$x^{\frac{1}{3}} -3 =y$
#help-33
novel juniper
elfin berryBOT
f(why am i here )= I don't know
knotty trellis
what is the inverse function?
knotty trellis
**f(why am i here )= I don't know**
this also just an equation
I don't see f^-1(x) in there
novel juniper
I just replaced y with x
knotty trellis
okay, and how do we get inverse function from that?
novel juniper
**f(why am i here )= I don't know**
This is it
knotty trellis
where is f^-1(x)?
novel juniper
$f^{-1}(x)= x^{1/3}-3$
elfin berryBOT
f(why am i here )= I don't know
knotty trellis
**f(why am i here )= I don't know**
okay, this works
now apply the same method above
elfin berryBOT
f(why am i here )= I don't know
knotty trellis
(or if you wanna do less work, use this equation and do substitution x = h^-1(u))
novel juniper
$f^{-1}(y)=cx$
elfin berryBOT
f(why am i here )= I don't know
novel juniper
so $x =\frac{f^{-1}(y)}{c}$
elfin berryBOT
f(why am i here )= I don't know
novel juniper
or $y= \frac{f^{-1}(x)}{c}$
elfin berryBOT
f(why am i here )= I don't know
elfin berryBOT
f(why am i here )= I don't know
novel juniper
so $h(x) = \frac{f^{-1}(x)}{c}$
elfin berryBOT
f(why am i here )= I don't know
novel juniper
which makes 0 ense
knotty trellis
**f(why am i here )= I don't know**
Yeah, the switch isn't really a legal algebraic operation
novel juniper
I think sticking to $y=. \frac{f^{-1}(x)}{c}$
elfin berryBOT
f(why am i here )= I don't know
novel juniper
may be a better idea
novel juniper
I mean
knotty trellis
sticking to it as a method or as an answer?
elfin berryBOT
f(why am i here )= I don't know
knotty trellis
this doesn't define a function though
whole this x-y swapping thing is a method to help you find the inverse
not a proof or anything
novel juniper
how is $y = h^{-1} (x)$
elfin berryBOT
f(why am i here )= I don't know
knotty trellis
how is $y = h^{-1} (x)$
this is what fucked up during the x-y swap
we can work from here though
before the swap
y = h(x)
so x = h^-1(y)
so $h^{-1}(y) = \frac{f^{-1}(y)}{c}$
elfin berryBOT
MæthIsAlwaysRight
knotty trellis
**MæthIsAlwaysRight**
and this gives us the expression for the inverse function
without doing the swap of variables x and y
knotty trellis
np
novel juniper
.close
marsh citrusBOT
knotty trellis
btw all this is valid only if the inverse exists
marsh citrusBOT
Available help channel!
tame ore
marsh citrusBOT
tame ore
bit confused here
unborn condor
you need curly brackets for the full exponent
elfin berryBOT
Ishaan
tame ore
and then i need to find all the values
unborn condor
that doesnt look correctly differentiated
rough trout
remember that x^2/3 is part of the denominator !
tame ore
**Ishaan**
yeah i got it all right so far i just did some weird factorisation and cancellation and got this its definitely right but i just dont know what to do from here
rough trout
im going to differentiate it myself
tame ore
okay yeah
unborn condor
unborn condor
**Ishaan**
anyways, your inequation can be solved kinda easily if you separate it in parts
for the product to be positive, either both components are positive, or both components are negative. Obviously the equality is when either is 0
can you solve $x^{-5/3}>0$ and $2+x>0$?
elfin berryBOT
LordFelix
tame ore
tame ore
for the product to be positive, either both components are positive, or both com...
ohh
so for the whole function to be positive either both negative or both positive
cuz then it makes +ve
unborn condor
what does ve mean?
tame ore
elfin berryBOT
Ishaan
lunar bolt
i need help translating the math expression from slovenian to english. We call it "stekališče" in Slovenian. The definition is that the number a is the "stekališče" of a series an, when every epsilon (ε) in the area (a-ε, a+ε) there is an infinite sequence of numbers. What do you call "stekališče" in English?
marsh citrusBOT
i need help translating the math expression from slovenian to english. We call i...
pallid tapir
we'd call a the limit of the sequence / series, and say the sequence / series converges to a
lunar bolt
No thats not what I am looking for
pastel cypress
we call it progressions , they r divided into several parts like arithmetic and geometric
not sure if this would help u
pallid tapir
oh, you really did mean that there was just an infinite sequence of numbers
lunar bolt
A theoretical question from my previous exam is, translated from Slovenian to English: Write down every "stekališče" of the series an = whats underlined with red in the screenshot? Does the series have a limit?
How do you translate "stekališče" to English and what would the answer to the first part of this question be?
pastel cypress
that straigh up look like a arithmetic progression to me
it jus looks a bit more elborated
pallid tapir
it's not an arithmetic progression, it's just a sequence
pastel cypress
A theoretical question from my previous exam is, translated from Slovenian to En...
what does "zaporedje" mean
lunar bolt
what does "zaporedje" mean
series / sequence
pallid tapir
also interesting, a series is what we call a sequence of partial sums of another sequence
while I stall, $(S_n)=\left(\sum_{k=0}^na_k\right)$
elfin berryBOT
Flip
lunar bolt
this is what pons translates it to, I guess you were right
pallid tapir
I'm gonna look through my old notes and see if we ever did define this
lunar bolt
A theoretical question from my previous exam is, translated from Slovenian to En...
so then what would all the points of convergence be for this an?
pallid tapir
so then what would all the points of convergence be for this an?
5 and -5
sin(npi/2) alternates between 1 and -1
and 5 - 1/n approaches 5
copper raven
cluster point/accumulation point is prolly the better translation
lunar bolt
sin(npi/2) alternates between 1 and -1
how did you figure this out
how for example would I go about figuring this out myself
pallid tapir
how did you figure this out
ah ah, my fault, I'm a bit wrong
but you imagine what the sequence npi/2 does on the unit circle
if that makes sense
a sequence of angles
at each step, starting at (1,0), you rotate 90 degrees around the circle, counterclockwise
on each even step, you're at (1,0) or (-1,0)
lunar bolt
I get this but I dont get what exactly we are looking for
lunar bolt
at each step, starting at (1,0), you rotate 90 degrees around the circle, counte...
you mean 180?
pallid tapir
I don't
pallid tapir
that's not what that would mean
on each step in the seqeunce npi/2, you're just adding pi/2
lunar bolt
oh ok my bad
pallid tapir
pi/2 is 90 degrees, so you're just taking quarter steps around a circle
so for even n, you reach points with a vertical component of 0
lunar bolt
so the answer would be 5 and -5?
pallid tapir
for n odd, you're either at a point with a vertical component of 1 (if n = 1 mod 4) or -1 (if n = -1 mod 4)
pallid tapir
so the answer would be 5 and -5?
no, I was wrong, sorry
the accumulation points of sin(npi/2) are therefore -1, 0, and 1
corresponding to n being 1 mod 4, 0 mod 2, and -1 mod 4 respectively
the sequence is multiplied by the sequence 5 - 1/n, which just converges to 5, full stop
so it scales these accumulation points by 5
yielding 5, 0, and -5 as accumulation points of the sequence of interest
lunar bolt
but I dont get why these are accumulation points
how are valuest that are not n=-1, =0 or =1 not accumulation points
pallid tapir
it might help to try proving it for yourself
copper raven
you have subsequences of a_n converging to -5, 0, 5
that's what flip was saying with the 1 mod 4, 0 mod 2, -1 mod 4
copper raven
you have subsequences of a_n converging to -5, 0, 5
this implies that they are accumulation points
pallid tapir
the 3 points (1, 0, -1) are also the only points in the image of the sequence sin(npi/2)
each of which are hit infinitely many times
lunar bolt
sorry but I still dont understand
copper raven
for other points a, you can try and find an ε such that (a-ε, a+ε) has no terms of the sequence inside if you want
this shows pretty directly they're not accumulation points
lunar bolt
but it doesnt reach -5 and 5
oh I see, it always gets closer to -5 and 5 but never actually reaches those values
whilst in 0 it does
pallid tapir
right
pallid tapir
but it doesnt reach -5 and 5
desmos trick, write n = 50 or some other large number, k = [0, ..., n], and replace each of your x's with k's to get a more honest representation of your sequence
lunar bolt
so what would the answer be here, I think its sin (pi*n)/3 or /2
not sure which one it is, would you maybe now whats more likely that the teacher would put?
lunar bolt
can I write my thinking process?
pallid tapir
of course
lunar bolt
so we ignore (1-1/n) and only look at sin at the start right?
pallid tapir
sure
lunar bolt
so when we look at the circle for sin, pi/2 would be 1 if n=1, 0 if n=0 and 0 if the n=2
if were looking for the division to be /2 and not /3
pallid tapir
it really does look like a 3 because it doesn't have the tail of a 2, but alright
lunar bolt
ill do for /2 first since its kinda easier hahah
pallid tapir
fair alright lol
lunar bolt
so when we look at the circle for sin, pi/2 would be 1 if n=1, 0 if n=0 and 0 if...
is this correct?
pallid tapir
yeah that's all true
lunar bolt
so the answer would be n = 1 mod 4, n = 0 mod 2?
or n = 1 mod 4, n = 0 mod 4 and n = 2 mod 4
i think its the second
since in the first we can go to 3pi/4 right?
pallid tapir
that's not an answer, but also you're missing consideration for n = 3 mod 4 (equivalently -1 mod 4)
also, 0 mod 4 or 2 mod 4 is equivalent to 0 mod 2
lunar bolt
but I dont understand why these are the answers, arent all points on this graph points of convergence?
since if you were to take any point and look at its left limit and right limit it would be equal to f(A) if A were the point
pallid tapir
I'm confused by what you're saying; what are we doing here again?
are we determining the image of the sequence sin(npi/2)?
and/or its accumulation points?
lunar bolt
were looking for the points of convergence aka. accumulation points (im guessing "points of convergence" and accumulation points" are interchangeable) of (1-1/n)*sin(npi/2)
pallid tapir
ok, but we began by saying "we can just look at the sin part first"
lunar bolt
yes
pallid tapir
what are we doing with the sin part?
lunar bolt
sorry, looking away from the sin part, I am confused as to what even are accumulation points aka points of convergence
I know the theoretical definition but I dont understand it
pallid tapir
imagine an unruly sequence of points that does not converge
I'll produce an image brb
lunar bolt
also the duck on your pfp is cute
I like looking at it (no weird sh#*, he just cute thats it)
pallid tapir
hell yeah lol
ok by produce an image I apparently mean haphazardly sketch something
suppose this is an unruly sequence
an accumulation point of this sequence is a real number for which infinitely many terms in the sequence can be arbitrarily close to that number
i.e., every neighborhood of this number admits infinitely many terms inside it
here I've tried to allow two accumulation points indicated by the dashed line
no matter how close we zoom in to one of these points (lines, I guess), we should always find infinitely points of the sequence in our scope
even if there's like, gaps between the terms themselves
lunar bolt
so basically an infinite number of points go towards a value but never reach it and these points are called accumulation points or points of convergence?
pallid tapir
so basically an infinite number of points go towards a value but never reach it ...
you can even drop the "but never reach it" quantifier. it can reach it, why not. the terms of the sequence just have to get arbitrarily close to the accumulation point infinitely often
I'm a big fan of calling them accumulation points because of topology
lunar bolt
so points that appear an infinite amount of times but are also = to that value are also accumulation points?
pallid tapir
limit points is also fine but in this context it feels wack to call them that lol
lunar bolt
for example 0 in the last task?
pallid tapir
so points that appear an infinite amount of times but are also = to that value a...
it depends on what you mean... for example, (1, 1/2, 1/3, 2, 1/5, ..., 1/(3n+1), 1/(3n+2), 2, 1/(3n+4), 1/(3n+5), 2, ...) is a sequence that does reach "1", but it doesn't get arbitrarily close to 1 infinitely often. it was just that one time
it also has 0 as an accumulation point, because the subsequence (1/2, 1/3, 1/5, 1/6, ...) converges to 0
lunar bolt
ohh ok I think I get it
pallid tapir
it also has 2 as an accumulation point, because (2, 2, ...) is a subsequence that converges to 2. we don't care that it equals 2 always, just that it gets arbitrarily close to 2
hopefully that gives a bit more intuition on the actual definition. the definition is safest to use
lunar bolt
so what would the answer be here, I think its sin (pi*n)/3 or /2
ok so looking only at sin, the result of sin cannot be 0 right? because then the point is always 0 and doesnt approach a value infinitely
pallid tapir
what do you mean by result?
lunar bolt
as in sin(...) cannot be = 0
pallid tapir
sin(0pi/2) = sin(0) = 0 though
pallid tapir
oh but of course you can't start your indexing at 0
sorry
well, sin(2kpi) = sin(4kpi/2) = 0 for all nonzero k
in fact sin(kpi) = sin(2kpi/2) = 0 for all nonzero k
lunar bolt
yes, so one of the answers isnt n = 0 mod 2 right?
pallid tapir
and 1 - 1/(2k) is defined for such points
wheat moth
pallid tapir
what do you mean by answer?
lunar bolt
so what would the answer be here, I think its sin (pi*n)/3 or /2
answer for this
pallid tapir
we don't talk about the congruency classes of accumulation points of a sequence
so n = 0 mod 2 isn't even in the right space to be called an answer
lunar bolt
oh so we talk about the number it approaches?
pallid tapir
the reason why I ever brought up congruency classes of your input n is because it produces an infinite family of values for n for which sin(npi/2) is some constant
sin(npi/2) = 0 whenever n = 0 mod 2
hence 0 is in the range of the sequence, and is hit infinitely many times
lunar bolt
sorry for interrupting with this but my exam is tomorrow thats why I'm in a bit of a hurry. Whats the answer for the last question?
lunar bolt
so what would the answer be here, I think its sin (pi*n)/3 or /2
to this
pallid tapir
we've actually already talked about the accumulation points of sin(npi/2) -- it's the same as the image, which is {1, 0, -1}
corresponding to inputs n whenever n = 1 mod 4, n = 0 mod 2, and n = -1 mod 4 respectively
we know that's all of them, because these congruency classes partition the set of integers in the domain of the sequence
lunar bolt
we know that's all of them, because these congruency classes partition the set o...
woah woah woah, can you dumb this down a bit for me?
pallid tapir
yeah: if you look at any integer that's at least 1, it's either 1 mod 4, 0 mod 2, or -1 mod 4
there's no other options, and no overlap between the cases
we've described all of the integers
lunar bolt
so an examle of n = -1 mod 4 would be 7, 3, -1, -5, ...?
pallid tapir
yes
however we're actually not considering the negative integers, probably
because it'd be weird to randomly skip over n = 0, which we'd have to do since 1/0 isn't a number
so they're {3, 7, 11, 15, 19, ...}
lunar bolt
ok I am very sorry about this but its gonna feel like your unteaching me things but I promise I'm learning from this since I didnt know what a accumulation point is b4 this and now I know
lunar bolt
ok I am very sorry about this but its gonna feel like your unteaching me things ...
and its gonna feel like this because im going to ask the same question I did b4
pallid tapir
this is fine
lunar bolt
but how did you figure out that -1, 0 and 1 are the accumulation points
for example how did you figure out that -1 is a accumulation point
not for all of them, please tell me only for -1
or just a single one out of these 3 so I can understand how to find all of them
pallid tapir
but how did you figure out that -1, 0 and 1 are the accumulation points
we know this because:
- -1 is in the range of the sequence sin(npi/2) : it corresponds to the value of n = 3
- the number -1 is hit infinitely many times : if n satisfies sin(npi/2) = -1, then n + 4 also satisfies sin[(n+4)pi/2] = sin(npi/2 + 2pi) = sin(npi/2) = -1
lunar bolt
we know this because:
- -1 is in the range of the sequence sin(npi/2) : it corre...
ok wait, I'm lost at the first point
ohhh ok
lunar bolt
and so n = -5 = sin(-5pi/2) = -1 right?
pallid tapir
and so n = -5 = sin(-5pi/2) = -1 right?
again, outside of range, but as numbers this is true
these two bullet points yield the existence of a subsequence sin((4k+3)pi/2), defined on all k, which is just the constant sequence (-1, -1, ...)
this constant sequence certainly converges to -1
thus -1 is an accumulation point of the sequence sin(npi/2)
lunar bolt
again, outside of range, but as numbers this is true
what do you mean by outside of range
pallid tapir
actually I mean outside of domain lol
in context we're not considering values of n less than 1
lunar bolt
what do you mean by outside of range
whats the answer to this question?
pallid tapir
-5 is outside of the domain of the original original sequence, (1 - 1/n) * sin(npi/2)
I'm nitpicking your example of choice for n
lunar bolt
oh because we are looking from n = 1 onward right?
pallid tapir
right
lunar bolt
ok
pallid tapir
outside of the context of the sequence, however, it is true that sin(-5pi/2) = -1
lunar bolt
we know this because:
- -1 is in the range of the sequence sin(npi/2) : it corre...
so why wouldnt n = 0.5 be one of the points that is for an accumulation point? I'm looking for disproving counterarguments to this thats why this is my question
and n = 0.5 mod 4
pallid tapir
n = 0.5 is not in the domain of the sequence
pallid tapir
and n = 0.5 mod 4
and this simply doesn't mean anything, sorry
the sequence is only defined for integers that are at least 1
0.5 is not an integer
lunar bolt
i know
pallid tapir
here's a problem I'll propose
Suppose you have a finite, nonempty set $X\subseteq\mathbb{R}$ and a sequence $(a_n)_{n=0}^{\infty}$ for which $a_n\in X$ for all $n\in\mathbb{N}_0$. Show that $(a_n)$ has accumulation points, and every accumulation point is a point in $X$.
lunar bolt
ok
elfin berryBOT
Flip
lunar bolt
is this a question?
lunar bolt
I have no clue its not a concrete problem like the one I gave where there are numbers
pallid tapir
there are numbers! there's a whole set of them, they all live in X
you can give them names even
pallid tapir
ok fine I'll backpedal
pallid tapir
so why wouldnt n = 0.5 be one of the points that is for an accumulation point? I...
you probably meant to ask "why are 1, 0, and -1 the only accumulation points of the sequence sin(npi/2)" which I do know you asked before, sorry
lunar bolt
yes
pallid tapir
the reason is because sin(npi/2) can only ever take on values 1, 0, and -1
suppose that x is a limit point of sin(npi/2) that differs from 1, 0, and -1
lunar bolt
why can it not take the value of (root of 3)/2?
pallid tapir
define $\epsilon$ to be the minimum of the set of half-distances between these points, $\epsilon=\min\left{\frac{|x-1|}{2},\frac{|x|}{2},\frac{|x+1|}{2}\right}$
elfin berryBOT
Flip
pallid tapir
why can it not take the value of (root of 3)/2?
this is a trigonometry question; refer to the unit circle
lunar bolt
yea, for sin its at 60 degrees, I dont get why this cant be the answer
pallid tapir
we're talking about sin(npi/2), not sin(npi/3)
it starts at an angle of 0 and each step corresponds to a counter-clockwise rotation of 90 degrees
you only ever reach the right, top, left, and bottom points of the circle
lunar bolt
oh ok so n cant be = (root of 3)*(1/pi) since its not in Df
pallid tapir
with horizontal components 0, 1, 0, -1 respectively
hoary jolt
is this channel only for one person only
marsh citrusBOT
To ask for mathematics help on this server, please open your own help channel or help thread. See #❓how-to-get-help for instructions.
lunar bolt
oh ok so n cant be = (root of 3)*(1/pi) since its not in Df
aka in the definition area
f: Df -> Zf
pallid tapir
that's correct
lunar bolt
ok
lunar bolt
I think I get it now
pallid tapir
it depends
lunar bolt
- I dont have a lot of time so i gotta go, thank you very much for the time and help you've offered
pallid tapir
no problem, good luck on the exam
hoary jolt
flip
frank mist
Hey, am I using the right formula for figuring out the volume of this triangular pyramid?
This seems to be an irregular triangular pyramid, and if that's the case, shouldn't the formula be a bit different as this formula is mainly used for regular triangular pyramid?
If this is not irregular, then can you please explain why so? What would make this pyramid irregular?
marsh citrusBOT
Hey, am I using the right formula for figuring out the volume of this triangular...
raw hawk
Hey, am I using the right formula for figuring out the volume of this triangular...
1/3(base area)*height of pyramid
frank mist
1/3(base area)*height of pyramid
is it same for the irregular ones?
or the one in the image
raw hawk
is it same for the irregular ones?
This is the general formula for volume of pyramids 👍🏿🙃
frank mist
This is the general formula for volume of pyramids 👍🏿🙃
yeah i understand now, thanks
was confused as to how to input the numbers
marsh citrusBOT
marsh citrusBOT
Available help channel!
forest dune
someone plz help
the greatest prime that divides :
1^2 - 2^2 + 3^2 - 4^2 + 5^2........ -98^2 +99^2
marsh citrusBOT
someone plz help
the greatest prime that divides :
**1^2 - 2^2 + 3^2 - 4^2 + 5^2...
forest dune
i think it uses the a^2 - b^2 property but not sure
knotty trellis
i think it uses the a^2 - b^2 property but not sure
Try using it then
broken dome
This can be simplifies so nicely
knotty trellis
what sum do you get if you apply a^2 - b^2 = (a-b)(a+b) for every consectuive pair?
knotty trellis
This can be simplifies so nicely
indeed
knotty trellis
what sum do you get if you apply a^2 - b^2 = (a-b)(a+b) for every consectuive pa...
(for simplicity, you can start at the pair 3^2 - 2^2)
forest dune
what sum do you get if you apply a^2 - b^2 = (a-b)(a+b) for every consectuive pa...
-4948
somehow
broken dome
Take mind of the last member being a +
knotty trellis
I expected you to just write out the sum
forest dune
Take mind of the last member being a +
yes i did not consider the first 1^2, started from 2nd, its coming an an ap of d=-4
knotty trellis
yes i did not consider the first 1^2, started from 2nd, its coming an an ap of d...
if you start from 2nd, then it should be (3^2 - 2^2) + (5^2 - 4^2) + ... + (99^2 - 98^2)
so the overall sum should be positive
forest dune
if you start from 2nd, then it should be (3^2 - 2^2) + (5^2 - 4^2) + ... + (99^2...
exactly
wait asec
knotty trellis
yes i did not consider the first 1^2, started from 2nd, its coming an an ap of d...
it should be d = 4
forest dune
it should be d = 4
oh
yes
still how tosolve tho
it goes on from 5,9,.......,197
so 49 total numbers
so n/2(a+l)
for sum
knotty trellis
okay, so what do you get?
knotty trellis
just do the factorization
it's divisible by 10
so 405 * 5 * 2
now it's still divisible by 5
forest dune
is 4050 correct tho?
elfin berryBOT
Result:
4949
Result:
4050
hmm
forest dune
so 5's the ans?
knotty trellis
4950
broken dome
550 / 2 = 225
forest dune
ig ans is 11 then
broken dome
It's the same kind of brainfart
forest dune
,calc 2 * 5 * 5 * 3 * 3 *11
elfin berryBOT
Result:
4950
thanks!
marsh citrusBOT
@forest dune Has your question been resolved?
red phoenix
marsh citrusBOT
red phoenix
Guys I got 4 hours for question 4, but the answer key says the answer is about 4 hours and 9 hours?
wooden tartan
i think you have to do 4 + 9 / 2
red phoenix
Wdym?
wooden tartan
4 + 9 = 13 / 2 = 7,5
red phoenix
Does it mean for how long is she away?
red phoenix
Yea?
wooden tartan
no, she stayed 4 - 9 hours 3 miles away
red phoenix
Yea
wooden tartan
no, it is asking approximately how much time did she stayed 3 miles away
wooden tartan
4 + 9 = 13 / 2 = 7,5
this would be the answer
red phoenix
For 5 hours
wooden tartan
she stayed 5 hours 3 miles away but the book is asking approximately at what time she stayed
red phoenix
Ok
red phoenix
she stayed 5 hours 3 miles away but the book is asking approximately at what tim...
So what is it
wooden tartan
So what is it
wdym
wooden tartan
we know that tanya stayed 3 miles away from home for 5 hours right? 4 - 9
marsh citrusBOT
@red phoenix Has your question been resolved?
marsh citrusBOT
marsh citrusBOT
Available help channel!
bright badger
marsh citrusBOT
bright badger
I am so lost pls help
opal shuttle
guys how did it change
granite dew
I want to find the domain of the following function:
1/tgx
How do you usually tackle such problems?
granite dew
I know that the domain of tgx itself is: (-pi/2 + pi * n; pi/2 + pi * n) where n is any whole number.
We also need to make sure that sinx is not equal to 0 and find the intersection with the former. That is the most confusing part for me.
Also, what should I generally use a guidance when it comes to finding the trig domains, the unit circle or the graph of the given function itself?
Like, I don't want to just memorize the domain of tgx for example,
brave marsh
So tan(x) is not defined anywhere cos(x) = 0.
Since we're taking its reciprocal, we also want to avoid places where tan(x) = 0, that is, where sin(x) = 0.
The domain of 1/tan(x) will then be the anywhere cos(x) != 0 and sin(x) != 0.
brave marsh
So no integer multiple of pi/2, essentially
granite dew
The answer from my textbook is ( (pi * n)/2; (pi)/2 + pi *n).
How do we get this range? Should I use the unit circle?
Let me show you what I mean visually
sorry it took me FOREVER
basically I try to look at this when I want to determine the domain / signs etc
so here we need to make sure that it is not equal to 2pi pi/2 pi or 3pi/2, right?
The part that confuses me the most is how we can represent that with one specifc range
or should we try to look at this graphically instead?
brave marsh
Just to be sure, your textbook suggests that the domain of $f(x)$ is $$\bigcup_{n=1}^\infty (n\frac{\pi}{2}, \frac{\pi}{2} + n\pi)$$?
elfin berryBOT
Azyrashacorki
granite dew
((pi * n)/2; pi/2 + (pi * n)/2)
brave marsh
Or maybe it would be $$\bigcup_{n=1}^{\infty}\left(\frac{n\pi}{2}, \frac{\pi + n\pi}{2}\right)$$
granite dew
yeah
brave marsh
Yeah ok that makes more saense
elfin berryBOT
Azyrashacorki
brave marsh
Well since the only disallowed inputs are the axes, you want to write down the "strict" quadrants.
So (0, pi/2), (pi/2, pi) , etc.
granite dew
yeah
brave marsh
So essentially, that means you can pick any multiple of pi/2 as your lower bound, and add pi/2 to it
granite dew
Any multiple of pi/2 will make those intersection points, right? yeah so the range of 2 such points
Did you look at the unit circle as a helper? or we can figure it out just like that
brave marsh
Yes, the intervals are made up of all consecutive multiples of pi/2
It's always a good idea to refer back to the unit circle for guidance
granite dew
yeah it makes more sense now as I look at it, ty!
So like, can we derive say, the values of sin that make it positive? using the unit circle
sorry I just want to make sure I get it
so for positive sin we would have first and second quadrant
brave marsh
You can express the domain on just [0, 2pi] from the unit circle, and once you have a general idea, you can just add multiples of 2pi to that
granite dew
yeah cuz sin has a period of 2pi, if we were dealing with tg/ctg I
brave marsh
Yep
granite dew
I'd add pi instead
so what would we get for positive sin
our starter range is (0, pi)
that is where sin is positive
so we get (0 + 2pi * k; pi + 2pi *k)
sorry can I ask u one more question
say we need to find all the values where cos is 0
these values are of kind pi/2 + pi * k
but cos has a period of 2 pi * k
so for some we do add 2pi * k and for other values we don't? like it is just not as intuitive for me to add pi here
I'd expect to add 2 pi k all the time but graphically it does make sense that we should add pi
brave marsh
cos has a period of 2pi, but it crosses the x axis twice in one period, once at pi/2 and once at 3pi/2
granite dew
oh I see, so before I write any range I need to check the period's graph first
elfin berryBOT
Azyrashacorki
brave marsh
So it all comes back to the period being 2pi. That's why I suggested you do the work for [0,2pi], i.e. the whole unit circle, and then add multiples of 2pi to your findings.
granite dew
I see, thank you so much for the detailed explanation.
You know, initially I wanted to solve it as a system of inequalities. Could we possibly get the right answer that way?
Like, I tried to add them together lol
Or maybe that is a wrong approach in general
oh sorry in the orange one it should be 2x
not x
There are some problems where I need to intersect\ multiple ranges like this
but yeah ig unit circle is the best way
granite dew
I see, thank you so much for the detailed explanation.
You know, initially I wa...
but like from here I don't really get anywhere I get a completely off solution lol
brave marsh
Systems of inequalities won't work, because x will never be in two quadrants simultaneously.
granite dew
yeah that too, I forgot about it lol
You definiely helped me clear up the confusion. Thank you so much, have an awesome day!) I will close the chat now ig cuz all my questions are answered:)
do you have anything to add?)
brave marsh
If you feel like you understand then that's good I've got nothing to add 🙂
granite dew
I will try to do similar problems now on my own and see if I can do them all, if I still have any doubts, I'll let you know!:)
.close
marsh citrusBOT
marsh citrusBOT
Available help channel!
strong belfry
help😭 how do i do no. 2?
marsh citrusBOT
help😭 how do i do no. 2?
wary kite
hello sushi
let x= a number
what is its multiplicative inverse
the multiplicative inverse of x
strong belfry
who's sushi😭😭
naive comet
help😭 how do i do no. 2?
try setting up an equation that represents this situation
wary kite
who's sushi😭😭
😹😹
wary kite
who pinged
still temple
help😭 how do i do no. 2?
x + 16/15 = 1/x
strong belfry
oohh
hollow sparrow
x + 16/15 = 1/x
Hey thanks for the help, however, we discourage providing the answer as we wish students to derive the answers on their own.
wary kite
!nosols
marsh citrusBOT
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.
next raft
hello sushi
lol
still temple
Hey thanks for the help, however, we discourage providing the answer as we wish ...
my bad. just wanted to help them asap as an ambassador to this community
wary kite
ambassador
next raft
ambassador
still temple
lol
what are you doing here? go back and do some topology
as an ambassador to the community 😎
next raft
D:
still temple
imagine trolling on a math server
wary kite
as an ambassador to the community 😎
nice bio
wary kite
imagine trolling on a math server
couldn’t be you
wary kite
ahh i’ve never been there before
i’ve always wanted to go there
never had enough time
still temple
how old are you 😭
wary kite
always busy working ya know
still temple
hyatt are you a troll
still temple
never had enough time
bro studies 24/7
wary kite
how old are you 😭
as young as i’ll ever be
strong belfry
😭😭🫶
wary kite
might be in the same class
still temple
just one of them asking should work
wary kite
yes because were sitting next to each other .
we are not
strong belfry
we are not
😭😭😭
still temple
😭😭😭
strong belfry
HWHAHAHAH😭
strong belfry
kind off 😭
still temple
shut up knief
still temple
kind off 😭
bro
still temple
do you really not understand what multiplicative inverse is
or you just don't study at all
strong belfry
i have the answer😭(i think)
wary kite
do you really not understand what multiplicative inverse is
your impudence is unwarranted sir
wary kite
or you just don't study at all
no need to be condescending
still temple
69 times 1/69 is 1. so 1/69 is the multiplicative inverse of 69
wary kite
nice
still temple
your impudence is unwarranted sir
lmfao
strong belfry
iron ocean
Is that r8ght for no.2
still temple
I did not expect you to know how to factorize ngl
iron ocean
still temple
you're smart, good
iron ocean
What
strong belfry
you're smart, good
tenk u por ur hilp☺️🙏 (thanks for ur help)
next raft
gyatt what are you doing 😭
gyatt
still temple
tenk u por ur hilp☺️🙏 (thanks for ur help)
thank god you get it 😭 you're really smart though
strong belfry
HUH😭😭
next raft
u got this sushi
next raft
sushi dont disturb someone elses channel
iron ocean
Im not were sitting next to each other and I got pinged here
next raft
:0
strong belfry
Im not were sitting next to each other and I got pinged here
HAHAHA
iron ocean
Samee
next raft
sameeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee
marsh citrusBOT
Available help channel!
thick solar
<@&286206848099549185>
2 4 8 32 256___ anyone help me with this question please
still temple
<@&286206848099549185>
2 4 8 32 256___ anyone help me with this question pleas...
@wary kite help him
it's going to be 2^1, 2^2, 2^3, 2^5, 2^8, 2^13
royal imp
<@&286206848099549185>
2 4 8 32 256___ anyone help me with this question pleas...
What question
marsh citrusBOT
Please only use the <@&286206848099549185> ping once if your question has not been answered for 15 minutes. Please do not ping or DM individual users about your question.
still temple
fibonacci
wary kite
<@&286206848099549185>
2 4 8 32 256___ anyone help me with this question pleas...
what’s the pattern look like
cold wolf
yeah fibonacci
thick solar
I was right
still temple
I am the ambassador to the community
wary kite
he needed reassurance
still temple
and then ping
wary kite
I was having dorito
what flavor
wary kite
But my teacher giving false answers xd
what did they say
still temple
what flavor
Sour Cream and Onion
wary kite
Sour Cream and Onion
ahh you can leave
still temple
🤓
wary kite
I also have original,cheese one too
i don’t eat doritos
still temple
i don’t eat doritos
ahahhaha why
wary kite
don’t eat chips
wary kite
to who
still temple
<@&286206848099549185>
2 4 8 32 256___ anyone help me with this question pleas...
8,192
thick solar
I was having dorito
Im soeeyyy i was checkin
still temple
ok
marsh citrusBOT
@thick solar Has your question been resolved?
marsh citrusBOT
marsh citrusBOT
Available help channel!
small stream
Given that the function ( f(x) ) is twice differentiable at ( x = 0 ), determine the number of correct statements among the following:
① If ( \lim_{x \to 0} \frac{f(x)}{x^2} = 1 ), then ( \lim_{x \to 0} \frac{f'(x)}{x} = 2 ).
② If ( \lim_{x \to 0} \frac{f'(x)}{x} = 1 ), then ( \lim_{x \to 0} \frac{f(x)}{x^2} = 1 ).
③ If ( \lim_{x \to 0} \frac{f(x)}{x^3} = 1 ), then ( \lim_{x \to 0} \frac{f(x)}{x^2} = 3 ).
④ If ( \lim_{x \to 0} \frac{f'(x)}{x^2} = 1 ), then ( \lim_{x \to 0} \frac{f(x)}{x^3} = 1 ).
Options:
(A) 1
(B) 2
(C) 3
(D) 4
marsh citrusBOT
Given that the function \( f(x) \) is twice differentiable at \( x = 0 \), deter...
elfin berryBOT
riyobi
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
severe owl
f(x) is twice differentiable
So we remove cases 3 and 4
1,2 both can be correct
So I guess it's 2 correct statement
small stream
So we remove cases 3 and 4
Why?
severe owl
In 3 and 4 we have x³ in denominator, a simple twice differentiable function is a quadratic, so if the denominator is cubic , it should result in the value infinity
small stream
So I guess it's 2 correct statement
Why
thorny bison
So I guess it's 2 correct statement
Why can't 1 be correct?
small stream
The correct answer is only 1
thorny bison
Why
f(x) is twice differentiable function so it may or may not be thrice differentiable
thorny bison
The correct answer is only 1
Was thinking that only
thorny bison
The correct answer is only 1
Take the limiting case for twice differentiable function as quadratic polyn
And solve the limits
thorny bison
For case 1 let's consider y = x²
It doesn't satisfy 2nd part
severe owl
2x/x =2
small stream
For case 1 let's consider y = x²
We can't prove its true or false only by making examples, cuz you can't list all examples exhaustively to make sure every of them all satisfied the condition
severe owl
For 2nd case let's consider x²-xsinx-cosx
small stream
Take the limiting case for twice differentiable function as quadratic polyn
Btw only quadratic polyn is twice differentiable? What about other functions? Idk but maybe some other special functions?
thorny bison
Btw only quadratic polyn is twice differentiable? What about other functions? Id...
U can find other functions but our aim is to play with the options and eliminate them we are not here doing a case study on twice differentiable functions
If qudratic wasn't enough for eliminating the options we would then have opted for other functions
small stream
Aah yeah
small stream
Take the limiting case for twice differentiable function as quadratic polyn
Then can you give an example?
small stream
For 2nd case let's consider x²-xsinx-cosx
It involved sinx, can we still call it quadratic polyn?
thorny bison
Then can you give an example?
Hmm let me think
Tbh rn I can't think of any other function other than quadratic 
Maybe I'm just dumb
marsh citrusBOT
@small stream Has your question been resolved?
small stream
I want explanation of (2) 😭 Anyone
mellow steppe
You want to know why the second statement above is wrong, right?
Is so, try f(x)=x^2/2. The first limit is equal to 1, so that's okay, but the second one is equal to 1/2
So that's a counterexample to the second statement, if that's what you were asking
small stream
Aah ty btw how can we rigorously prove that (1) is true
mellow steppe
Yeah I was trying that and then I got distracted, my first instinct was to prove it using taylor's theorem, and that's fairly easy to do, but I think there should be a more simple way
Ah yeah you can do something like this: If f(0) where anything different from zero, that the first limit would be equal to infinity (because of the x^2 in the denominator). Then, having an indeterminate form lie 0/0 you can use l'hopital's rule, and conclude
mellow steppe
why?
vestal forge
you can't use L'H in any of these cases though
we don't know whether the limit is indeterminate. It's best that you go by counterexamples for these
mellow steppe
Ooh wait I misread, I mean 2 and 4 can't use LH
You actually can't if you do not show that the second limit exists
small stream
Could you elaborate
mellow steppe
Yeah sorry, I got distracted
anyway: LH says that, under certain hypotesis, if the limit of f'(x)/g'(x) exist, than it is equal to the limit of f(x)/g(x)
what I wanted to do was something like: we know that the limit of f(x)/x^2 is one, it is an indeterminate form (for the reasons I explained before) so, "by LH" we have: the limit of f'(x)/2x=1 and then you can say that the limit of f'(x)/x=2
But we do not know whether or not the limit of f'(x)/x exists
the best you can say is that f'(0) is equal to zero
and that the limit of f'(x)/x should be equal, if it exists, to the limit of f''(x)
but we do not know if that exists
small stream
yea so we can use LH in 1, isn't it
mellow steppe
yeah
but only because you proved that the limit of f'(x)/x exist in a different way
small stream
yeah, anyway it feels safer to use Taylor instead. L'Hôpital's rule has various cases where it can fail.
.close
mellow steppe
How can f be a twice differentiable function with a third derivative? There is an error in the question? If not, condition 4 is false
small stream
yes it's false
marsh citrusBOT
@small stream Has your question been resolved?
near elbow
marsh citrusBOT
near elbow
guys this is the question
near elbow
the yellow highlight is the answer key on my book but it might be wrong
cobalt sedge
uhh, on right side 5th step, your 10^12 somehow became x^12
cobalt sedge
the yellow highlight is the answer key on my book but it might be wrong
that one IS right
limber condor
You just need to solve this system
near elbow
uhh, on right side 5th step, your 10^12 somehow became x^12
ohhhhh you're rightt!!
near elbow
You just need to solve this system
ohhh
okkk thank you sososo much @cobalt sedge @limber condor
.close
marsh citrusBOT
marsh citrusBOT
Available help channel!
torn night
what did i do wrong?
marsh citrusBOT
what did i do wrong?
torn night
the correct answer should be $u_{\pi/2}(t)(1-cos(t-\frac{pi}{2}))+sin(t)$ according to wolfram alpha
elfin berryBOT
nevergreen
torn night
where $u_{\pi/2}(t)$ is the unit step function at $\frac{\pi}{2}$ (0 if $t<\frac{\pi}{2}$ and 1 if $t>=\frac{\pi}{2}$)
elfin berryBOT
nevergreen
torn night
so just replacing the (t-pi/2) with a 1 would make it the correct answer, but that would also break the logic
marsh citrusBOT
@torn night Has your question been resolved?
storm meadow
I think you made an algebraic mistake when moving the f'(0) over and combining fractions?
Correct me if i'm wrong
Also inverse laplace of 1/s is 1
cuz L^-1(n!/s^n+1) where n =0 == t^0
@torn night
marsh citrusBOT
@torn night Has your question been resolved?
torn night
Also inverse laplace of 1/s is 1
yea but I did a modification to the substitution where I shifted it by pi/2
oh wait
tysm
.close
granite bone
can someone check my answer?
unique bough
can someone check my answer?
How did you get this result?
granite bone
How did you get this result?
i think i made a mistake, was i suppse to do y=3(sin(x))?
unique bough
3^sin(x)
Determine the range of sin(x) then 3^sin(x)
Since f(g(x)) = f(sin(x)) = 3^sin(x)
granite bone
3^sin(x)
this doesn't even give an option as an answer
thanks for the help tho
.close
marsh citrusBOT
marsh citrusBOT
Available help channel!
grim hemlock
The question is to find the inverse laplace transform on this
marsh citrusBOT
The question is to find the inverse laplace transform on this
grim hemlock
Here is the answer in the memo (they used partial fractions)
Here is the answer I got. I tried completing the square to end up with this answer
I don't know what I did wrong
round furnace
I havent completely followed your derivation to conclude if you did anything wrong but did you check if the answer really is something different? Cosh and sinh can still be written in terms of e^..
grim hemlock
I havent completely followed your derivation to conclude if you did anything wro...
I haven't checked this
grim hemlock
I havent completely followed your derivation to conclude if you did anything wro...
turns out they the same I think
.close
marsh citrusBOT
marsh citrusBOT
Available help channel!
alpine wharf
I'm stuck at the bottom line. Rlly don't know how to isolate this exponent..
cunning fiber
Logarithms: $n=\log_2 (2732)$ (however, $n$ isn't an integer, so idk if that's inconsistent with the broader context)
alpine wharf
Oh I'm not allowed to use logs
elfin berryBOT
Civil Service Pigeon
cunning fiber
Oh I'm not allowed to use logs
What's the original question?
b/c my guess is that n is an integer, but something went wrong along the way
alpine wharf
In a geometric series, the first term is 23 and the third term is 92. The sum of all the terms of the series is 62813. How many terms are in the series?
alpine wharf
How can I find the common ratio for this?
cunning fiber
well how did you find the common ratio?
alpine wharf
I sort of guessed like 23x2 is 46 and 46x2 is 92
cunning fiber
do it properly with the explicit formula
If we let the first term be a and the common ratio be r, what are the first and third terms in terms of a and r?
cunning fiber
What's the formula for r?
Remember $ar^{n-1}$?
elfin berryBOT
Civil Service Pigeon
cunning fiber
92=23r^3-1
92/23
4=r^2
Root4=r
2=r?
$r^2=4 \implies r=\pm 2$
elfin berryBOT
Civil Service Pigeon
alpine wharf
ohhh right
Icccc
I'll try it with - 2
Omg that number is so much nicer thx LOL
I got 13 terms
cunning fiber
marsh citrusBOT
marsh citrusBOT
Available help channel!
marsh citrusBOT
https://discord.com/channels/268882317391429632/1275579342281506876
fleet sable
Can someone coach me precalc?
marsh citrusBOT
@tender spruce Has your question been resolved?
marsh citrusBOT
https://discord.com/channels/268882317391429632/1275579342281506876
If you are done with this channel, please mark your problem as solved by typing .close
Available help channel!
marsh citrusBOT
https://discord.com/channels/268882317391429632/1275579342281506876
tender spruce
Help
marsh citrusBOT
@tender spruce Has your question been resolved?
tender spruce
Not yet
marsh citrusBOT
@tender spruce Has your question been resolved?
marsh citrusBOT
@tender spruce Has your question been resolved?
rich cairn
@tender spruce I believe it would be less time-consuming if you repost the question(s) you need help specifically here.
marsh citrusBOT
@tender spruce Has your question been resolved?
mild sentinel
You spin a 6-sided die on one of its vertices on a flat table. What is the angle between the table and one of the bottom faces of the die?
marsh citrusBOT
You spin a 6-sided die on one of its vertices on a flat table. What is the angle...
mild sentinel
I am having trouble visualizing the geometry of the problem
frail orbit
think about looking at the dice straight on at one of its vertices. you can simplify this problem
frail orbit
ye
mild sentinel
Ok then hmm
still temple
it's easy to visualize
frail orbit
what class is this for?
mild sentinel
it's easy to visualize
I’m sorry but that doesn’t help me lol
still temple
mild sentinel
what class is this for?
This isn’t for a class, it was just a problem I came across and wanted to do since geometry is my weakest subject in terms of competition math
still temple
are people that lazy here
mild sentinel
It’s why I said I was having trouble visualizing the geometry because my diagram wasn’t helping me visualize
mild sentinel
are people that lazy here
ok whatever
still temple
It’s why I said I was having trouble visualizing the geometry because my diagram...
what did you make
show me
mild sentinel
This was how I tried to draw it as if i was looking at a vertex
But then I didnt really know what to do with that
mild sentinel
Yep 😃
I also tried to think of how the cross section would look
But it wasn’t very intuitive to me
still temple
the easiest way to find out will
write the equation of the plane of the bottom of the die
and then find the angle between it and the xy plane
simple
mild sentinel
How would I find that equation
mild sentinel
Not really, like I said geometry is probably my weakest subject
still temple
then use intuition
there's no mathematical tool that can help you to analyze a 3d angle then
still temple
Not really, like I said geometry is probably my weakest subject
learn 3d geometry
mild sentinel
why would it be a 3d angle? Cant I treat it as a 2d angle?
still temple
no
mild sentinel
Sorry if I’m being a monkey but this stuff isn’t intuitive to me🥲
still temple
2d angle is made by the bottom side of the die
the bottom side is a plane
3d angle is completely different
solid angle
no use of it though
but we need 3d geometry nonetheless
mild sentinel
Hmm ok
mild sentinel
It’s not very constructive for you to repeatedly say it’s intuitive when I am clearly having trouble with the intuition part of the solving process
Anyways
I’ll come back to this when my brain has a few more folds I suppose
.close
marsh citrusBOT
still temple
It’s not very constructive for you to repeatedly say it’s intuitive when I am cl...
it's bad without 3d geo
Place the balancing vertex at the point (0,0,0) and then, on one face, the two vertices adjacent to this one at (2,2,0) and (2,0,2).The centre of this face is (2,1,1) and so the fourth vertex of this face is (4,2,2). It would be fair to consider desired angle to be the angle made by the table this the diagonal of this face joining (0,0,0) to (4,2,2). Perform this dot product
do you get it?
marsh citrusBOT
Available help channel!
void elm
marsh citrusBOT
https://discord.com/channels/268882317391429632/1260424130822541403/127568670208...
void elm
.solved
novel ice
marsh citrusBOT
still temple
18*14 + 7*9 ez
novel ice
huh
still temple
bruh
novel ice
gonna need this to be dumbed down, it's my first time learning this :(
still temple
novel ice
I see, but why 18-11?
still temple
gonna need this to be dumbed down, it's my first time learning this :(
treat them as two rectangles
14 and 18 feet rectangle
novel ice
right, I know that part
still temple
and (18-11) and 9 feet rectangle
novel ice
but why 18-11?
novel ice
the bottom...
novel ice
ged math test
still temple
just treat it as two rectangles
still temple
ged math test
no but what's the point of bottom
or saying that
makes zero sense
just say length and breadth
novel ice
cause there's a big empty bottom
novel ice
breadth?
still temple
yes
novel ice
are you american?
still temple
still temple
are you american?
no I am not, thankfully