#help-10
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you wanna find derivative of this
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okay sowe need
function horizontal and vertical asymptote
and function local min and max
i hope google translated it right
local min max f'(x) and asymptote
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For horizontal asymptotes: what happens to f(x) when $x \to \pm \infty$
tofu
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I was wondering if there are word problems for power series I have a project to make a situation involving power series but I cant find any exampels
what is power series?
a infinite series
ohh
of infinite series
lemme give you an example
@primal river
see this if it help you
I think I was supposed to think of something like a word problem
you can create a word problem of this
also that's not a power series and it's not totally clear what the pattern is but it doesn't seem to equal 1
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distinct positive integers b, c, d
apologies for the disruption.
ok so you do realise the question boils down to $c^3=d^4$
ed
to reiterate what i said that got purged,
a needs to be a cube and a fourth power at the same time
where c and d are integers and you're trying to minimise the values of c and d
yes what ann said
thats cool and all, but please don't just give the man the fish, teach them how to fish
jigglyproff be quiet.
@nocturne quail use fundamental theorem of algebra
every positive integer can be written as a unique factorisation of prime factors
i think that's the wrong FTA
arithmetic haha
same
mood
I think it would be more helpful on why the approach is the right one, instead of just giving it
dude
he might not even know FTA
also its obvious why its the correct approach
thats how math is
once a solution is presented its hard to forget
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are you sure that this inequality is in the right direction
bc as-is it feels like a contradiction is about to surface
or maybe not a contradiction but the problem might become a little too trivial
can you show your proof?
it's strange that your proof would depend on a_2 itself being nonnegative as opposed to say the difference between a_2 and a_1
$a_n = (n-1)a_2 - a_1$?
Ann
er
≥ not =
that looks odd to me though
you could say $a_n - a_1 \geq (n-1)(a_2 - a_1)$
Ann
are you sure though
sorry, got distracted
Using this I managed to get
a(n)>=(n-1)a(2)-a(1)
i think this is nonsense, but given how you jump from the givens to this with nothing inbetween i can't really comment on where you went wrong
here's how i would've done this: let $d = a_2 - a_1$, then for all $n$ you have $a_{n+1} \geq a_n + d$, and from this it is transparent that $a_n \geq a_1 + (n-1)d$
Ann
from this you can conclude that if d > 0 then the sequence is unbounded, so d has to be ≤0
because $a_{n+1} - a_n \geq d$
Ann
are you sure the inequality at the top is right?
If the sequence is monotonically increasing, then we have $\lvert a_{2} - a_{1} \rvert \leq \lvert a_{n+1} - a_n \rvert$. Hence we have some $\varepsilon$ so that $\varepsilon < \lvert a_{2} - a_{1} \rvert \leq \lvert a_{n+1} - a_n \rvert$` for any n, therefore it is not cauchy, and cannot converge.
let $b_n = a_{n+1} - a_n$, then you're told that $(b_n)$ is an increasing sequence
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can anyone help with this
Calculate every surface area and add them together.
Calculate for each figure. Rectangle, 2 of 4x6, 4 of 15x4. And do it the same way with the triangle.
Take your time and calculate each side of the figures.
i still dont understand D
Do you know what surface area is?
i not very goof with it D:
Like it is, the area of the surface. You got a table that is 4cm long and 6cm wide, the areal of the surface is 24cm^2.
6x4 ok
If we include the height, then it's just area.
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324 dosen't work?
Ah, I saw my mistake. It should be ||4* 6 + 2(15* 4)+2(15* 6)+(3* 4)+5* 6+3*6||
I mean, does the question just ask for surface area of one pyramid?
like 20 questions
of this but it gets harder
If it's just one, then 21.
no..?
no
hi you need help?
Alright, I'm in a doubt. You should probably find other helpers.
yes pls
you see those shapes they gave you?
how can you break them up into
(smaller shapes)
u mean this
pyramid\
can you see a square in the middle?
yes and fyi cool pffp
and four other triangles that connects the square?
ty
yh
okay now what you need to do is, in order to calculate the entire surface area, you calculate the surface areas of the smaller shapes (square, triangle), and add them up
make sense?
not at all
gimme a sec
ok
nice drawing mine is ass
do you understand this or not?
yh i think
alright
so we just find the area of each individual shapes that we just got
tell me how do we find the surfacd area of a square?
is the answer 5`
51
yep
is it 51?
how did you get it?
you break them up into smaller shapes
so we calculate the small shape's area and sum it up
for the square, its 9inches squared
for one of the triangles, its 10.5 inches squared
bruh i just did 7 x 3 / 2 and times it by 4 and i did 3 x 3 and added it
i have things to do, if i can ill come back but no guarantees..
how old are you btw?
we dont have the same timezone yea..
jk jk
i'm 13
<@&286206848099549185>
<@&286206848099549185>
<@&286206848099549185>
@heady tendon Has your question been resolved?
<@&286206848099549185>
@heady tendon
done with that
i need help with this
wait pls try and solve it with me pls pls
is the answer 314
is 489.84 @timid silo
umm pls can I get another <@&286206848099549185>
@heady tendon come in vc?
@heady tendon sorry?
@heady tendon so can't come in vc
i think i can
?
what vc?
Wait a sec I am inviting you
@heady tendon
@heady tendon come in 384kbps
@heady tendon
@heady tendon fast
@heady tendon why?
@timid silo
@timid silo why?
.
do u know surface area of composition figures
Yes
@heady tendon it is the same type of sum I helped you now try on your own
@heady tendon donno
I manage to finish that one but an even harder one
<@&286206848099549185>
pls can anyone help me
stop
no one will help you if you behave like that
its only the green part right
2 x base x height + 5x4^2
so 2 x 5 x 4 + 5 x 16
if i am not mistaken
poor @jaunty stream
Please don't spam the helpers
It does the opposite of the desired effect
,, 3(12\times3)+2(3\times4)+13\times4+2(5\times12)+5\times4
Vortism
@heady tendon
<@&268886789983436800> . And i got a random ghost ping here
.close
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hey
.close
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How?
Try substitution
It isn't
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having some issues with this problem
in the center figure, ABC is a right triangle with BC = 2AB, BCD is a right isoceles triangle right in D and E is on AC s.t AE = AB.
i need help with b) prove BDE is equilateral
i got 1 60° angle down but i just cant see what to do to get the 2nd
or prove 2 sides are congruent
cube people i am hereby reaching outto any of you
<@&286206848099549185>
been 15 minutes
i counted
come onnnnn 😭
if im not mistaken abc should be a 30 60 90 triangle
yeah ik
did that already
i tried messing with the angles a whole lot and all i got was ebd is 60
anything else involves proving edc is isosceles
which idrk how to do cause all i can rly compute is bec to be 135
if anyone finds a solution ping me
ill be skipping and doing the next problems
ACB = 30, BEC = 135, (solve for third angle of triangle BEC) 180 - 135 + 30 = 15, DBE = CBE + DBC, DBE = 15 + 45, DBE = 60
isoceles right triangles are 45-45-90 triangles so DBC = 45
so thats one angle of the equilateral triangle if that helps @loud tangle
its just 1 anglei
angle
i already did that
i need to find a 2nd one
just cause 1 angle's 60 doesnt mean its equilateral
ebd is dbe
literally said here ebd is 60
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the answer has to be 84, but i didn't understand how to get that
I mean I got 36 so if anyone else can pitch in and verify before i give out any misleading info that'd be great
oh have u drawn the altitude from the vertex A? i also did that and got 36 as well, but it is only part of it
@stark hollow Has your question been resolved?
PLS HELPPP
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how did they solve this,?
It's written there
it says subtract 14x, but from where?
From both sides of the equation
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is this savvas?
yes😭
i hate savvas sm
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that's the definition of polar to cartesian
wait wdym
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Hello
well the first one gives you b = 10 - a, try substituting that into the second equation
A×10-a=36
10a - A^2 = 36
so far so good
Quad quad
Ye quadratic formula
I don't get what I need
Where
I keep getting wrong numbers ffs
@stone flint Has your question been resolved?
How?
Send me your working
(Ping me if u do send me the workin)
@stone flint Has your question been resolved?
@copper spade
Slide
Can u tell me precisely what are a b and c as numbers
C 36
A is 1
B is -10
Am I right or wrong about it
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What can I do for this
$x^{2009} + 2009x + 1=0$
beijing
Prove it only has one solution in [-1, 0]
in what field are you learning this?
did they teach you with modulo or with cyclic sets?
Calculus
I see
I think I’m supposed to use derivatives or something related
with this it means that the graph is only considered from -1 to 0 right?
Yea yea
I just have a hard time explaining that without telling the answer right away ;-;
do you already have an approach or are you stuck at the beginning?
The derivative is 2009x^2008 + 2009
mhm yes but do you already know what to do with that information?
I know how to prove its always growing but I can’t prove the solution is exactly in interval-1,0
mhm... if you know that its always growing in some interval, how could you check if it passes some certain y value inside that interval?
Do you know first and second order derivatives? Or how fo calculate maxima and minima and input interval values on the derivative?
i think they already did that to find out its always growing in the interval
since its basically required to have no min or max inside that interval then
hint: ||plug -1 and 0 into f(x)||
Second order derivative is 4034072x^2007
Odd power
Idrk but try rolle's theorem or other ways to find.. wish i could help
I think they already figured out there is no extreme locality in that interval though...
since thats required for steady growth
Is your function continious? Do you have a theorem that says smth about what points a continious function attains? You have shown its increasing so it has at most 1 solution
Figure out what theorem gives you it has 1 solution
Bro what are u doing
@sharp pecan help
lets try to not overtake someone else's channel
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is there more to the task?
if I get it right, you may have level 1 with 29 people, then the other levels have 0 and its still true
I don't understand what ur tryin to say
I don't quite know how to interpret the task.
there are a total of n objects and 8 containers
One of the containers must have at least 29 objects
is that correct?
yes
so, the minimum n is 29 right?
yes
the others have to be more than 29
sorry, but where does that condition come from?
yes
so if level 1 has 29 people the condition is fullfilled.
how do you know the other levels need 29 too?
I get you
at least 29 people in at least one level I think that really means that ONE level needs to be 29+
I think
something like {29 ,0 ,0 ,0 ,0 ,0 ,0 ,0} could be a distribution over the levels
since you have at least 29 people in at least one level
so min(n)=29
so the function for n(x) could be n(x)=29+x, x is a natural number
u reckon we have a max value too
do we?
in practice yes, but there is nothing said that you cannot have infinite people in a single level
and you have 8 levels
something like {12321321312312321 ,12312321321312, ...} would be possible
so there are many possible answers but the lowest it can get to is 29
Jigglyproff
alright, that helps, thx
yw
do u go to uni
yes
no wonder
im here to prognosticate on my own math tasks XD
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Am currently stuck at example A the last part
How did we get that( -xe^-x )+(-e^-x) | 0 to infinity equals 1
take limits as x -> 0 and x -> inf
might be easier if you write as a fraction
How
@tribal iris Has your question been resolved?
(-x-1)/e^x
Kk thanks alot
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|x-1|+|x-2|+|x-3|+...|x-n|>=59
Please don't occupy multiple help channels.
n is the smallest possible natural number
to find n
and x is any real number
Wat
|x-1|+|x-2|+|x-3|+...|x-n|>=59
send screenshot
how do i calculate it?
I dont understand you writing send pic
no its in another language
Send idc
no its not 1
Idk 1
consider |x-1|+|x-n|, what's the minimum value of this
in terms of n, if x can vary
ye
well it has to at least be bigger than the middle term of the sum
so its at the very least positive
no wait
its mod anyway
what do you need to find from this
n?
i think its 16 but i cant prove it without a graph
yes yes
is there any specific things for n?
its natural number
its any real number
yeah but you wanna find the min value of n
umm so
this is always correct if you said n is natural number cause everything on abs is always +, so n is real number?
nvm wait
wait
not really
x can be any real number i have to find the n (smallest) that is possible to prove thin inequality
im sorry for lettin yall comfused its more or equal to 59
the Arithmetic progression of "|x-1|+|x-2|+|x-3|+..." goes until it hits |x-n|
for all x?
like for every value x can have you wanna find n?
or just the smallest n
x is any real number
yes
honestly it try induction
set x to like the minimum value then see if u notice anything that is the same for x is 0 and 1
then by induction works for all x
yes
yea i think so
hmm
can't we say its = 59
cuz we need to find smalled n natural number to get it right
yea
but if its =59 and if n has to be natural number it wont be possible
if x is 0 it would just be 1 + 2 + 3 + ... + n >= 59 => ((n)(n+1))/2 >= 59
=> n(n+1)>=118
if x = 1 => 1 + 2 + 3 + ... + n-1 >= 59
=> n(n-1) >= 59 => n²-n >= 118
.
.
.
meaning for all x the series reduces to 1 + 2 + ... n-x>= 59 => (n-x)(n-x+1)>=118
maybe that general formula does something
x is real right?
meaning it can be irrational
yea
umm how did u get ((n)(n+1))/2>=59
i don't understand about n(n-1)
idk about that
(n-1)(n-1+1) = n(n-1)
@balmy scaffold can u give me the question pic?
um can u copy paste it?
can u translate it with google translate?
or u may copy the question
sure
For any real number x, find the smallest natural number n such that |x-1|+|x-2|+|x-3|+...|x-n|>=59 is satisfied.
@ebon remnant
thanks
ye
ye
start from 1
working on that final equation and assuming its 118 cause its minunum
i got $n = \frac{2x \pm \sqrt{437}}{2}$
Kel.plush
the n can't be decimal
how did u got the 437?
its -b +- thingy
oh
i dont really think there is a minimum value that satisfies all values of x
cause x is dependent on n
so if we have to define n automatically x gets restrained
hold on i fucked up
i realized i did
where
ok so here i assumed for the values where n > x it would be negative and cancel with a following term
but its in mod
so its always +
ye
ok so u can split it in 2 sums
1 for x>n and one for the opposite
and if x=n its 0 anyway
and?
and that may partlemy be why it resulted that goofy ahh result
oh you literally just put a - in front of the first bit
fisrt bit?
yes?
lemme write it out in latex
if x is 0 it would just be $1 + 2 + 3 + ... + n \ge 59 => \frac{n(n+1)}{2} \ge 59
=> (n)(n+1)\ge 118 =>n²+n \ge 118\\vdot\
\vdot\
\vdot\$
if x is 2, it would mean $|2-1|+|2
-2|+|2-3|+...+|2-n| = 1 + 0 + 1 + 2 + 3 + ... + |3-n| = 1 + \frac{|3-n|(|3-n|+1)}{2}\
\vdot\
\vdot\
\vdot\$
for a general scenario, it could be interpreted as 2 sums, one of x-1 terms and one of n-x+1 terms (the +1 being the case where x = 0)
therefore, a general formula would be:
$|x-1|+|x-2|+...+|x-n| = \frac{x(x-1)}{2} + \frac{|x-n|(|x-n|+2)}{2}$
im still editing hold on
oke
Kel.plush
still editing?
if x is 0 it would just be $1 + 2 + 3 + ... + n \ge 59 => \frac{0(0-1)}{2}+\frac{n(n+1)}{2} \ge 59
=> (n)(n+1)\ge 118 =>n²+n \ge 118\\vdot\
\vdot\
\vdot\$
if x is 2, it would mean $|2-1|+|2
-2|+|2-3|+...+|2-n| = 1 + 0 + 1 + 2 + 3 + ... + |3-n| = \frac{2(2-1)}{2} + \frac{|3-n|(|3-n|+1)}{2}\
\vdot\
\vdot\
\vdot\$
for a general scenario, it could be interpreted as 2 sums, one of x-1 terms and one of n-x+1 terms (the +1 being the case where x = 0)
therefore, a general formula would be:
$|x-1|+|x-2|+...+|x-n| = \frac{x(x-1)}{2} + \frac{|x-n|(|x-n|+2)}{2}=\frac{x(x-1)+|x-n|(|x-n|+1)}{2}$
i think thats it
i think
this is very confusing tbh
it is
wait lemme write the first sum in a clearer way in the first 2 examples
Kel.plush
umm
i might have fucked up somewhere
so the general formula right
$\frac{x(x-1)+|x-n|(|x-n|+1)}{2}$
water bottle
yes
its equal to the sum
on
if were taking the minimum its 59
its from other server
i don't understand the whole thing (I'm on grade 10)
why theres sigma
same tbh
yo
is that the same exercise?
but still getting the grasp of it
cause idk where the floor came from
i asked the same question
Anonymous
I'd probably approach the problem by finding the value of x for which the sum is minimal, which is x=(n+1)/2
then I'd split the summation into two parts, one for k less than (n+1)/2 and one for the rest
then I'd make the second summation negative and remove the absolute values
we have summation formulas that would apply to this scenario
and then you should be able to solve for n
wait that's actually almost right
but the minimal value of n for which this is true is 15
but it should be 16
what server did u ask tho
its a mathematics server too
can u like inv me
sure
Anonymous
I said it was floor((n+1)/2) to simplify the algebra, but I think it was supposed to be floor((n-1)/2) or floor((n+1)/2)-1
hm
also this problem is from 10th olympiad
but i think thats it
thank you for your time
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Why am I getting p = .5
@wintry sundial Has your question been resolved?
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wait but what did you do to calculate p
I just expanded
should be $\frac{1}{2}(y+4) = (x-2)^2$
asakura_akio
and got this
asakura_akio
did you forget the 4 part
yeah the 2 is supposed to go to the other side
you should have nothing except (x-2)^2 on the right side
you're in this form because you have x^2 not y^2 so 4p is on the y-k side
Is (y+4)/2=(x-2)² same as 1/2(y+4)=(x-2)²
yes
Im really sorry but can you explain why it is the same
yeah, you just divided by 2 on both sides of the equation
1/2 times something is the same as dividing that something by 2
it's just a different way to write it
its like saying 1/2 times 2 is the same as 2/2
or 1/2 times 5 is the same as 5/2
you just have y+4 here instead of a number
np
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Where does the 1/11 and 11 infront of the e come from
Application of reversing the chain rule
Basically the direct result of a u-sub
If you let u = 11x
then du = 11 dx
=> dx = 1/11 du
But they instead wanted to keep dx so they rather split 1 = 11 • 1/11
Wait what?
I have a formula sheet with integration of e and derivative of e, but neither show or explain why that is done
Same thing with like sin1/7 x, idk why it’s -7cos1/7x
$$\begin{align*}
\int e^{11x}dx &= \frac{1}{11}\int e^u du \
&= \frac{1}{11}e^u \
&= \frac{1}{11}e^{11x}
\end{align*}$$
Umbraleviathan
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They basically did this but in one step
Okay but the very first step, why is there a 1/11 infront of the integral
If you're familiar with the chain rule, deriving e^(11x) gets you 11e^(11x). They manipulated so that you're only integrating 11e^(11x) -> e^(11x) + C, but you need to balance out that 11 with a 1/11
I thought i was familiar with chain rule but I guess not
$$u = 11x$$
$$du = 11 dx$$
$$\implies dx = \frac{1}{11} du$$
Umbraleviathan
Doesn't change anything
Okay
If you do this
$$e^{11x}dx \rightarrow \frac{1}{11}e^udu$$
Umbraleviathan
So e^ what ever just stays the same but because we have a dx we need to have a value for the dx
Wdym a value for dx
Like you’re taking the derivative of 11x to get du/dx = 11, then multiplying 11 by dx to get du = 11dx, then dividing du by 11 to get du/11 = dx
Yeah
So the value of dx is du/11 which is the same as the 1/11 x du
Right so when we have the formula of integral e^u du = e^u + c
We actually need to re arrange what ever were taking the integral of to have a du there as well
well you can't just forget about the 1/11
You're integrating 1/11 • e^u du
That becomes 1/11 e^u + C
You had let u = 11x so just place it back in
Right but what I meant was like we’re integrating e^11x dx
There initially is no 1/11 there
But that dx has a value of du/11, so we take the 1/11 out infront, but idk why we keep the dx, unless it’s because we’re also putting an 11 infront of the e
Well what they're doing is similar. You would agree that $e^{11x} = 1 \cdot e^{11x}$?
Umbraleviathan
Because what they basically said is "let's do a u-sub but fuck showing any of that"
@vestal niche
Rofl yes
Their logic is similar to u-sub. The only reason why I'm showing you the u-sub is to show their logic behind doing that (basically expanding a step)
I agree 11/11 is 1, it’s just dumb to make it complicated for no reason but I guess there is a reason.
And 11 • 1/11 = 1
Okay
So they split the 1 into 11 x 1/11 to do a u sub randomly
They understand that if you were to derive e^(11x), you would get 11e^(11x) right
Right cause it’s e^u x u prime
But the integrand is e^(11x), not 11e^(11x)
Which would be 11e^11x
So they slapped a 1/11 to match the integrand
Right right right okay
But the same intuition is used for u-sub
Same as integral of 6x^2 is 2x^3
Yeah
Okay so even with e
It’s going to be the same integration rules as anything else like x
You need to cancel out the constant if there’s one there
Okay
That makes sense
So the formula I have for e^ du, I need to plug a du value in before
If need be yeah
Lol np
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How would I rationalize the numerator
Rationalise the Numerator? There is not much else you can or should do to that tbh
Ok
Maybe provide context on what you're trying to do
thats all
but that the answer
I got it wrong
You can multiply with (sqrt(r)-sqrt(7))/(sqrt(r)-sqrt(7)) if you want to make the thing be rational ib the numerator then
Oh hm
how about this one
Yeah idk should be right, maybe expand the thing
Multiply by the conjugate
That ain't the conjugate
Where did the 2 come from
Also the conjugate is a fraction
thats what I meant
Multiply by $\frac{\sqrt{x^2 + 3} + x}{\sqrt{x^2 + 3} + x}$
Umbraleviathan
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yo
These are "past" exam papers that im doing, not homework.
.6 == 0.6%?
60%
Are you referring too the chance of a server being busy?
how did you reach (ii)?
(i) and (iii) seem to be correct
but on the (ii) is giving me another answer
are you using binomial distribution?
for part (ii) i just done .6^3 * .4^7
no, my logic was "less that 4 busy" = 0 busy + 1 busy + 2 busy + 3 busy
Aight give me a min, ill try it again
and im assuming order doesn't matter, right?
i dont think so
bro?
.05476
it didn't give me that value, what was your way to thinking, i may be forgetting something
oh yeah, okay, i forgot the binomials
but instead of <= i done <
you are write if you did from x=0 to x=3
yes
its been a while since i have done this 😅
Well i think im right
yeah, i agree!
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yo
Yeah
(i) == 1/5
(ii) == 1/25
(iii) == 1/625
(iv) == 24/625
(v) == 11/25
(vi) == 1/25
give me a sec
Ok i know why i got q2 wrong, but what about iv
idk what i done wrong
vi is correct
well if theres no balls of the same colour
then theres 5 possible colours for the first ball
4 possible for the second
3 2 1
5 * 4 * 3 * 2 * 1
= 120
1/120
you have 5 spots for 5 different colours
so they can all hjave different colours
so not 0
how did tyou get that
wait actually
oh you cant type that lmao
,calc 1 * 4/5 * 3/5 * 2/5 * 1/5
Result:
0.0384
,calc 1/120
Result:
0.0083333333333333
Result:
0.0384
I got it by 5/5 * 4/5 * 3/5 * 2/5 * 1/5
yeah
soi this
I also got another question, how would I be able to approach this
These questions are also not homework, they are past exam papers which im doing as revision
Can anyone give me a hint?
12
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hi my teacher sent us this for a task and i’m confused by the task like what am i suppose to do and how if you could give hints
@timid silo Has your question been resolved?
@timid silo Has your question been resolved?
