help-27

[hoary anvil]

double checking

[grand siren]

we can prune a few 2T in the first two trees

HH: P1 1T P2 1H, P1 2H, P2 bluff, P1 P2 2H, P1 bluff, P2 P2 bluff, P2 P1 1H, P2 2H, P1 bluff, P2 P1 2H, P2 bluff, P1

HT: P1 1T P2 1H, P1 2H, P2 bluff, P2 P2 2T, P1 bluff, P1 P1 1H P2 2T, P1 bluff, P1 P2 bluff, P1 P1 2H, P2 bluff, P2

[hoary anvil]

Sorry I'm behind, the first four tables seem right though

yeah those are right

we can probably prune more

[grand siren]

i don't think we can actually prune anything else but also this is small enough to basically just analyse the whole game from here

P2's strategy in response to P1 playing 1T is to essentially guess what P1's coin is, and either play 1H or not

P1 is therefore given the choice of either playing 1T to let P2 guess, or guessing P2's coin

i think it's just a more complicated 50/50 though

[hoary anvil]

yeah it seems like the choices don't matter again

D:

P1 1T
  P2 1H, P1 2H, P2 bluff, P1
  P2 2H, P1 bluff, P2
  P2 bluff, P2
P1 1H, P2 2H, P1 bluff, P2
P1 2H, P2 bluff, P1

HT:

P1 1T
  P2 1H, P1 2H, P2 bluff, P2
  P2 2T, P1 bluff, P1
P1 1H... P1
P1 2H, P2 bluff, P2

TH:

P1 1T
  P2 1H... P2
  P2 2H, P1 bluff, P1
  P2 bluff, P1
P1 1H, P2 2H, P1 bluff, P1
P1 2T, P2 bluff, P2

TT:

P1 1T
  P2 1H... P1
  P2 2T, P1 bluff, P2
P1 1H... P2
P1 2T, P2 bluff, P1```

I combined the branches that yield the same winner

[grand siren]

oh yeah that's a good idea

[hoary anvil]

that's what the ellipses represent

[grand siren]

after 1T in HH/TH we could also combine P2's choices of 2H and bluff

since P2 knows they're equivalent

[hoary anvil]

yeah just noticed that

not sure how to notate it

slash maybe

[grand siren]

yeah we can just like, P2 2H/bluff... P2

or something

[hoary anvil]

P1 1T
  P2 1H, P1 2H, P2 bluff, P1
  P2 2H/bluff... P2
P1 1H, P2 2H, P1 bluff, P2
P1 2H, P2 bluff, P1

HT:

P1 1T
  P2 1H, P1 2H, P2 bluff, P2
  P2 2T, P1 bluff, P1
P1 1H... P1
P1 2H, P2 bluff, P2

TH:

P1 1T
  P2 1H... P2
  P2 2H/bluff... P1
P1 1H, P2 2H, P1 bluff, P1
P1 2T, P2 bluff, P2

TT:

P1 1T
  P2 1H... P1
  P2 2T, P1 bluff, P2
P1 1H... P2
P1 2T, P2 bluff, P1```

[grand siren]

so yeah it really is just, P1 either guesses P2's coin, or lets P2 guess P1's coin

[hoary anvil]

yeah

Ok I'm satisfied with that answer now

thanks for your help

A bit disappointed though I was hoping we would get mixed strategies

since I think that's what we would have in the actual game with dice

[grand siren]

i think we did get a mixed strategy, just not a very interesting one

[hoary anvil]

(I guess we are technically getting mixed strategies, just not ones that matter)

yeah

it's a mixed strategy that can be played as a pure strategy

so

[grand siren]

i don't think P2 has an optimal pure strategy

...or maybe they do

hm

[hoary anvil]

pure given the move history and their coin

[grand siren]

the issue is that if they're too predictable, then P1 can play 1T (to let P2 guess the coin) only if P2 will get it wrong

but i don't know if P2 can use their own coin to work around that without needing external randomness

[hoary anvil]

oh I see

you're right

so not a pure strategy

[grand siren]

ok but yeah if P2 just guesses that the game is either HH or TT then i don't think there's any strategy for P1 that defeats that

[hoary anvil]

yeah

after 1T

go 2T if they have T, and go 2H if they have H

and it's 50/50

well I'm content to close this now unless you have anything else you want to add

[grand siren]

alright so basically yeah no mixed strategies lol

i'm curious now what the analysis would look like for like, a 3-sided dice

[hoary anvil]

me too

[grand siren]

at that point it would be really tedious to do by hand but we can start writing code to generate game trees

[hoary anvil]

Should I open another help channel for that or?

It's a separate question technically

[grand siren]

idk

it's probably just going to be us two either way so i'm not sure it matters

[hoary anvil]

lel

fair enough

Ok I'll leave this open for now but I'm gonna take a break, might come back later for 3 sided dice variant

[devout snow]

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[neat oasis]

I understand that noise canceling involves trigonometric functions, motsly sin i think? but

[neat oasis]

I cant seem to find a formula for it? I can't find any source that explains the math behind it

I know sin func is used but idk how and why

[safe fractal]

Noise canceling?

[neat oasis]

yes

[safe fractal]

As in, sound wave noises?

[neat oasis]

i think so?

[safe fractal]

Can you be more pacific?

[hoary anvil]

I think it's just the same wave shifted by half a wavelength

[neat oasis]

pacific?

[hoary anvil]

probably specific

or the same wave multiplied by -1

so they cancel

[neat oasis]

yeah

but like

the formulas

[hoary anvil]

f(x) -> -f(x)?

[neat oasis]

idk

[eager nova]

Basically, you have to find the function of the wave noise

[neat oasis]

yeah i understand

[eager nova]

And create one that

How to say…

Eliminates it

[neat oasis]

but generally speaking, if i were to explain this to someone I need like examples and explain why trigonmetric functions is used in noise canceling

[neat oasis]

destructive interfernce

i understand that

[eager nova]

Yeah

[neat oasis]

.

[safe fractal]

https://www.desmos.com/calculator/mj7vt3w4js

if green is noise, purple is the counter function and black is the resulting wave

see how purple changing changes black

[neat oasis]

yeah

[safe fractal]

at a = 0 it has the greatest effect

so the point of noise canceling is to detect the green wave

determine it's frequency and phase

and then create a counter sine wave that "eliminates" it

[neat oasis]

ye

but how is math, trigonometric functions specifically used here?

[safe fractal]

Well all sound waves can be described as a sum of sin waves

and so we take that sound wave

we split it into sin waves

we determine which of these sin waves are noise

we create counter sin waves

to eliminate them

and then we get clean sounds

that's the gist of it

[neat oasis]

oooooh

so like

sound wave itself

can be describe as a sin function

or seen as a sin function

so no formula is needed?

like this is a normal sin function formula

[safe fractal]

no, rather as a sum of sin functions

like this

https://www.desmos.com/calculator/mj7vt3w4js

here, play around with the sliders to see how they affect the resulting sound function

basically we can manipulate it

and then people figured out how to detect which sin waves make noise and which don't

so they engineer counter waves

[neat oasis]

ahhhh

okay

so is that like the only part

where it uses trigonometric function

the sound wave itself and the counter sound wave

@safe fractal

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[thin sequoia]

I'm having a hard time understanding the highlighted sentence, especially "is only defined on A". May someone please help me?

[acoustic leaf]

basically we only allow f to give a valid output if the input is in A

so for example, if we have f(x) = x², we normally allow x to take in negative numbers as input

but we could restrict f to only positive numbers, so that only positive numbers can be taken as input for f

[thin sequoia]

Okay, but what does it mean when they say 'the restriction of f to A is the same function as f, but is only defined on A'

I'm having trouble understanding 'but is only defined on A'

Is it saying that the new function has a defined subset for its original domain?

[acoustic leaf]

that's the same thing as saying the domain of f restricted is A, which is a subset of the domain of the original function f

[thin sequoia]

So is that what it means?

[devout snow]

@thin sequoia Has your question been resolved?

[thin sequoia]

@acoustic leaf

[thin sequoia]

I think i figured it out; is the article stating that such equality between the two functions given their domains only holds such that the value of the original function's argument is strictly defined as therein common values within A that it shares with the original function's domain?

<@&286206848099549185> ?

[limber ice]

It means the domain (input set) of the restriction $f|_{a}$ is only set A which is a smaller part of the domain or input set of the original function f, which was E

[woven radish]

Tangent

[thin sequoia]

it seems like you only defined what A and E was; assuming A was a proper subset of E.

[thin sequoia]

is my proposition here correct?

[limber ice]

yep

[thin sequoia]

thank you

.closee

.close

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[covert nexus]

I try to do this,I am using cylindrical coordinates, in the question says the answer is 4/3,When I do the procedure I find that the result is pi/2
The intervals I have when I intersect the surfaces are:
0 < theta < pi
0 < r < 1
0 < y < (1-r(Cos(theta))
this is correct?

[restive river]

i don't speak

much

spanish

[covert nexus]

im using a transalor because my english is bad,the important part of the problem is this, it is a semi-cylinder whose equation is x^2+z=1 and a surface x+y=1,that intersects the planes y=0, z=0

<@&286206848099549185> ,I need to find the volume generated by these surfaces, but I can't get to the supposedly correct result.

[devout snow]

@covert nexus Has your question been resolved?

[devout snow]

@covert nexus Has your question been resolved?

[hot ferry]

the volume of a semi-circle is 0.5 (πr^2h)

[covert nexus]

Yes, but it is a problem of integrals, of an infinite semi-cylinder that intersects other surfaces of which I must find a volume

[hot ferry]

uno minuto, im translating the question

find the volume of the solid limited by the cylinder x^2 + z = 1 and by the planes x+ y =1, y = 0, and z = 0

Lo siento, no he aprendido integrales 3D pero este video me ayudará, hay subtítulos en español.

https://www.khanacademy.org/math/multivariable-calculus/integrating-multivariable-functions/triple-integrals-topic/v/triple-integrals-1

o esperar para un otra persona

[covert nexus]

thanks for the try too

[hot ferry]

de nada

[devout snow]

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[devout snow]

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[dire slate]

Can someone explain to me how |f(x) - L| < $\epsilon$ is true for $\epsilon > 0$

[woven radish]

TheKingPin

[dire slate]

for the precise definition of a limit

its nonsense

it should equal episilon

epsilon represents the distance between f(x) and L

and im looking at the triangle inequality proof for it

which is true up to the point the absolute value is added

[uncut crow]

can you be more clear. no “it” etc, write out the definition in detail

[dire slate]

$ - \epsilon$ < (x-c) < $\epsilon $

[fierce remnant]

I can offer a counterexample to this notion

[dire slate]

please

[fierce remnant]

Consider a function that is identically equal to 1 apart from at the point x=a

at x=a, say that it's equal to 2

[woven radish]

ekafeman

[dire slate]

so a is 1 away from x?

[woven radish]

ekafeman

[dire slate]

so it doesnt exist at f(1)?

im confused

[fierce remnant]

What doesn't exist?

[dire slate]

the function

[fierce remnant]

Do you know what the graph looks like?

[dire slate]

im not sure what im looking at

[fierce remnant]

Have you tried drawing it?

[dire slate]

again, im not sure what im looking at

[fierce remnant]

What are you unsure of?

[dire slate]

the entire piece wise function

[fierce remnant]

Let me draw it

do you agree that this is the graph of f?

https://cdn.discordapp.com/attachments/903486480075354182/1245316103647789078/1112288312535810160.png?ex=66584e7f&is=6656fcff&hm=8cfd4b00243547eb312bc92a9398cae4342e9d5d27add18f514d2c4590415aa7&

[dire slate]

ya, ive never seen a piece wise be used like that

really cool

[fierce remnant]

ahhhhh ok

[dire slate]

im used to it only being |x|

but yes i agree

[fierce remnant]

yea that's why we allow for an epsilon of room when it comes to the vertical line test

[dire slate]

ya, i know for limits thats why you usually dont let x=a

[fierce remnant]

nice

so back *to the OG question

on the triangle inequality stuff

which bit looks strange for you?

[dire slate]

|f(x) - L| < $\epsilon$ is true for $\epsilon > 0$

[woven radish]

TheKingPin

[dire slate]

this

its nonsense

the part before this point makes sense

[fierce remnant]

I think maybe that's not the full statement

(because it doesn't refer to what x can be)

do you have the OG passage where it comes from?

[dire slate]

yes

Book of Proofs

precise definition of limits

let me see if i can screen shot it

i hope thats visible

[fierce remnant]

the diagram they use is a bit confusing

we can use our diagram tbh

[dire slate]

do you not recommend reading this book for calculus? cause i do agree its a bit confusing

im kinda delaying reading understanding analysis by abbott

but i think its just hurting me

[fierce remnant]

most books are fine and you're doing the sensible thing by asking here

[dire slate]

ok

[fierce remnant]

do you see that 0 < |x - c| < delta bit?

that 0 < part is what we discussed

[dire slate]

yes, i understand why it has to be ggreater than 0

[fierce remnant]

the idea of limits is that you consider the behaviour around the point (not necessarily at the point)

ah ok

[dire slate]

my confusion is that the right side should be = not <

cause delta is the distance between x and c

[fierce remnant]

ok so for the real numbers, both are correct

[dire slate]

the diagram does confuse things but from what i can tell c is delta away from x such that -delta < x-c < delta

[fierce remnant]

you can use either strict or non-strict inequalities

(they're equivalent)

[dire slate]

wait, really?

[fierce remnant]

💯

I remember there being a nice stackexchange answer I read a while back that really dug into it

I'll try link it

(I believe using strict inequalities is because there's some theory later on where you generalise from open intervals like (c, c+d) to open sets e.g. a ball of radius in R^3 around some point c and it just matches up nicer with conventions)

https://math.stackexchange.com/a/3240989/

yea this answer

[dire slate]

sweet

one more small question

for definition 13.2 in the book

why is f: X -> R

why is it not R to R?

[fierce remnant]

probably just to demonstrate that this works for functions defined on subsets of the real line

sometimes you'll have like the positive branch of y^2 = x

[dire slate]

gotcha, that makes sense

[fierce remnant]

https://www.medcalc.org/manual/functions/sqrt.png

yea

only defined for x >= 0

just for generality's sake

you're doing great btw

good questions and understanding

don't get FOMO from not reading other books

the concepts are all the same

[dire slate]

thanks, im scared to take this course in the fall

but ill keep trying

this has been helpful

thank you

.close

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[crisp anchor]

Does "(...) T> (...)" mean the same thing as " '(...) -> (...)' is a tautology"?

Let T> denote tautological implication, -> be implication, and (...) just be some arbitrary formula

[devout snow]

@crisp anchor Has your question been resolved?

[acoustic helm]

I’d assume so based on the name ‘tautological implication’

[crisp anchor]

It makes sense to me too

[devout snow]

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@crisp anchor Has your question been resolved?

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[tawny knot]

hey guys, is there a good free website to prepare for igcse?

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[fallow tangle]

i have a question about my math credits

[devout snow]

@fallow tangle Has your question been resolved?

[lost laurel]

[devout snow]

@fallow tangle Has your question been resolved?

[fallow tangle]

ok

sorry im in class

ok so

last year in algebra 1

i was in 8th grade but i got injured and missed half the semster then got moved into normal level math raht year

but i got the credit for it

this year i failed 1 semester for geometry but passed the other semester

do i have to worry about retaking geometry as i already have the required 1 math credit for freshman year

[devout snow]

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[onyx loom]

how would i do this question as dy/dx = ∞

[hexed wind]

you should practice A. Das books

these are basic stuff of ellipse

ok are you listening ?

[onyx loom]

im on part b

[hexed wind]

?

what

A. Das is the author of a good maths book

[onyx loom]

what are you on about

[hexed wind]

This is an ellipse question no?

take x^2 + 4xy as )

as 0

you will get mod of tan alpha as 1/4

[onyx loom]

huh

[upper schooner]

Ignore him

[upper schooner]

And well, it's more that dy/dx is "dividing by zero", and sooooo...

[lost laurel]

well, no

use implicit diff

[onyx loom]

ye i did part a, im on b

[lost laurel]

oh

[onyx loom]

sorry forgot to mention

[lost laurel]

yes

the slopes are infty

what does that tell you

[onyx loom]

no solutions?

[lost laurel]

no

3y=x

[onyx loom]

how?

[lost laurel]

makes sense?

what's the slope when 3y=x

[onyx loom]

1/3?

sorry idk where u got the 3y=x from

[upper schooner]

[hint: it's been mentioned somewhere here already]

[lost laurel]

substitute that back into the og equaion

[onyx loom]

where did you get this from?

[upper schooner]

Where are you dividing by zero, for dy/dx?

[onyx loom]

oh when x=3y

[hexed wind]

dude it's a problem from that exact book

[onyx loom]

this feels like hackin maths like it shouldnt work

[hexed wind]

what's up with bro reacting with weird emojis

[upper schooner]

Well, remember you want the gradient at the points furthest east and west to be undefined, after all, and those happen when you do something "naughty"

[onyx loom]

so y = sqrt(50/6)

[hexed wind]

atleast it's located on the directrix

[onyx loom]

+-?

[upper schooner]

Well, considering you're making mention of ellipses, when they're not explicitly taught in the course that OP is doing...

[hexed wind]

it can be solved that way

differential calculus and ellipse are not that far apart here

[onyx loom]

im not doing these questions from a book anyways, its from Additional Assessment Materials for Summer 2021

[upper schooner]

Anyways-

[hexed wind]

I am not really telling you with booked solutions

it's my general opinion but alr

anyways see ya

[upper schooner]

Seems about right

[lost laurel]

yeah

[onyx loom]

thank you again

[devout snow]

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[devout snow]

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[mild saddle]

Someone help PLEASE

how do i know the range on phi?

and theta too pls

Like in this they found the boundary on B to be in between 0 and pi for phi

idk why thats the case

this is spherical coordinates

[acoustic leaf]

that is the "default" range on theta and phi which allows it to reach all coordinates

theta should be between 0 and 2pi in order to get the "full circle" in the xy-plane, and phi should be from 0 to pi to cover the full range from +z axis to -z axis

[mild saddle]

wait so

gimme a sec lemme draw something

ok my drawing is ugly as hell 😭

but

[mild saddle]

which side is phi??

is it the one that gives the smallest value of phi?

while theta always goes counterclockwise?

[acoustic leaf]

phi is the angle with the z-axis

[mild saddle]

but does it go counterclockwise up to the point? or clockwise

or depends?

[acoustic helm]

What are the spherical coordinates you are using?

[mild saddle]

i had another question

i figured it out the other one

for triple integrals

or double integrals

we are allowed to just group the integral with their corresponding variables like that and integrate them instead of doing one integral at a time?

.close

[devout snow]

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[acoustic helm]

Yeah as long as the bounds don’t contain rho, theta or phi

[devout snow]

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[upper meadow]

Does anyone personally know of a good SAT practice app that offers relevant SAT practice questions and/or has like a daily question to do or whatnot?

[viral rapids]

blue book

[upper meadow]

Does blue book offer daily things?

Also I don’t think it’s for mobile or at least I just haven’t found it

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[lament schooner]

i determined the interval to be 39/10 < x < 41/10 already but idk how to find the borders

if i replcae x theres still n+1/n+9 idk hwo to remove

[upper schooner]

As a sequence, what does that converge to?

[lament schooner]

The (n+1)/(n+9)?

My intuition says 1 but

I think with séries a lot of times I use intuition 💀

“Idk why it works but Ik its right”

[upper schooner]

I'll let you have that one

e.g. divide both top and bottom by the highest power

[lament schooner]

Ah that means I have to divide (10x -40)^n as well right?

[upper schooner]

Well I thought e.g. putting in that 41/10 did that for you, no?

As long as you just put them in, you're mostly alright, anyways-

The fact you got 1 as a limit for the sequence, what does that tell you about the series?

[lament schooner]

No I meant like let’s say I divide the nominator by n

Oh wait

Nvm I can isolate (n+1)/n

And thats just 1

Division is too hard 😞

[upper schooner]

Ah, I see what you mean-

[upper schooner]

I meant, in here, divide the numerator and denominator of this expression with, in this case, n

[lament schooner]

Long story short these two are thanos snapped

Therefore whatever value x is except 4 means it diverges

[upper schooner]

Not quite-

[upper schooner]

You get convergence for everything in here, but

You wanna check if you converge when x = 39/10 and when x = 41/10

For which, when $x = \frac{41}{10}$, you're analysing the convergence of the series $\sum_{n = 1}^\infty \frac{n + 1}{n + 9}$, but you found that the $n$th term, $ \frac{n + 1}{n + 9}$, has a limit of 1 as $n\to\infty$...

[woven radish]

@upper schooner

[lament schooner]

Oh yeah sorry I meant whatever is in ()^n that’s smaller than 1

Wait

https://tenor.com/view/skeleton-im-dead-falling-over-gif-2711800222951691028

I confused myself

[lament schooner]

These two diverge

[upper schooner]

They do indeed

[lament schooner]

Wait no I’m right here

Rip

[upper schooner]

(unless you've changed the expression?)

[lament schooner]

(X)^n

If x is smaller than 1 does it converge

[upper schooner]

Well, erm-

[lament schooner]

It doesn’t

What am I even saying

Anyway if 39/10 or 41/10

Whole thing inside = 1 or -1

Therefore diverge 😭

[upper schooner]

Yep, when you're at x = 41/10, the sequence converges to 1, when you're at x = 39/10, your sequence doesn't converge (of course, see the nth term test, to even have the possibility of convergence, you need that sequence limit to be zero and exactly zero)

[lament schooner]

WAIT NO I WAS RIGHT

I meant like

(10x-40)^n

If stuff between 39/10, 41/10 converge

And that’s between (-1)^n and (1)^n

Everything between -1 and 1 converge

When ^n

That’s what I meant

[upper schooner]

Ah, so you did change the expression

[lament schooner]

Oh yeah sorry I wasn’t clear 😭

2 hours of sleep isn’t helping much

[upper schooner]

-Whaaaaa

Make sure you rest when you can

[lament schooner]

Coffee = rest it’s fine

I rested 600mg today

anyway tysM!!

.close

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[main goblet]

Hey guys this is a really hard excersise, how do I determine the convergence of this?

[devout snow]

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[main goblet]

<@&286206848099549185>

[kind tiger]

can you clarify the notation. does ln^2 n mean ln(ln(n)) or (ln(n))^2

[kind tiger]

assiming the latter i think the trick (or a trick) is to change the logarithms to base ten (wont effect the convergence as its just a constant multiple)
then turn the sum into a double sum, sum n = 0 to inf sum i = 10^n to 10^(n+1) - 1
then use some crude bounds, eventually you should be able to show that its less than some constant + sum 1/10^n n = k to inf

i believe

if you need some more guidance feel free to ping

[devout snow]

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[restive river]

The second one

Say there is f^2(x)

Which equals (f(x))^2

[kind tiger]

yeah ok

i would try to avoid using that notation

it is more common that f^n (x) means f(f(...f(x))) but often just leads to confusion

it is only really with trig functions that it is totally accepted for e.g. sin^2(x) = [sin(x)]^2

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[gray walrus]

Hey i was wondering if i could just get a check on this? Thanks

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[jade oak]

Looks good

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[hearty remnant]

can someone help me with B

I dont know how they did that

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[gray walrus]

Could i get a check on this

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[vivid estuary]

I am kind of confused on 6a

[vivid estuary]

Not sure how to go about proving this

[sonic smelt]

Have you tried induction?

[vivid estuary]

Induction wasn't mentioned in this book yet though, he only mentioned the rules of < with addition and multiplication

I got this point one sec

So I know that if x < y, and c > 0 then cx < cy, so in the case where x is not 0 then we can use this I think, after doing looking at google for help there was only one post about it and they mentioned using the fact that since
x < y then y-x > 0, and they used the fact y^n-x^n = (y-x)(x^(n-1) + x^(n-2)y...y^(n-2)x + y^(n-1) which was a problem a couple of questions before in the book, so I imagine he set it up for us to use this I think

but I was confused on something

because if we do y-x > 0

then multiply both sides by (x^(n-1)....)

to get y^n-x^n > 0

we would have multiplied by 0 but c has to be greater than 0

and thats where Im kind of confused

ohhhh

yeah just before this question we had a problem to show 0 <= a < b and 0 <= c < d then ac < bd

but I didn't understand that proof too well and moved on

[vivid estuary]

I imagine this allows us to multiply both sides by (x^(n-1).....)?

[upper schooner]

[in particular, x is nonnegative and y is strictly positive, so this is strictly positive]

[vivid estuary]

I thought it was just 0, this x^(n-1) + x^(n-2)y...y^(n-2)x + y^(n-1) is positive?

[upper schooner]

it is [strictly] positive - y^{n - 1} is strictly positive and all other terms are nonnegative

[vivid estuary]

ohh ok

I see now

Thank you I think I can write the solution now

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[spring oasis]

How many numbers are in the list $-36, -29, -22, \ldots, 41, 48?$

[woven radish]

938c2cc0dcc05f2b68c4287040cfcf71

[solid osprey]

you use the same thing

[spring oasis]

arithmetic sequence formula?

[solid osprey]

ye

[spring oasis]

an = 48

a1 = -36

d = 7

,w solve 40 =-36 + (7n-7)

[woven radish]

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Requested by __melomaniac

[spring oasis]

its not

[upper schooner]

40

[upper schooner]

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[fair oracle]

given that 11x - 3 >= 8x + 10, state the least possible value of x if
a) x is an integer
b) x is a prime number
c) x is a rational number

[fair oracle]

I've managed to get that
x >= 4 1/3

[soft umbra]

How did you get this?

[fair oracle]

by solving

[viscid ivy]

,w 11x-3 ≥ 8x+10

This is solution 🥲

[soft umbra]

You mistyped

[woven radish]

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Requested by itzmecairo

[viscid ivy]

This correct

[soft umbra]

May I see your attempt?

[fair oracle]

yeah but why 5 for (a)?

[fair oracle]

好的

[viscid ivy]

Integer

Since integer consists of whole numbers positive and negative

For b) its 5 also

[viscid ivy]

For c its 4 1/3

A prime number is a +ve integer

So for b its also 5

And when we speak of rational numbers it can be any fraction

So the least possible value becomes 13/3

[fair oracle]

ah, I see, thanks y'all

[fair oracle]

@soft umbra thanks to you too

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[primal shadow]

2log(3x+2)=log121

[primal shadow]

I have no idea where to start

[bold halo]

what is your idea

[tender cobalt]

you here?

[primal shadow]

yea

I have it on my homework but I don't remember how to do it

[tender cobalt]

first, theres a detail you can find helpful which is that 121 = 11^2

second, do you remember your log rules?

[primal shadow]

um I think there was like adding 10 or something but not really

[tender cobalt]

thats about how logs work, not the rules of how to turn one log() into another log()

[primal shadow]

oh like how log is the opposite of exponents or no

[tender cobalt]

thats still about how logs work
the log rules are how to turn something like log(mn) into a sum of logs

for now you only need the log rule that $\log(\text{this}^{\text{that}})=\text{that}\cdot\log(\text{this})$

[woven radish]

mtt

[tender cobalt]

for example, $\log(2^3)=3\log(2)$

[woven radish]

mtt

[primal shadow]

oh i think it's too complicated

I will do it another time

thanks for help tho

.close

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[tender cobalt]

I havent even shown you how to use the rule

what makes you think its already too hard for you?

you also didnt get to ask for why the rule would make sense

[long pasture]

i feel you

[devout snow]

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[primal shadow]

oh well if you want you can try and teach me but just warning you i have literally 0 idea about anything in that question

[primal shadow]

so you’d have to explain it from the very beginning and I didn’t want to waste your time

[solid osprey]

???

[primal shadow]

hi

[solid osprey]

could you post the original question

[primal shadow]

2log(3x+2)=log121

[solid osprey]

$2\log(3x+2)=\log(121)$

this?

[woven radish]

Skill_Issue

[primal shadow]

yea

but i have completely utterly absolutely no idea how to do it

[solid osprey]

alright, do you know $\log(a^b)=b\log(a)$

[woven radish]

Skill_Issue

[primal shadow]

ya

[solid osprey]

you can use that

[primal shadow]

i learnt that at least

oah ok i will try rn

um ok wait im confuse abt the format

confused*

is the part in brackets like the base?

or is it just multiplying

[drowsy compass]

This is basically the same thing as $$ (3x + 2)^2 = 121 $$

[woven radish]

Reuben

[solid osprey]

the brackets is the log

[drowsy compass]

Solving the quadratic is easy enough

[primal shadow]

ya i can do that but

bit*

it’s the log part that confuses me

[solid osprey]

normally when there is no number its $log_{10}(a)$

[woven radish]

Skill_Issue

[primal shadow]

oh wait so it’s just 3 then

[solid osprey]

wdym

[drowsy compass]

if you have $$ log x = log something $$ you can cancel the logs out

[woven radish]

Reuben

[drowsy compass]

because theyre the same base

[primal shadow]

or is it different with the log

[drowsy compass]

for example

$$ log 3x = log 9 $$

[woven radish]

Reuben

[drowsy compass]

since both the logs are base 10

they can be cancelled

this expression is the same thing as 3x = 9

[primal shadow]

bbut how do i know if the log is base 10

[drowsy compass]

if there is no base written at the bottom

its base 10

[primal shadow]

ok and how do i know 3x = 9

[drowsy compass]

because both the logs are of the same base

so hence I can just cancel them out

[primal shadow]

soo if the logs have the same base then does x just equal the same number like 3x3

or am i missing something

[drowsy compass]

not sure what you mean, could you elaborate

[solid osprey]

$$\log(\text{something})=\log(\text{something else})$$
$$\text{something}=\text{something else}$$

[woven radish]

Skill_Issue

[drowsy compass]

[primal shadow]

umm like you said 3x=9 so how do I know that x is 3

[solid osprey]

what

[drowsy compass]

3x = 9
x = 9/3
x = 3

[primal shadow]

what do you mean by cancel them out

ohhhh

my bad sorry

[drowsy compass]

Not to be rude but but if youre having trouble with one step equations, perhaps dont approach logarithms yet

[drowsy compass]

got it now?

[primal shadow]

yeah

omg sorry i just realised the 3x=9 thing is obvious

okay and one last question

so the equation is 2log(3x+2)=log121 but i got the answer from just the quadratic equation (idk if that’s the right name) than what is the 2 at the start of the equation for ?

[drowsy compass]

its just multiplying

2 * log(3x+2)

[solid osprey]

it means $2 \cdot \log(3x+2)$

[drowsy compass]

which happens to be the same thing as log(3x+2)^2

[woven radish]

Skill_Issue

[drowsy compass]

due to laws of logarithms

[solid osprey]

log((3x+2)^2) actually

[primal shadow]

ok so if there is a number multiplying the log it’s an exponent?

i have not done my khan academy lessons in a while so i forgot most of what i learnt

[drowsy compass]

yea, my bad

you may wanna revise properties of logarithms

[primal shadow]

yessir im going to do my lessons rn

thank u for the help guys

[drowsy compass]

np

[primal shadow]

im going to close the channel now

.close

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[glass root]

i am doing some laplace exercises could someone explain me from where did that f(t) appeared ?

[devout snow]

@glass root Has your question been resolved?

[midnight echo]

it's the right hand side of your ODE expressed using heaviside functions

[glass root]

okay but how did it get to f(t) = 10sin[u(t-0)-u(t-2pi)] - 0

[midnight echo]

u(t-2pi) means that u = 0 if t is less than 2pi and u = 1 if t is more than 2pi. Likewise, u(t-0) means that u = 0 if t is less than 0 and u = 1 if t is more than 0. You want to have a function that is equal to 10sint if t is [0, 2pi]. Thus you do:

10sin(t)u(t-0)-10sin(t)u(t-2pi)

If t is less than zero, both terms are 0, so the rhs is zero, good.
if t is between zero and 2pi, the left term is 10sin(t) while the right term is zero, making their difference 10sin(t).
if t is more than 2pi, both the left and right terms are 10sin(t), making their difference 0, as desired.

[glass root]

so for something like this i would have:
1u(t-0) - 1u(t-1)

[glass root]

and the whole f(t) would be f(t) = 1u(t-0) - 1u(t-1) - 0 ?

[midnight echo]

yeah

I don't really get what the -0 represents, but the principle is correct

[glass root]

well the one above has one as well

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[glass root]

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[muted goblet]

can this identity be proven? im having trouble tryna simplify it

[eager nova]

What problem?

[devout snow]

@muted goblet Has your question been resolved?

[muted goblet]

I was just trying out stuff to see if I can get the LHS as sin/cos

But I’m not sure if I’m even on the right track

Or if I messed up somewhere

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@muted goblet Has your question been resolved?

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[orchid arrow]

Did I use trig sub correctly here?

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[bright ibex]

hello, how can I prove this identity for all x element of R

[feral agate]

Just input the value of sin(3x)

[bright ibex]

hmm, what does it do?

[feral agate]

do you know of any formula for sin(3x)?

[bright ibex]

nope, not given

[deft star]

do you have a formula for sin(a+b) ?

[feral agate]

I didn't ask if one was given, I asked if you knew of one

whether one had been taught to you or not

[bright ibex]

yes

[feral agate]

yeah otherwise you prove it from this

[bright ibex]

not yet

[feral agate]

$\sin(3x)=\sin(x+2x)=...$

[woven radish]

kheerii

[feral agate]

go nuts with that formula

[bright ibex]

i will try, thank u!

[spring oasis]

thank you

[feral agate]

!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.

[restive river]

chill

[devout snow]

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[spring oasis]

How many odd perfect squares are between $5$ and $211$?

[woven radish]

938c2cc0dcc05f2b68c4287040cfcf71

[restive river]

there might be a rule that you have to wait until your other channel closes

but idk

[spring oasis]

all other channels are closed

[restive river]

whatever

well

an odd number squared is ofd right

[spring oasis]

yes

[restive river]

so similar strategy

find the lowest odd perfect square above 5

which is uh 9

[spring oasis]

what does perfect square mean

[restive river]

and find the highest below 211

a number where its square root has no decimal point

,calc sqrt 211

[woven radish]

The following error occured while calculating:
Error: Unexpected type of argument in function multiplyScalar (expected: number or Complex or BigNumber or Fraction or Unit or string or boolean, actual: function, index: 0)

[restive river]

,calc sqrt(211)

[spring oasis]

?

[woven radish]

Result:

14.525839046334

[restive river]

so yeah

i have bad wifi rn its really laggy for me

also for simple calculations ,calc also saves msg space

[spring oasis]

after that how do I count them all

[restive river]

so what's the highest odd number below 14.52...

13

[spring oasis]

ohhh

,w sqrt(211)

[woven radish]

Results provided by WolframAlpha

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

Requested by __melomaniac

[restive river]

yep

so the odd numbers here are 5, 7, 9, 11, and 13

the even numbers when squared give even perfect squares

[spring oasis]

okay

[restive river]

your answer is just the number of those odd numbers

[spring oasis]

between 5 and 169

[restive river]

you mean 13

13^2 = 169 btw

just assuming that here lol

[spring oasis]

but we need to count the numbers between 5 and 169 or 5 and 13?

[restive river]

5 and 13

only the odd ones

do you understand?

oh shit

its 3 and 13 in fact

do you understand this fully?

i can give a deeper explamation if you want

[spring oasis]

I didnt understand a single thing

I only know what an perfect odd number is

the rest I didnt understood

[restive river]

ph ok

oh ok

so here

you know what odd perfect square means

(perfect odd number is totally different btw)

[restive river]

right?

[spring oasis]

odd perfecty square yeah

basically that if you take the sqrt its a whole number

[restive river]

and that the number itself is odd

[spring oasis]

yeah

[restive river]

even numbers have even squares and odd numbers have odd squares

because of this

you can pretty much take the square root of the whole problem

[spring oasis]

okay

[restive river]

How many odd numbers are between sqrt(5) and sqrt(211)?

,calc sqrt(5)

[woven radish]

Result:

2.2360679774998

[restive river]

,calc sqrt(211)

[woven radish]

Result:

14.525839046334

[restive river]

so how many odd numbers are between 2.23 and 14.52?

[spring oasis]

between floor(sqrt(5)) and floor(sqrt(211))

2 and 14

[restive river]

i mean that works

[restive river]

it's actually ceil(sqrt(5)) and floor(sqrt(14))

[restive river]

you don't need this...

so what's the answer?

[spring oasis]

3, 5 ,7, 11,13

[restive river]

so the answer is 5!

[spring oasis]

no

i ommited 9

mb

[restive river]

its 6 then

also if you have another hyper elementary NT type question please paste it here to avoid clogging up the channels

[spring oasis]

?

[restive river]

if you have a related question just ask it here now

[spring oasis]

How many perfect squares are there between 20 and 150?

,calc sqrt(20)

[woven radish]

Result:

4.4721359549996

[spring oasis]

,calc sqrt(150)

[woven radish]

Result:

12.247448713916

[spring oasis]

4

.close

[devout snow]

Channel closed

Closed by @spring oasis

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[devout snow]

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[lapis pelican]

can someone explain why tangent graphs has a period of 180 degrees even though tangent is a ratio of sin(x)/cos(x) which has a period of 360 degrees

[stone stump]

in the fraction it can happen that either both sin and cos are positive or both are negative. but either way the fraction would have the same value as the - would cancel out

similarly one of them be negative or the other one but still same value for the fraction

[lapis pelican]

but why does the period needs to be 180 degrees

<@&286206848099549185>

[brisk escarp]

The period needs to be 180 degrees as the values every 180 degrees, the values repeat

But you already know this.

Let's say a tan(x) period ends/begins every time the value is undefined.

This happens every 180 degrees.

Tan(x) is undefined when cos(x) is 90 or 270 degrees,

And there is 180 degrees inbetween those angles.

And if the period ends/begins everytime the value is undefined,

[brisk escarp]

Then the time between period start (90) and period end (270) is 180 degrees.

[devout snow]

@lapis pelican Has your question been resolved?

[lapis pelican]

oh

ok

tysm

.close

[devout snow]

Channel closed

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[open rock]

i feel its e but i like someone to confirm as i need the right answer, or give me a tool to check

im looking for the particular solution of this equation

might be called general solution also in english, not certain

[frozen aurora]

you can try plugging them in and checking which of these work

[stable storm]

e looks about right

try solving the homogeneous equation first

it should become more clear why

[open rock]

pluging them in where

im geting told its b , alright

[stable storm]

b would be the solution to the homogeneous differential equation

one of the solutions at the very least

[open rock]

alright tnx, think i got it

.close

[devout snow]

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[worldly ridge]

hello, i got a question about this one slope intercept question

[pine kernel]

send a screenshot

[worldly ridge]

kk

[pine kernel]

,rccw

[worldly ridge]

its based on the

question before

[pine kernel]

can you show the one before

[worldly ridge]

kk

here

[pine kernel]

oh that one

did they tell you what the x variable was for

like

does it correspond to lessons

or cost

[worldly ridge]

ohh

look

[pine kernel]

typically x is for lessons and y is for cost for this kind of question

so cost of 4 lessons

[worldly ridge]

7 lessons is 82

[pine kernel]

means you plug in 4 for x

[worldly ridge]

and 11 is 122

dollars

[pine kernel]

yep

[worldly ridge]

or whatever currency

lemme send pic of the one before again

[pine kernel]

so y = 10(4)+12

find

y

y=52

just remember ur units

[worldly ridge]

yes

1sec lemme understand

how did u get that?

like

how did u get the equation

?

[pine kernel]

which equation

[worldly ridge]

.

[pine kernel]

oh

i took the equation u got from #11

y=10x+12

[worldly ridge]

yes

oo

[pine kernel]

and you said 4 lessons

so you put x as 4

[worldly ridge]

ooooooo

ok bro

tysm

i got it

gn

.close

[pine kernel]

np

[devout snow]

Channel closed

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[devout snow]

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[lament schooner]

what is

[topaz beacon]

the question remains: what isnt?

[lament schooner]

honestly i thik we shoudl ask ourselves why we exist

also i figured out the question LOL oops

.close

[devout snow]

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[eternal tapir]

i ask myself that every moment

[topaz beacon]

math is philosophy with too much logic

[devout snow]

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[prime narwhal]

how do you do this style of question

i haven't done any of this type before

[sharp root]

This is a question thats pretty much just asking which of the systems seem like they describe this situation?

[frosty cradle]

yeah, it's asking you to convert the word problem into equations

[prime narwhal]

what is blue eyes

x or y?

let me just think this through

[lean plover]

What are you leaning towards and why?

If you're just scribbling this down, this shouldn't matter too too much at first!

let's just start the problem from scratch, how about that?

suppose the multiple choice isn't given

[prime narwhal]

im not totatly sure where to start

browneyes*2-6=blueeyes

[lean plover]

I think you are at a great start tbh! The intuition to represent blue eyes as a variable is a great start

[lean plover]

Yup! this is right!

and (i think this is the tricky part imo lol) how many people are there in total, assuming that there are only blue and brown eyed people on the train?

[prime narwhal]

42?

[lean plover]

yup! so that would be
browneyes + blueeyes = 42

[prime narwhal]

so its this option

[lean plover]

i believe so!

[prime narwhal]

wow thanks

can we do a few more examples

so i get the hang of this?

gst and pst are tax's btw

[lean plover]

for sure!

also feel free to ping if i don't respond for a long time

[prime narwhal]

ok

[lean plover]

so there are two items, x and y

just by the first sentence

and then we start collecting taxes from these items!

do you have any idea what to do?