only indicates if the roots are real or not

[undone flicker]

I see yes you are right but there are many sites which claims it gives about rational irrational too

which is wrong

[undone flicker]

I got the values here you can see by the formula

[sullen gyro]

The conditions of irrationality and rationality only holds true when a, b, and c are rational

Why? x= {-b+√(b²-4ac)}/2a

As you can see, when a, b, and c are rational, the only factor that makes x irrational is the √(b²-4ac) part

[undone flicker]

$b^2-4ac$ only tells us about roots will be real, same or different right?

[sullen gyro]

But here, b is irrational

[thorny flame]

arjunn5589

[sullen gyro]

It can conditionally tell if the roots are irrational at certain circumstances as explained here

[undone flicker]

no

the part b

[sullen gyro]

Any objections?

-b/2a is rational

[undone flicker]

√(b²-4ac) this is rational here

b is irrational here

[sullen gyro]

This is why I assumed when a, b, and c are rational

I'm explaining why they explained that b²-4ac can explain rationality although they're wrong

[undone flicker]

no?

[sullen gyro]

They're right with a condition

Say if it's like 5x²+2x-4

Then b²-4ac can check rationality as well

[sullen gyro]

Due to this formula

[undone flicker]

here b^2-4ac is not rational

[sullen gyro]

-b/2a is rational because we assumed the case where a and b are rational. In this case, if √(b²-4ac) is rational, so is x

if √(b²-4ac) is irrational, so is x

[undone flicker]

check it once

squrtroot 84/5

[sullen gyro]

However as you can see if there's no word on a, b, c to be rational, this no longer works

Because even if √(b²-4ac)/2a is rational, -b/2a can be irrational

[undone flicker]

here a,b,c are rational

but still roots are irrational

√(b²-4ac)/2a is irrational here in this example

-b/2a is rational

[sullen gyro]

Yes

Which is why I said IN THIS CASE we can determine the rationality with √(b²-4ac)

[sullen gyro]

But NOT in this case

[undone flicker]

ohh

[sullen gyro]

The situation is DIFFERENT

[undone flicker]

I understand your statements now

I think it is good to check properly with the formula would be better

or just focus on a,b,c

thanks again for enriching my knowledge deeply

[sullen gyro]

Just focus on the basics

Use formulas when you understand the basics and want to make it short

[undone flicker]

i will surely join tutle gang

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[rough yew]

could anyone explain this question?

[abstract wadi]

How familiar are you with, say the period of tan(x)

[rough yew]

i feel like i forgot most of it

[abstract wadi]

That's not very good.

Do you know anything like what tan(2pi-x) , tan(pi-x) or tan(pi+x) would be?

[rough yew]

um....

[neon iron]

maybe think about what sectors of the unit circle tan(x) is positive

is tan(x) an even or odd function?

[rough yew]

like quad 3 n 1 smt like this?

[neon iron]

yeah

[rough yew]

odd

[neon iron]

no yes

[rough yew]

ok thx

[rough yew]

I checked answer. so if tanx = 1.5, why x= pi+0.98

wow nice drawing

[rough yew]

180..?

[neon iron]

yeah

$$\pi$$

[thorny flame]

brandon_hu

[neon iron]

if the red angle on the graph was 30o

you would have to go around the circle $$\pi$$ times before the slope is the same

[thorny flame]

brandon_hu

[rough yew]

ok

[neon iron]

if you draw a line through the unit circle like in the picture you can easily see why

[rough yew]

ok ty

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[neon iron]

I'm stuck midway with the following problem. I'm going to attatch my work.

This is it simplified but I don't know hot to move on from here

[bitter sluice]

You may factor x out

Sum of roots = -b/a

[neon iron]

As in for a quadratic equation like ax^2+bx+c?

[bitter sluice]

Yes

[neon iron]

Okay

[neon iron]

Oh

answer is -25/4

correct?

i totally forgot that rule sorry

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[solar stirrup]

the probability of rolling a 5 is now 30%. the die is rolled 10 times. find: the probability of rolling exactly 8 "5"s.

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@solar stirrup Has your question been resolved?

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[vernal vale]

what have you tried?

[karmic zealot]

F(x+3) for the first one? Since it’s being moved left?

[vernal vale]

sure

[karmic zealot]

And f(x)+1 for bottom since it’s being moved up one unit?

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[rocky flame]

o

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Please don't occupy multiple help channels.

[rocky flame]

.close

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[neon iron]

Hey how would i solve for side x?

[fringe egret]

mistake nvm

[neon iron]

your good

[fringe egret]

sin50 = x/12

[neon iron]

wait

sin (50 degrees)

=x/12

right?

[fringe egret]

yea

[neon iron]

okay hold on

[fringe egret]

im holding'

[neon iron]

sin(50 degrees)(12)=x

so that side length is

approx 9.2

correct?

[fringe egret]

if tahts what the calc says

[neon iron]

yup

it has to be in degree mode too right?

[worthy storm]

matlab agrees

>> sin(50*pi/180)*12
ans =
          9.19253331742774

[fringe egret]

yes bro

[neon iron]

thank you sir

i apprecaite your help

[fringe egret]

@worthy storm could u take a qucik look at #help-4 tysm

[fringe egret]

have a good day/night

[neon iron]

you too

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[sick torrent]

get some help omfg

[thorny spire]

Ok so

In the second to last line

It says since l_t(i) > 1/n

For i not in S_t

But it never explains why that is true

Like at the top it says for i in S_t i is less than 1/n (which i don’t see them using)

[odd pagoda]

definition of S_t

[thorny spire]

But how does that imply this

[thorny spire]

Where

[odd pagoda]

in the lemma

[thorny spire]

No

that says l_t(i) must be less than 1/n

[odd pagoda]

for the i in S_t

which means its > for the i not in S_t

[thorny spire]

I’m talking about i not in s_t

[thorny spire]

No bruh