Cuz in this case: s and t must both be the same (Condition 2)
#help-23
shut aspen
therefor s = t = 5/2 ?
But if you plug it it, you will find that x1 = x2 and y1 = y2, but z1 != z2
z1 does not equal z2
Therefor the crossection of f and g is Ø
@brazen nimbus Im gonna sign of for today, glad to have helped! Good luck!
brazen nimbus
Therefor the crossection of f and g is Ø
oh
wait so if they have an intersectio
t and s MUST be the same?
shut aspen
No no, in this case (according to condition 2: 2t + 2s = 0)
S and t can be any real number, but the conditions confine it to 1. But, if you plug in, s and t in their respective places, you will find that the x an y coordinates are the same, but the z coordinate differs.
brazen nimbus
hmm
shut aspen
Good, now plug it in f and g respectively
You ll find 2 different points
Therefor these lines cross but dont interdect
brazen nimbus
Therefor these lines cross but dont interdect
wait
omg
mb english isnt my language
is there a difference between cross and intersect?
brazen nimbus
i meant like the point where they meet each other basically
shut aspen
And crossing is where they dont interdect and Arent parallel. Take 2 rulers and wave them around, they dont necessaraly intersect
shut aspen
But they cross (the distance between the lines is the smallest there)
brazen nimbus
wait the point for 5/2 why do we get that for both i dont understand
it bascially tells us the common point?
shut aspen
Therefor the crossection of f and g is Ø
Refering back to this:
brazen nimbus
ohhhhh
wait so i can imagine it like 2 rulers
one point being above another but paralell basically?
shut aspen
wait the point for 5/2 why do we get that for both i dont understand
In this case we get it since we only considered the first 2 conditions: x point must match and y point must match. Z, however does not.
brazen nimbus
In this case we get it since we only considered the first 2 conditions: x point ...
oh damn so
shut aspen
Imagine 2 rulers floating in the air, extending to infinity. They only intersect if they are on the same plane. But they cross otherwise. Unless they are parallel. Then they never cross
brazen nimbus
Imagine 2 rulers floating in the air, extending to infinity. They only intersect...
oh i get it yeah
like basically
the point where they cross
they dont actually touch there right? the one point is basically above the other?
shut aspen
Yes
If you measure distances between lines, you do it at those crosspoints
Since they are the closest to any other point
shut aspen
No problem man
Now, i dont want to confuse u, so if its confusing just ignore. But these rules about crossing and intersecting and parallel only apply in euclidian geometry. You also have spherical and hyperbolic geometry, where for example, 2 parallel lines on a sphere, still intersect. (Imagine the globe and its poles)
brazen nimbus
Now, i dont want to confuse u, so if its confusing just ignore. But these rules ...
nah i dont need this atm
i gotta learn about distances between lines, planes, planes and lines etc
all that
isnt too hard yet tho
shut aspen
Its worth a mention. But good luck man! Close if you are done?
safe radishBOT
safe radishBOT
Available help channel!
supple basin
i gotta prove if the equality is true
but i dont know what i'm doing and why i didnt get the answer
supple basin
*parte da esquerda = left side
*parte da direita = right side
i gotta prove if the equality is true
but i dont know what i'm doing and why i didnt get the answer
safe radishBOT
*parte da esquerda = left side
*parte da direita = right side
i gotta prove if t...
@supple basin Has your question been resolved?
quiet plume
It still requires some more work than just rewriting the LHS and the RHS.
Try to show that some x in the LHS is in the RHS and vice versa.
i.e. show that if x satisfies the condition you wrote down for the left side, then it satisfies the equation for the right side.
And similarly for the other direction
supple basin
no idea how to to do that
can u show me?
because i neved did this kind of exercise b4
quiet plume
Well suppose $x$ satisfies "$(x\in A \land x\not\in B) \lor (x\in B \land x\not\in A)".
flat frigateBOT
Azyrashacorki
Compile Error! Click the 
reaction for more information.
(You may edit your message to recompile.)
quiet plume
Then look at the first case. If $x\in A$ and $x\not\in B$, can you conclude that $(x\in A \lor x\in B)$?
flat frigateBOT
Azyrashacorki
quiet plume
Can you conclude that $(x \not\in A \lor x \not\in B$)?
flat frigateBOT
Azyrashacorki
quiet plume
Well let's just look at the left side. If you pick x in the left side, then there are two cases
1- either x is in A-B
2- or x is in B-A.
In case 1, x is in A-B, so x is in A and x is not in B.
Then your goal is to show that x is in the right side, which means x is in AuB and x is not in AnB.
So in case 1, is x in AuB? Is x in AnB?
supple basin
So x has to be in A and not in B on the right side too?
quiet plume
To show two sets are equal, you show that any element of the LHS is in the RHS, and that any element in the RHS is in the LHS.
So for LHS -> RHS, you pick some arbitrary element in the LHS and you show that it's in the RHS.
In this exercise, being in the LHS means that x is in A-B or that x is in B-A, so you have two cases to check.
In the first case, x is in A and x is not in B. You have to conclude that x is in the RHS.
In the second case, x is in B and x is not in A. You have to conclude that x is in the RHS.
quiet plume
For the first case, you know x is in A and x is not in B. To conclude that x is in the RHS, you have to show that it's in AuB and that it's not in AnB.
safe radishBOT
@supple basin Has your question been resolved?
vague phoenix
Suppose I have $\lim_{n \to \infty} \sum_{i=1}^n \frac{8i^6}{n^7} = \int_a^b f(x)dx$, how would I find a,b, and f? Don't give me answer, I just want to know how would I approach this problem
safe radishBOT
Suppose I have $\lim_{n \to \infty} \sum_{i=1}^n \frac{8i^6}{n^7} = \int_a^b f(x...
flat frigateBOT
❤ Hu Tao ❤
median vigil
i would start out with the riemann sum of a generic function (make some simplifying assumptions e.g. equal length intervals and evaluating at an endpoint) and then compare to the sum given
vague phoenix
Not Darboux sum?
median vigil
well it would be much easier if the sum involved evaluating the function rather than taking supremum or infimum
and we're not setting out to find all possible riemann sums really, just show that this is one possible riemann sum corresponding to that integral
vague phoenix
But I haven't really touched Riemann sum yet, we mostly covered Darboux sum, hang on, let me copy the Riemann sum here
(\text{Area}\approx \sum {i=1}^{n}f(c{i})\Delta x_{i})(n)
flat frigateBOT
❤ Hu Tao ❤
median vigil
yeah so we're going to make some choices about what delta x_i and c_i are
vague phoenix
So $\Delta x$ here is $\frac{b-a}{n}$
flat frigateBOT
❤ Hu Tao ❤
median vigil
assuming equal length subintervals, yes
vague phoenix
But $\Delta_{x_i}$?
flat frigateBOT
❤ Hu Tao ❤
median vigil
we're assuming that all Delta x_i are equal and therefore dropping the index and calling it Delta x
vague phoenix
Is it for i-th intervals?
Ah okay
So what can I do first with n^7?
It kinda related to the Delta x
median vigil
i would also make a simplifying assumption about c_i
vague phoenix
Assumption?
Well I know that if we convert this into the Riemann sum formula, which is height x width, then width is when divide the whole length from a to b by n sections
which length could be 1/n if suppose it run from [0,1]
Then height would be $8 \cdot (\frac{i}{n})^6$
flat frigateBOT
❤ Hu Tao ❤
safe radishBOT
frigid lion
How did they get 6 squared times 2??
safe radishBOT
How did they get 6 squared times 2??
frigid lion
Isnt the answee supposed to be 3 squared and 2 cubed?
patent smelt
Do you know how you can swap numbers in factorization?
karmic merlin
2^2×3^2=6^2
patent smelt
Like say we have 2*3*5, this is also 5*2*3
patent smelt
2^2×3^2=6^2
Or yeah you can just shortcut everything and not explain why...
Basically, we make two 6's using a two and a three within that factorization, giving us 2*6*6
Because 2*3 is 6
Because we can just place the factors wherever, we can group them like this
frigid lion
I understand now
patent smelt
Isnt the answee supposed to be 3 squared and 2 cubed?
And also, yes, these are equivalent ways to write it
opaque fern
Or yeah you can just shortcut everything and not explain why...
What they said serves as an explanation too. There is no need for the attitude though
frigid lion
And also, yes, these are equivalent ways to write it
I understand now, also the answer is actually reasonable for what its going to be used for in the example
Thanks
.close
safe radishBOT
patent smelt
What they said serves as an explanation too. There is no need for the attitude t...
Oh my god I forgot ellipses come off as passive aggressive
naive dragon
green Car when
patent smelt
green Car when
Never if I can't fix my language icl
safe radishBOT
Available help channel!
shut gust
i lost my khan academy qudratics lesson link can anyone gimme link
shut gust
@vague phoenix i lost my khan academy qudratics lesson link can anyone gimme link
small sandal
errant ravine
-# did u try going to khan academy and seraching for it? you should find it pretty easily i belive..
https://www.khanacademy.org/search?referer=%2Fmath%2Falgebra%2Fx2f8bb11595b61c86%3Aquadratic-functions-equations&page_search_query=qudratics (here is alll lessons about quadratics idk which one u want)
safe radishBOT
safe radishBOT
Available help channel!
blazing kindle
safe radishBOT
brisk shard
can i get a hint
maybe use the fact that the distance between centre and the line which touches the circle is equal to the radius
brisk shard
yes... do you know how to do that?
blazing kindle
yes... do you know how to do that?
i forgot
i searched about it on the internet and they are just saying the formula
i wanna know how it works
errant ravine
yes... do you know how to do that?
-# also looking at the choses- i feel like there is no answer wehre the circle touches the line once-
errant ravine
i wanna know how it works
okay i can try to explain it if u want- but it might take a bit-
blazing kindle
okay i can try to explain it if u want- but it might take a bit-
sure man
btw i got the answer
lean otter
hey there
blazing kindle
errant ravine
btw i got the answer
yup thats right!
lean otter
i would like some books on abstract algebra as well as vector algebra
errant ravine
i would like some books on abstract algebra as well as vector algebra
-# sorry this helps channel is occupied check the category above and u should be able to find open helps channel! sorryy!!
brisk shard
In case of circle, touch means just cutting at once
errant ravine
In case of circle, touch means just cutting at once
okay then D is correct i belive
errant ravine
sure man
okay so first things off we need to know the basic distance forumla do you know the basics distance formula?
i can explain that too dw (if u dont know)
tawdry plover
i would like some books on abstract algebra as well as vector algebra
!occupied
safe radishBOT
i would like some books on abstract algebra as well as vector algebra
Someone else is already using this help channel. If you need help with a question, please open your own help channel/thread (see #❓how-to-get-help for instructions).
errant ravine
sure man
brb 1 sec but dw if u would like the explination just ping me when ur back and i will be here!
blazing kindle
brb 1 sec but dw if u would like the explination just ping me when ur back and i...
yes
if u can provide me resources
i can look at that up
myself
blazing kindle
yes
errant ravine
if u can provide me resources
well its really basic i can explain it quickly- if u have the time rn
errant ravine
okay so first things off we need to know the basic distance forumla do you know ...
^^?
blazing kindle
okay so first things off we need to know the basic distance forumla do you know ...
yes
i know that
errant ravine
okay know i want you to imagne any point and a line-
if u draw the shortest pathbetwheen them what angle would it form?
do u have any guesses?
blazing kindle
okay know i want you to imagne any point and a line-
if u draw the shortest pat...
90
degree
line
perpendicular!
blazing kindle
know did u ever take how to find perpendicular lines?
umm
like in geometry yes
slope
i guess
errant ravine
YES!
errant ravine
YES!
blazing kindle
we find the slope
errant ravine
yes!
and you know what we can do when we have a slopeand a point?
we can draw a line there!
using the formula y - y1 = m (x- x1)
errant ravine
yes so know we have a line that intersects our point and intersects the original line
and so by findning the point of intersection between our 2 lines
we can use the distance formulw with our original point and this new intersection point
does that make sense?
errant ravine
i want to ask here too
!status
safe radishBOT
i want to ask here too
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
so i need to find the another equation on h and k
so that i can find the center of the circle
and then i can write the equation of the circle easily
errant ravine
okay oaky actually finding the cneter here is really easy-
okay i am gonna give u a small hint
blazing kindle
so first step i did is equating the radius
errant ravine
so first step i did is equating the radius
yes that is correct
errant ravine
if h ave lets say a tree and a trash can
if we assume that the distance i want is 5
at how many points can i stand where the distance between both of these object compared to me is 5?
errant ravine
yes yes but there is a simpler approach to find the center-
blazing kindle
if we assume that the distance i want is 5
at how many points can i stand where...
idk
errant ravine
if we assume that the distance i want is 5
at how many points can i stand where...
wait i will draw u visualization
okay so lets imagne the distance between them was a total of 10?
at how many points could I stand wehre the distance between each object for me would be 5?
cause if i stand too cloose to the tree i will be too far away from the trash can-
blazing kindle
mid point?
blazing kindle
how is that going to help with my solution
errant ravine
say i want to be 6 distance away instead?
errant ravine
how is that going to help with my solution
wait trust me-
blazing kindle
section formula
errant ravine
say i want to be 6 distance away instead?
at how many points could i stand where the distance between me and the trash can are both 6
blazing kindle
dont know
errant ravine
okay imagne ursef likeactually i nthat situation-
and you have a rope that u hold in ur hands tied to each object- above
errant ravine
to fully extend the rope you would keep awalking away either up or down until ti fully stretches right?
errant ravine
okay can u imagne there bieng any other points other than this one?
what if u had went up instead until the rope stretches?
u would get the same exact traingle but upwards correct?
errant ravine
no nono- wait wait-
blazing kindle
and since we already know the two ends we can find the lengith of the chord
but how will we find h,k
errant ravine
wait wait-
errant ravine
u would get the same exact traingle but upwards correct?
so how how many places can u stand in where u would be 6 far away from both points?
blazing kindle
this is the middle point of the chord that we can easily find
now we can use pythagoras theorem to find that
blazing kindle
u were hinting that right?
errant ravine
okay so nitce how only 2 points could possible exsist- where both ropes are = 6
if we take You (person as pulling the rope)
as a center of the circle
and the rope as R
that means how many places could the center of the circle be?
errant ravine
u were hinting that right?
hoenstly i am impressed that u though of that cause i honestly didnt so that meanas u jsut outsmarted me! 
blazing kindle
hoenstly i am impressed that u though of that cause i honestly didnt so that mea...
nah i am stupid person
errant ravine
that means how many places could the center of the circle be?
-# would u like a hint-
blazing kindle
i know nothing man
errant ravine
that means how many places could the center of the circle be?
no no come on think about it ^^
blazing kindle
that means how many places could the center of the circle be?
the line that cut the middle through
errant ravine
the line that cut the middle through
yes but if we place our circle there each side will only be 5 long!
we need the rope fully strecthced
so 6 long!
errant ravine
^^
errant ravine
thats okay so what i am trying to say is that bassicly the circle center can only exsist on 2 points
either below or above
why you ask?
because when u calculate the distance between any 2points-
blazing kindle
so how is i am going to find the center
errant ravine
okay so notice how here there is only 2 Possible points where the lines are equal 2?
becauase if u go any further to the right it gets too long
and any closer to the left its too short
so this means we know that circle center can only exsists in 2 places-
i hope this made sense-
if it doesnt its okay we can sovle it with pythagerous its oaky-
blazing kindle
hmm
errant ravine
just nvm sorry for wastng ur time countinue as u were we can do it the other way since its probably easier for you!
blazing kindle
btw i did used that pythagoras solution it is too long
and i might messed it up somewhere
cuz i am getting fraction values
@errant ravine
errant ravine
wait are u sure u graphed that line correctly?
it should have a downward slope no?
blazing kindle
i randomly draw it
errant ravine
oh okay makes sense-
blazing kindle
i will try it again after eating
blazing kindle
will probably find a good way
errant ravine
its correct
blazing kindle
why do uthink this solution is wrong?
i think the points are wrong
they shouldnt be fraction
see the equation
blazing kindle
lemme find the equaiton
errant ravine
okay so here is the thing- about the circle equation
its sually written as
(x-h)^2 + (y-k)^2 = (radius)
blazing kindle
how
errant ravine
the center of the circle is (1 , -3/2) which is (1 , -1.5)
wait how much did u calcualte the radius to be?
blazing kindle
radius
blazing kindle
ohh so i did messed it up
errant ravine
this graph is wrong-
because point (-1,3) lies directly perpenducilar to the line-
and therfore distance from the two points is the diamater-
sorry i was really confused i should have realized this earlier
errant ravine
okay but this question is insaely easy now-
because distance from (3,0) to (-1,-3) is the diamater
and therfore the mid point is the center!
@blazing kindle
errant ravine
hey no worries-
tbh its aslso my fault for relying on the drawing so much that i didnt realise this earlier sorry iwasted alot of ur time 
but i think know that u can very easily prove that point (-1,-3) lies on the perpedincular line its pretty easy to get everything else probably-
blazing kindle
this was hella lot of adventure of a question
errant ravine
well i hope i hlped somehow,,
i wish you best of luck! and if u need anymore help feel free to ask!
errant ravine
mid point?
i should also menton that u were absoultuly correct here btw^
blazing kindle
safe radishBOT
blazing kindle
how can i proceed
errant ravine
okay so so
we know that since the circle touches that line! Its center will be on the perpendculaiar line correct?
blazing kindle
errant ravine
-# the slope here isnt equal so this is porbably not true-
karmic merlin
Do you know the equation for the distance between a point and a line
blazing kindle
blazing kindle
Do you know the equation for the distance between a point and a line
yes
karmic merlin
now think about the distance between the center of that circle and the line, what do you think that would be
blazing kindle
now think about the distance between the center of that circle and the line, wha...
errant ravine
wait how do you know the circle is in the inside?
blazing kindle
since its tocuhing all three lines
thats the most possible cases
these are all three lines
so the circle must be inside
@errant ravine
if i guessed right
karmic merlin
Shouldn't there be an absolute value there
errant ravine
because it also had the answer- ;3
oop-
but uget the idea
it doesnt Have to be inside-
errant ravine
well there is one thing we know-
if u have a circle touching both x=0 and y =0
burnt notch
Shouldn't there be an absolute value there
Yeah that's the reason for which you will get two solutions: the circle inside and the one outside
karmic merlin
Yeah that's the reason for which you will get two solutions: the circle inside a...
Makes sense
errant ravine
well there is one thing we know-
if u have a circle touching both x=0 and y =0
that means that the center of the circly can only exsist on line x = y
makes sense?
errant ravine
nice!
errant ravine
also uwere correct about ur assumption here!
errant ravine
Yeah that's the reason for which you will get two solutions: the circle inside a...
||i should mention though that looking ath the possilbe solution all hese circles are too large to fit insde-||
errant ravine
no i mean line x =y as in the daignoal line-
so if we know that it always is touching x = 0 and y = 0
and that its center is on line x = y
how can we use the line it give us to find our solution?
(think about R)
low canyon
that means that the center of the circly can only exsist on line x = y
or y=-x
blazing kindle
(think about R)
errant ravine
okay so we know then that there must be a point (on the purple line)
where the distance from x=0 and y=0 and the red line is the same correct?
errant ravine
or y=-x
-# there is actualyl a soltuin for this too lol nice! (although its not in any of the optoins)
errant ravine
what are the cord u found of the point for the center of the circle?
blazing kindle
i am confused
errant ravine
what to do after
well after u find the center of the circle u simply calculat the radius-
errant ravine
is it inside or outside
(both could be correct but the asnwer u have in ur book is only outside)
blazing kindle
how can i find center
errant ravine
okay so
to find center-
we know that the center is on line (x = y)
and we know that the circly is always touching the x =0 and y =0
blazing kindle
errant ravine
we know that the center is on line (x = y)
so we can go ahead and say that the radius of the the circle increasing proportianly to its x and y value?
errant ravine
okay the daigram you drew is true- except for on tine thing..
blazing kindle
what?
errant ravine
notice how here
the point that connects the line to the circly is not- the same point where xy intersct wit hthe circly-
blazing kindle
UGGHHHH
blazing kindle
now i understand why some ppl dont like maths
blazing kindle
wtf
errant ravine
<@&268886789983436800>
blazing kindle
mr beast
errant ravine
okay so so so lets stop the visualiation for a minute
and i want u to just think about this algeberlicly-
blazing kindle
why is this so complicated bruhh
errant ravine
its not-
the solution is really simply and just relies on finding R
so here is the cool thingabout the circle touching y =0 and x =0
blazing kindle
okay so so so lets stop the visualiation for a minute
so even if those two points are different
we can still find R
errant ravine
so even if those two points are different
yes!
blazing kindle
family of cute lil circles
errant ravine
grow porpotianlly to how far away they are on the x axis-
so bassicly the more to the right we go the bigger the circle gets
makes sense right?
errant ravine
YES!
blazing kindle
so the circle gets bigger and bigger and bigger
thats why the center point is called origin
errant ravine
so lets say i put a point at idk (2,1)
blazing kindle
cuz it lays baby circle
blazing kindle
so lets say i put a point at idk (2,1)
alr
blazing kindle
go on
errant ravine
u can imagne that our circle keeps getting bigger and bigger
until it touches that point
right?
blazing kindle
what if the point is inside
errant ravine
what if the point is inside
the circle?
blazing kindle
we have to look at the bigger picture
errant ravine
YESSS
okay so distance R
at the point of when we touch the "point" we want
will asso be the distance the same distance away from y=0 and x =0
and it is that Fact that makes us able to solve this!
does that make sense?
blazing kindle
no
errant ravine
look here-
notice how the distance from the center and the point we choose
is the same distance from the center and y?
blazing kindle
i can see square
errant ravine
okay okay oaky say Any random postive point
blazing kindle
yes
errant ravine
like you choose any random point
blazing kindle
errant ravine
yes nicE!!
errant ravine
now this circle can grow or get smalleracording to our graph here^
so bassicly we want it to get big enough whereit just TOUCHES the line it gave us
Does it make sense now?
blazing kindle
okie
errant ravine
so ofcourse with just the fact that it touches x =0 and y =0
we can get inifite small and big circles-
so what is the stopper in the end?
blazing kindle
with that line
errant ravine
brb- try to solve it you got this!! i will be back soon!!
safe radishBOT
Available help channel!
austere heart
Hey is this right formula for power set of all polynomial till degree of m
austere heart
x <= m, **
warm warren
Hey is this right formula for power set of all polynomial till degree of m
$\sum_{i=0}^n a_i x^i$, I feel.
why did you keep $a_i x^{n-i}$?(i feel both are correct, but the $a_ix^i$ is more common is more common)
flat frigateBOT
Annie Maqionde
azure harbor
Hey is this right formula for power set of all polynomial till degree of m
reddit logo for the set , damn is 😹 🎉
flat frigateBOT
Annie Maqionde
austere heart
why did you keep $a_i x^{n-i}$?(i feel both are correct, but the $a_ix^i$ is mor...
From certain book
warm warren
Hey is this right formula for power set of all polynomial till degree of m
yk instead of doing the entire reddit thing, you cold have just done $n \in \mathbb{N}, n \leq m$
flat frigateBOT
Annie Maqionde
austere heart
Oh yeah meeee
warm warren
Hey is this right formula for power set of all polynomial till degree of m
your current usage of 'x' in the second line may confuse people. Consider using a different variable
austere heart
Thankuu
austere heart
your current usage of 'x' in the second line may confuse people. Consider using ...
Oh yeah I’m stupid I got used to variable scope
Okay byeeeee
.close
safe radishBOT
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Available help channel!
fringe dock
hi
fringe dock
i need some help on this question on part b
as i dont get the working out
none of this makes sense to me
karmic merlin
As the ball goes up, gravity goes against it, reducing it's speed
That is until the velocity reach 0 as then velocity goes up and gravity increase the speed
fringe dock
As the ball goes up, gravity goes against it, reducing it's speed
That is until ...
so what does the speed get increased to?
a very big number?
foggy fossil
No it has a limit due to ait resistance
Air
It gets increased to √2gh in ideal case where there is no sir resistance where h is the maximum height it goes up and it is proved in kinematics class 11
foggy fossil
What the question?
fringe dock
part b
foggy fossil
No for that like it is asking when the speed is less than 24.5 so basically in projectile motion the speed decreases while going up and increases while coming down so the there will be two times when the speed will be 24.5 once when going up and once when coming down
So basically we need to find the time required by ball to travel from the first to second point
fringe dock
its ok i have the answer, i just needed help with understanding the concept
errant ravine
its ok i have the answer, i just needed help with understanding the concept
okay
so the concept of part b right?
well think about it this way
when u first "throw" the stone its pretty fast but grabity pully it doen so ti s pseed slowly decreases until it reaches the apex where gravity starts acelerating it again UNTIL it hits the ground
errant ravine
a very big number?
well if its at the same hieght it goes back to the same number u threw it at
lets say i drink 20 liters per second
after 5 seconds i finished the enitre bottle
then i start pouring water again at the same rate
cause gravity rate never chanes
it just goes from working against to with
and if the hieght it can fall is infiite then yes it wil keep accelrating infinetly
safe radishBOT
@fringe dock Has your question been resolved?
fringe dock
it just goes from working against to with
i see
so the max speed itll ever be
is the speed it was thrown at?
errant ravine
is the speed it was thrown at?
if it is the same hieght yes..
fringe dock
if it is the same hieght yes..
so it is like symmetry?
fringe dock
okay
errant ravine
and iget dizzy the closer i get to it
fringe dock
okay
errant ravine
i start 5 away and start moving towards it
every step i move towards it i become 1 more dizzy
fringe dock
yes
fringe dock
yes
errant ravine
and ofcourse the amount it takes me to travel to it
fringe dock
ohh i think i get it now
errant ravine
is the same amount it takes for me to get back
BUTT-
if there is no wall to stop me
i can just keep moving away from it
fringe dock
ya
fringe dock
does that make sense now?
a lot more yeah
this part
at the same point when going up vs going down, it is 24.5
errant ravine
at the same point when going up vs going down, it is 24.5
yes because the amount of dizzniess here is its speed
and its telling u how much time it spends not bieng more or less dizzy than 24.5
errant ravine
yes!!
ifu want u can test this out
throw a water bottle into the air
film it
and catch it at the same hieght
u wuld notice the time it took for it to go all the way up
is the exact same amount of time it took to go back down
errant ravine
this is called prjectile function in physics!
fringe dock
same logic to a bullet being dropped to the ground vs a bullet is fired until it hits the ground will be equal in time
right?
errant ravine
YES!
becaue the amount of time it takes to fall is unrealted to how fast ur moving horzotnally
safe radishBOT
errant ravine
you too!
safe radishBOT
Available help channel!
nova plover
can anyone help me solve this?
safe radishBOT
can anyone help me solve this?
outer pollen
for future helpers: what have you tried and what do you not get?
opaque fern
can anyone help me solve this?
identify the conditions for NOT playing any game
nova plover
wdym
opaque fern
wdym
student playing cricket are those with odd numbers
students playing football are with numbers divisble by 5
students playing hockey are those numbers divisible by 7
students NOT playing any games are........
fill in the blank
opaque fern
thats what im asking to help me out
im not telling you to tell me what the value is
just what conditions are necessary to meet to be not playing any game
buoyant shadow
actually you should draw venn diagram
and fill inside out
i will do both
one student is in the middle, number 70
opaque fern
even, not div by 5 and not div by 7
yes
nova plover
i mean cricket intersections
buoyant shadow
then additional 1 is football hockey
so 5 − 2 = 3 go into cricket hockey
so 8 − 1 = 7 are cricket football
etc.
darn
i read odd as even
there's a chance it's the same answer
ok, not the same
11 people go hockey, so 6 of them also play cricket
nova plover
there's a chance it's the same answer
umm
buoyant shadow
it's not, 4 became 5
looks ok now
oh i see
you can read it 2 ways
of → for or of → out of
i bet 16 cents it's intentional 
let's finish it
28/4 = 7
so of is for
safe radishBOT
@nova plover Has your question been resolved?
barren fjord
Why is the area 0
safe radishBOT
Why is the area 0
barren fjord
I thought area was always positive
brave wolf
( @twilit whale )
opaque fern
barren fjord
I don’t really know what that means to be honest
severe pond
I thought area was always positive
it isn’t asking for the area
opaque fern
there is a difference between geometric area and signed area
severe pond
it’s asking for the value of the integral
opaque fern
the integral is by default calculating the latter
twilit whale
I don’t really know what that means to be honest
any function f which holds f(-x) = -f(x) for all x in its domain
to put it simply, all values are mirrored along the x axis
barren fjord
Oh ok
twilit whale
Oh ok
so the area they have ABOVE the x axis will also exist BELOW the x-axis, and your main doubt: why is the area below x-axis negative? well because its signed area and not simple area which is literally defined to be positive above x axis and negative below it
im repeating myself here but basically its definition
what youre probably talking about is ordinary nonnegative area in which case its always greater or equal to 0
barren fjord
Ohhh ok thank u sm i get it now
Because it’s an integral rather than just area we don’t have to consider the fact the area is always positive then ?
twilit whale
Because it’s an integral rather than just area we don’t have to consider the fac...
yesyes
severe pond
if it asked for the area of a certain region you’d integrate the absolute value of the function
finding the value of the integral is a separate thing
barren fjord
Ok thanks guys I have an unrelated proof by induction question asw
twilit whale
Ok thanks guys I have an unrelated proof by induction question asw
yuh
barren fjord
Im doing q7
severe pond
twilit whale
Im doing q7
!status
safe radishBOT
Im doing q7
What step are you on?
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2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
LMAOOO im dying yo
severe pond
you mean 3k^3
barren fjord
Yh
barren fjord
have you tried difference of two cubes
I’ve heard of the difference of two squares but never cubes
open wedge
$a^3 - b^3 = (a - b)(a^2 + ab + b^2)$
flat frigateBOT
1 divided by 0 equals Infinity
twilit whale
I’ve heard of the difference of two squares but never cubes
$a^3 - b^3 = (a-b)(a^2+ ab + b^2)$
but i think thats not even necessary if uve already expanded
just rewrite $3^{k+1}$ as $3\cdot 3^{k}$
flat frigateBOT
esker
twilit whale
and then u expand the right hand side
since we're proving for $n\geq 4$ you can try proving that expression is actually lower than the rhs
flat frigateBOT
esker
open wedge
how about proving $3k^3 > (k + 1)^3$
flat frigateBOT
1 divided by 0 equals Infinity
open wedge
didn't expect that would work
barren fjord
how about proving $3k^3 > (k + 1)^3$
Yh how do we do this
twilit whale
because we already proved true that $3\cdot 3^k > 3k^3$
barren fjord
I don’t think it factorises well
flat frigateBOT
esker
twilit whale
I don’t think it factorises well
what did you get?
twilit whale
uh
open wedge
uhm
twilit whale
what
barren fjord
This was my equation
twilit whale
umm twin? i dont think itll factor completely i meant to just use the fact that u have the lower bound 4
barren fjord
umm twin? i dont think itll factor completely i meant to just use the fact that ...
What did you do?
open wedge
**1 divided by 0 equals Infinity**
we still prove this
you get something > 0
divide the LHS by k - 4
notice that we are considering k > 4
severe pond
i mean guys we can do algebra but it might be complicated or unclear
the simplest way is to just use induction
remember yesterday we had a similar problem with 2^n > n^2
twilit whale
What did you do?
i did:
$3(k+1)^3-(k+2)^3
=3(k^3+3k^2+3k+1)-(k^3+6k^2+12k+8)$
$=(3k^3+9k^2+9k+3)-(k^3+6k^2+12k+8)
=2k^3+3k^2-3k-5$
first and then because $k\ge4$ this is $\geq$ 0
flat frigateBOT
esker
Compile Error! Click the reaction for more information.
(You may edit your message to recompile.)
severe pond
i said we could use induction for that statement we had after rearranging
but the algebra was easy enough that we didn’t bother using induction
open wedge
i did:
$3(k+1)^3-(k+2)^3
=3(k^3+3k^2+3k+1)-(k^3+6k^2+12k+8)$
$=(3k^3+9k^2+9k...
the -3k does not work out really
severe pond
here, with cubes we might want to just use induction
open wedge
yea ik
barren fjord
here, with cubes we might want to just use induction
Like use induction again ?
twilit whale
its geq 0 because $2k^3+3k^2-3k-5 \ge 2\cdot4^3+3\cdot4^2-3\cdot4-5
=128+48-12-5=159>0.$ which implies $3(k+1)^3>(k+2)^3$, hence $3^{k+1}>(k+2)^3$ and tuhs by induction we proved this for all n geq 4
flat frigateBOT
esker
severe pond
Like use induction again ?
yes
open wedge
its geq 0 because $2k^3+3k^2-3k-5 \ge 2\cdot4^3+3\cdot4^2-3\cdot4-5
=128+48-12-5...
i don't think the $-3k$ works like that
flat frigateBOT
1 divided by 0 equals Infinity
severe pond
to prove 3k^3 > (k + 1)^3
open wedge
ill put this in spoilers so if you're not ready then don't open it
severe pond
yea let’s be real you’re probably not finding this
twilit whale
i don't think the $-3k$ works like that
urm no twin because
i just tried proving that for $k\ge4$:
$$2k^3+3k^2-3k-5 \ge 2\cdot4^3+3\cdot4^2-3\cdot4-5
=128+48-12-5=159>0.$$
so the $-3k$ term isnt affecting the ineq or anything and the cubic and quadratic terms r big enough to dominate for $k\ge4$, so i just exploited that and held the induction
severe pond
i’d just recommend induction
flat frigateBOT
esker
barren fjord
yea let’s be real you’re probably not finding this
Yh I was just abt to ask how he even saw that😭
severe pond
Yh I was just abt to ask how he even saw that😭
he is infinityiq
open wedge
he is infinityiq
lmao
open wedge
Yh I was just abt to ask how he even saw that😭
i saw k >= 4 so i just really did that lol
severe pond
yea guys how did y’all not see that
open wedge
yea guys how did y’all not see that
fr yall
safe radishBOT
@barren fjord Has your question been resolved?
open wedge
yea guys how did y’all not see that
also check the last term
fringe dock
safe radishBOT
fringe dock
i am not sure where to put theta on my triangle in blue
fringe dock
safe radishBOT
@fringe dock Has your question been resolved?
fiery remnant
this is how that comes about to be
where 90 - (90-theta) gives you the angle as theta
@fringe dock please close this channel first before moving on to your new other channel
safe radishBOT
@fringe dock Has your question been resolved?
icy coral
safe radishBOT
icy coral
How to do q9 b
dusk halo
How to do q9 b
!status
safe radishBOT
How to do q9 b
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
loud wren
that's a good start
icy coral
icy coral
that's a good start
Here is it
loud wren
ok, what happens when you plug in sqrt(3)
also try to find the angle that is formed by z with the x-axis after you plug in k=sqrt(3)
icy coral
also try to find the angle that is formed by z with the x-axis after you plug in...
I found the arg is 7pi/12
I plug into sqrt 3 in real part of z
Which is
loud wren
so now what can you do to relate that argument to z as a whole
icy coral
I think u use z equals cos theta plus i sin theta?
icy coral
No prob
icy coral
well there's also a modulus in front, but yes
Wait I get holy shit just divide the modulus with cos 7pi/12
whole hill
Wait I get holy shit just divide the modulus with cos 7pi/12
And what did you get for the modulus?
icy coral
And what did you get for the modulus?
Sqrt 2 /2
Re z equals (1-sqrt 3)/4
Divide re z by modulus and and u get cos 7pi/12
loud wren
hence proven
whole hill
Divide re z by modulus and and u get cos 7pi/12
YIPPIE!!
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Available help channel!
stone ore
safe radishBOT
stone ore
Would I start by checking continuity, lim as x approaches 2 from the left and right?
quiet plume
Yes. You check both sides of the limit, make sure they agree, then check the value of f(2) and make sure it also agrees with the limit.
stone ore
Yes. You check both sides of the limit, make sure they agree, then check the val...
When I check the limits of 2 from the right and left, I get k in my answer am I doing something wrong?
quiet plume
Yes that's normal. You'll need to choose k such that both sides agree.
dense furnace
For a function $f(x)$ to be continuous at a point $x = a$, all 3 must be true:
- $$f(a)$$ is defined
- $$\lim_{x \to a} f(x)$$ exists
- $$\lim_{x \to a} f(x) = f(a)$$
stone ore
I think I got it
flat frigateBOT
9k
stone ore
I set my asnwers from the limits equal to eachother to solve for k
wide ruin
When I check the limits of 2 from the right and left, I get k in my answer am I ...
u need to do LHL = RHL for x=2
equate both sides
stone ore
equate both sides
Thats what I did
dense furnace
what did you get?
wide ruin
Thats what I did
show
stone ore
Ill upload my work
dense furnace
i remember when i was in calc 1 and i took an exam and this exact question (of course, with different functions) was on it
but it was this exact question
and i had no idea how to solve it so i made sure after the exam i went home and studied it and made sure i understood how to do it lol
stone ore
wide ruin
its correct
safe radishBOT
safe radishBOT
Available help channel!
icy coral
safe radishBOT
icy coral
Need help with bottom left and right
safe radishBOT
What step are you on?
1. I don't know where to begin.
2. I have begun but got stuck midway.
3. I got an answer but I was told that it's wrong.
4. I got an answer and would like my work checked.
5. I have a question about someone else's work/solution.
6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
I don’t how to start
These are part a to b from left to right
I don’t how to part 13 c i and ii
How would p look like
tardy mango
This is just interpreting matrix multiplication in terms of the corresponding transformation
$(BA)^2=BABA$, so you're applying applying the transformation from $A$, then $B$, then $A$ again, then $B$ again
flat frigateBOT
Civil Service Pigeon
tardy mango
Surely you can just do this by direct computation?
$C=BA \implies C^2=(BA)^2=kI$ is hopefuly the easiest laziness cop out
flat frigateBOT
Civil Service Pigeon
tardy mango
but you could always just work with C and BA directly without squaring as well
@icy coral
safe radishBOT
@icy coral Has your question been resolved?
safe radishBOT
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Available help channel!
mossy gyro
safe radishBOT
mossy gyro
no matter how much i expand or try to escape the interdeterminate form, i keep getting 0/0
plucky elk
are you supposed to use l'hopital
plucky elk
did you use l'hopital twice
mossy gyro
wdym twice