Okay, I think I see it.
We can do (-1^2)^m = [(-1)(-1)]^m = [1]^m = 1, since 1 raised to any positive integer is 1.
#help-27
gilded coyote
pseudo basin
yeah sure
stark knoll
Prove that the integral from 0 to pi/2 of cos^n(t)dt is strictly > 0. We need a proof, we cannot just say since cos^n(t) > 0 then the integral > 0.
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Prove that the integral from 0 to pi/2 of cos^n(t)dt is strictly > 0. We need a ...
stone stump
if cos^n(t)>0, then you can find some interval [a,b] such that cos^n(t) > 1/k all t in [a,b] for some natural k
stark knoll
I’m sorry I don’t understand, but on the interval given (0 ; pi/2), cos^n(t) is > 0. I agree with that, but I don’t understand what does it says about the integral
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stark knoll
<@&286206848099549185>
stark knoll
I’m struggling to find a way to prove that the integral of cos^n(t) on the interval [0;pi/2] is positive
No :/
restive river
hmm im not sure i know tha try puting it in help-forum and wait
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pearl plume
I’m sorry I don’t understand, but on the interval given (0 ; pi/2), cos^n(t) is ...
Have you seen a proof that if $f(t) \geq 0$, then $\int_a^b f(t) dt \geq 0$? Here we only have nonstrict inequalities, but we can still use this fact to prove your fact if we're careful
woven radishBOT
ryc
stark knoll
We haven’t proven this, which is making the question kinda hard
pearl plume
(if you haven't seen that proved, it's because any riemann sum for f is positive - after all, every term f(x*) delta x is going to be positive. and the limit of a sequence of nonnegative numbers is nonnegative, so the limit o the riemann sums is nonnegative)
pearl plume
We haven’t proven this, which is making the question kinda hard
Okay. You can also directly compute the integral
It's a little tricky
stark knoll
I think I did it earlier, is it something along the line of this ?
Un = (cos^n+1(x)sin(x))/n - n-1/n * integral of cos^n-2(x) ?
Sorry don’t know the Tex commands
But i didn’t manage to use it since n isn’t defined
pearl plume
it's some kind of integration by parts thing like that
Well, you'd get a recurrence and then you'd need to prove the terms of the recurrence are always positive. but that might not be super easy
Let me think for a moment
If n is odd, I would write $\cos^n(x) = \cos^{n-1}(x) \cos(x)$. When you integrate by parts, you get
$$ \cos^{n-1}(x) \sin(x) \Big|_0^{\pi/2} - \int_0^{\pi/2} (n-1) \cos^{n-2}(x) (-\sin(x)) \sin(x) dx $$
woven radishBOT
ryc
stark knoll
Oh, that looks like something I did for the following question
pearl plume
First of all, the first term is 0 unless n = 1. When n > 1, we have cos^(n-1)(pi/2) = 0, and we have sin(0) = 0, so both the left and right endpoints give 0.
Then, we get
\begin{align*}\int_0^{\pi/2} \cos^n(x) dx &= (n - 1) \int_0^{\pi/2} \cos^{n-2}(x) \sin^2(x) dx \
&= (n - 1) \int_0^{\pi/2} \cos^{n-2}(x) (1 - \cos^2(x)dx \
&= (n - 1) \int_0^{\pi/2} \cos^{n-2}(x) dx - (n - 1) \int_0^{\pi/2} \cos^n(x) dx
\end{align*}
Ack
one sec
woven radishBOT
ryc
pearl plume
There we go
Im missing one more parenthesis somewhere. Whatever
So now, how do we find the integral of cos^n in terms of the integral of cos^(n-2)?
stark knoll
And you get $$\int_0^{pi/2} cos^n = {n-1}/n * \int_0^{pi/2} cos^{n-2}$$
woven radishBOT
zarblug
stark knoll
And that’s the part where I got stuck, I think I just forgot to to until n = 2
Yeah cuz cos^0 = 1
pearl plume
Yeah
so now integral of cos^2 is (2-1)/2 times integral cos, which is a positive number times a positive number
stark knoll
And we do that all the way, so we have only positive terms
pearl plume
Yep
This lets you deal with all the even n's
How about the odd n's? Where do you start there
stark knoll
I don’t really see why it matters tho
We can just integrate by part if n is odd no ?
pearl plume
Yeah but each time you go down by -2
Instead of by -1
so for even n's, we get down to cos^0
stark knoll
Oh we know the integral of cos^1
pearl plume
yeah. interestingly this tells you that the formula for the integral will involve pi when n is even and wont involve pi when n is odd. when n is even, it's (n-1)/n * (n-3)/(n-2) * ... * (1/2) * pi/2
and when n is odd, it's (n-1)/n * (n-3)/(n-2) * ... * (2/3) * 1
pearl plume
I think that’s it right ?
Yes, that's right. That should prove it
stark knoll
yeah. interestingly this tells you that the formula for the integral will invol...
Yeah I prove that at the end of the exercice
It was not that hard overall, I just didn’t think of using the answer we found afterwards ^^
Anyways thx for the help, I was stuck for quite a while there
pearl plume
If you want a formal proof, you would use induction to say "base case: for n = 0 and n = 1, the integral is positive because... (calculations)
inductive step: for n > 1, suppose integral cos^(n-2) is positive. the integral of cos^n equals (n-1)/n * integral of cos^(n-2). since n - 1 > 0, (n - 1)/n > 0, and by the inductive assumption, this means integral cos^n is a product of 2 positive numbers. so integral cos^n is positive.
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.close
lapis pelican
can somone help me to solve this question pls?
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topaz axle
write an equation
like one equation with 2 variables
though i don't know how to solve it, so maybe it's not helpful
lapis pelican
<@&286206848099549185> can someone pls help me
dull canyon
It’s cos^n (t), not cos(nt)
Oh my bad
@pearl plume
I don't know if it's in your syllabus or not but check out walli's formula and β functions for integral of (sin(x))^m(cos (x))^n in the internal 0 to π/2 (it doesn't work for other intervals)
lapis pelican
<@&286206848099549185>
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lapis pelican
<@&286206848099549185> pls help me
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lapis pelican
<@&286206848099549185> can someone help me pls?
i think rc = 12r+10c-1
right?
but idk what to do after this?
<@&286206848099549185>
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covert fox
how do i perform a reversal of the order of integration
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delicate fox
it would be really funny if you could just swap the dxdy to dydx and swqp the limits of integration
wooden veldt
Draw the region of integration first
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tranquil bloom
I'm quite new to complex numbers and im wondering what the signum function is/how to use or interpret. For example
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swift knoll
Sanity check. I'm reading Complex Analysis and Applications by Pathak. The following definition is made:
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sunside
swift knoll
Is this the correct definition of a function? Say, we replace "one or more" with "two". Do we then have a function?
.close
lapis pelican
can someone help me with this question pls?
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.reopen
devout snowBOT
✅
topaz axle
so you found rc = 10c + 12r + 1
rc - 12r = 10c + 1
c−12 = (10c+1)/r
r = (10c+1)/(c-12)
from this you can find a c by trial and error
but there's probably some smart way
but i don't know
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lapis pelican
from this you can find a c by trial and error
what do u mean by trial and error
do i have to try every number
topaz axle
yeah, i'm not sure why i suggested it
it would work though
you start with c=13
any less would make negative r, doens;t make sense
eventually it would divide
it's stupid i know
oh i googled it, it's a competish
due nov 3rd
lapis pelican
ye
thats why i need help
but not the actual solution
cuz im allowed to ask for help but not for solutions
pls someone help me
<@&286206848099549185>
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lapis pelican
<@&286206848099549185> pls help me
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<@&286206848099549185>
astral sigil
uh so 12b + 10g + 1 = rows x columns
astral sigil
yeah im doing it one sec
12r +10c+1=rc
12r + 10c < 1000
ok i think i got it
so this is basically
rc-12r-10c = 1
then
rc-12r-10c+120 = 121
(r-10)(c-12) = 121
so if we take (r-10) = (c-12) = 11
we get r as 21 and c as 23
21x23 =483
@lapis pelican uh idk if it's correct
best i could do
lapis pelican
how did u factorise?
astral sigil
how did u factorise?
add 120 to both sides
astral sigil
no there's a method
lapis pelican
what method
astral sigil
dm
lapis pelican
dm?
lapis pelican
k
astral sigil
basically adding 120 helped me find a pair to satisfy the eq
lapis pelican
lemme check
lapis pelican
can u summarise the steps for a diophantine equations pls
astral sigil
basically what happens is
rc-12r-10c = 1
now to represent the LHS as something like
(r-a)(c-b) = d
we can just try rearranging so that we get that
so
(r-10)(c-12) ==> rc -10c-12r-120
so we've got an extra 120 which we can add to the RHS
giving
(r-10)(c-12) = 121
astral sigil
can u summarise the steps for a diophantine equations pls
makes sense?
astral sigil
but whats next
just need to find a pair that equals to 121. just take 11x11 or 121 x 1
121 x 1 wont work as later it exceeds the 1000 limit
11x11 works just fine
lapis pelican
and that pair should be a natural number as we r dealing with peoples
right
?
@astral sigil ?
astral sigil
and that pair should be a natural number as we r dealing with peoples
yeah
astral sigil
how?
just plug in the values you'll see
for 121 x 1, it is either r=131 and c = 13 or r = 11 and c = 133
the values will exceed 1000 when plugged into the equation for number of boys and girls
so we use 11x11
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glossy thorn
gxdfchjbklm;'
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thorn sage
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thorn sage
Anyone who can hint what I am supposted to do here?
thin inlet
What must be true about a pmf?
about the values it takes and its probabilities overall?
@thorn sage
thin inlet
Well, I mean more in a general sense, like what should be true for a dice roll and the probability of all its outcomes
also
thorn sage
And that probability is > 0 and = 1
thin inlet
mhm
echo steeple
thats some high lvl maths
thorn sage
And P(X=2) = 2P(X = 2 - 1)=2P(X=1)
thin inlet
and X can be only 0,1,2 in your sample space yes?
thin inlet
And P(X=2) = 2P(X = 2 - 1)=2P(X=1)
also note you can rewrite this in terms of P(X=0)
using the previous equation you wrote
thorn sage
also note you can rewrite this in terms of P(X=0)
How do you know this?
Or how would I know this?
thin inlet
which is 2P(X=1)
thorn sage
Yes
thin inlet
and we have that P(X=1)=2P(X=0)...right?
so it would make sense to write
P(X=2)=2(2P(X=0))
thorn sage
P(X=1)=2P(X=1-1)=2P(X=0), yes
thin inlet
so we have that P(X=2)=4P(X=0)
thin inlet
sometimes it is easier to understand this more if you say it in words, because I know then notation can get a bit strange. What P(X=0) is saying, is 'the probability that our random variable X, is equal to 0'
and then do you see how you could solve for P(X=0) using what you know about pmfs?
then find P(X=1), P(X=2) from therE?
thorn sage
For X=2 we have 2P(X=2-1)=2P(X=1) and we know P(X=1) is equal to 2*P(X=0) so really we have 2(2)P(X=0) =4P(X=0)
P(X=1)=2P(X=1-1)=2P(X=0)
P(X=2)=2P(X=2-1)=2P(X=1)
So,
2P(X=1)=2(2)P(X=0)=4P(X=0)
Wait lmao
I forgot to insert the right ones
I assume it was
2P(X=0)+4P(X=0) = 1
instead of
P(X=0)+ 2P(X=0)+4P(X=0) =1
since it says
P(X=i) ... for i = 1,2
P(X=0)+ 2P(X=0)+4P(X=0) =1
7P(X=0) = 1
P(X=0)=1/7
aaah! and now we have P(X=0) and we can then insert into 2P(X=0) and 4P(X=0)
thin inlet
mhm
:)
good job!
.solved
.close
agh, you need to do it, unless you got any more questions?
thin inlet
when it specifically says i=1, 2
that's just for the equation
because if you plug in i=0 you get P(X=-1) and P(X=-1) doesn't...exist
does that make sense?
thorn sage
Yes, that makes sense, but I'd definitely end up writing without P(X=0) i feel like
And I'd get 1/6 instead 😢
I don't think when i=1, 2
I'd think to myself to write up P(X=0)
thin inlet
yeah, but in this case you had to solve for P(X=0) because everything could be expressed in terms of it :)
thin inlet
uhhhh
no
P(X=0)=1/7, you got that right
and then you can solve for P(X=1) and P(X=2) respectively
thorn sage
Yes, of course but
I still don't quite get why it's allowed for me to just wirte P(X=0) in the equation when it states i is only for 1, and 2
thin inlet
ok, it's bascically saying what i you CAN plug in for the equation to hold
the equation won't hold for i>2 because you'll have a P(X=i=3) and P(X=3) doesn't exist in this case, and same for i=0, you'll have 2P(X=i-1)=2P(X=-1) and P(X=-1) doesn't exist either in this case
@thorn sage
thorn sage
Aaahhh!
So for the values i=1, 2 the domain holds, but P(X=0) still exists
I think it makes sense now somehow
That obviously i=0 wouldnt work but P(X=0)
still exists
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thorn sage
hope that makes sense
Yes, indeed thank you for helping me! ❤️
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Yes, indeed thank you for helping me! ❤️
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civic oxide
i need help
devout snowBOT
i need help
civic oxide
frozen aurora
what's the problem
civic oxide
i understand how to work with triangles got no idea how to work with these equations
frozen aurora
this specific equation you need to remember
like half of your entire trig syllabus depends on it
not kidding
civic oxide
ok
frozen aurora
but i personally just memorised that lol
frozen aurora
yeah a triangle on a unit circle is a derivation technique for this
tame matrix
atb
civic oxide
thx
empty ravine
im taking a trigonometry course right now
so, it's good to understand how it works
why it's = 1
rapid grove
can someone help me with this plz
devout snowBOT
can someone help me with this plz
ancient ravine
!status
devout snowBOT
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
supple knot
i found a 4 pi over 5 its 8 but how do i find if its in degree 110 ?
180 deg = π rad
rapid grove
okay can u help more plz sry but its a bit hard so how do i solve it ?
rapid grove
180 deg = π rad
?
dry oxide
that is just a hint u need to solve your question
relation b/w radians and degree
and then u can use the relation b/w arc lenght, radius and angle to find the answer
rapid grove
but if its 110 not 180 what should i do ?
its been long since i did math and i have entrance exam thats why i am training a bit
dry oxide
potent salmon
this was asked in class and my teacher wasn't too sure but basically why is √4 = +2 when it's equally valid to say √4 = -2
quick cave
Hello i need help with matrices
devout snowBOT
this was asked in class and my teacher wasn't too sure but basically why is √4 =...
restive river
this was asked in class and my teacher wasn't too sure but basically why is √4 =...
In mathematics, it's conventionally accepted that when taking the square root of numbers we only consider the principal square root
meaning, the positive root only
why you might ask?
well, going back to the definition of a function
an input cannot output more than one output in functions
meaning,
[
\m f x = \pm x
]
isn't a function
woven radishBOT
restive river
as it is outputting two outputs for only one correspondening input
the same applies for square roots
frozen aurora
this was asked in class and my teacher wasn't too sure but basically why is √4 =...
2^2 is 4, just like (-2)^2 = 4.
BUT sqrt(4) isn't -2, it's only 2.
instead, we say that if
a^2 = b
then
a = ±sqrt(b)
restive river
if we have has made it [
\m f x = \pm \s x
]
then the function $\m f x = \s x$ suddenly loses its function status
woven radishBOT
frozen aurora
if we have has made it \[
\m f x = \pm \s x
\]
then the function $\m f x = \s x$...
sorry to interrupt but are those LaTeX shortcuts available to everyone or just you?
restive river
they are personally implemented for me yes.
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dense sail
guys, im having a real hard time understanding why the multiplicative modulo cycles containing only coprimes of the modulo have to be the same size... could someone please explain it to me in layman's terms? 🥺
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guys, im having a real hard time understanding why the multiplicative modulo cyc...
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quartz linden
Help please with part b?
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dark arrow
devout snowBOT
dark arrow
what now?
woven radishBOT
gatz
\begin{gather*}
(-6 + √4*3)/2
\end{gather*}
woven radishBOT
gatz
dark arrow
what now?
dark arrow
parentheses?
willow helm
$\frac{x+1}{2} - x \cdot (x + 1) = \frac{x+2}{3}$
woven radishBOT
USS-Enterprise
willow helm
This the original?
dark arrow
yeah
willow helm
Right, something went wrong simplifying
dark arrow
Right, something went wrong simplifying
https://cdn.discordapp.com/attachments/903486480075354182/1170344799304425492/rn_image_picker_lib_temp_259697e1-34d6-47fa-a04e-97bba7d6908e.jpg?ex=6558b388&is=65463e88&hm=0c010ece7f8723c222f8ad452ea8796a6ef29726e78e83455cc7a790f122090d&
here's the whole solve btw
dark arrow
First you distributed -x
wdym
willow helm
$\frac{x+1}{2} -x^2 -x = \frac{x+2}{3}$
You multiplied the parantheses by -x
Distributed -x through the parantheses
dark arrow
why -1?
willow helm
Oops
woven radishBOT
USS-Enterprise
willow helm
Then multiplied by 6?
dark arrow
yeah
worthy talon
I don't think anything went wrong
willow helm
We get $3 \cdot (x+1) - 6x^2 - 6x = 2 \cdot (x+2)$
woven radishBOT
USS-Enterprise
willow helm
$3x + 3 - 6x^2 - 6x = 2x + 4$
woven radishBOT
USS-Enterprise
dark arrow
yep, i also think i did the beginning correctly
willow helm
Then you collected like terms and moved everything to one side
willow helm
√12=√3*√4=2√3
Those are not the correct solutions.
dark arrow
Then you collected like terms and moved everything to one side
yeah
woven radishBOT
gatz
willow helm
Okay that's correct
willow helm
a = 6, b = 5, c = 1
worthy talon
rip
willow helm
$x_{1, 2} = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$
woven radishBOT
USS-Enterprise
willow helm
$x_{1, 2} = \frac{-(5) \pm \sqrt{5^2 - 4 \cdot 6 \cdot 1}}{12}$
woven radishBOT
USS-Enterprise
willow helm
$x_{1, 2} = \frac{-5 \pm \sqrt{1}}{12}$
woven radishBOT
USS-Enterprise
dark arrow
-6/12 and -4/12
willow helm
Yes, simplified
willow helm
$x_1 = -\frac{1}{2} \\ x_2 = -\frac{1}{3}$
woven radishBOT
USS-Enterprise
willow helm
Of course
willow helm
Yeah, so just a mistake in the quadratic formula
dark arrow
similar to these?
devout snowBOT
willow helm
No problem
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ocean haven
can anybody explain what they've done here for (x-2)^2(x-3) lesser than zero?
ocean haven
i did it w wavy curve method but its not the same answer
pseudo basin
show your work for wavy curve @ocean haven
molten spindle
what do i read this?
devout snowBOT
what do i read this?
pseudo basin
do you mean "how do i read this out loud" or "wtf does this mean"
molten spindle
do you mean "how do i read this out loud" or "wtf does this mean"
i mean how to write it in english
instead of math symbols?
molten spindle
do you mean "how do i read this out loud" or "wtf does this mean"
yea, i meant "wtf does this mean"
pseudo basin
$\bQ_{\sqrt{2}}$ is the set of all real numbers expressible in the form $a + b \sqrt{2}$ for $a$, $b$ rational.
woven radishBOT
Ann
pseudo basin
if you're ok with somewhat more fanciful language, it's the set of all rational linear combinations of 1 and sqrt(2)
molten spindle
.close
static pollen
how is this solved algebraic?
devout snowBOT
how is this solved algebraic?
static pollen
i cant get the right answer in my way
urban harbor
so you have unknowns V1, V2, R3, and R4? and what is that Vb at the top?
urban harbor
so youre trying to solve for what in terms of what?
static pollen
yeah
urban harbor
im asking you
static pollen
vb
urban harbor
oh ok
urban harbor
,rotate
woven radishBOT
turbid steeple
this feels wrong, can someone tell me what i did wrong here?
devout snowBOT
this feels wrong, can someone tell me what i did wrong here?
@turbid steeple Has your question been resolved?
sacred rock
Currently struggling with these questions
devout snowBOT
Currently struggling with these questions
sacred rock
For a) I have two answers
9C2 = 36
or
$\frac{(10+3-1)!}{3!(10-1)!} = 220$
woven radishBOT
CoshamGames
topaz axle
the second one is d
sacred rock
the second one is d
Ah okay, could you explain where the 9C2 came from? I sort of just got it from what I did in my lectures but I'm not aware how
topaz axle
X X X X X X X X X X
10 letters have 9 empty spots
X X X X . X X X X . X X
putting a divider in any two divides it into 3 groups
sacred rock
X X X X . X X X X . X X
putting a divider in any two divides it into 3 groups
how does this result in 9C2 ?
sacred rock
X X X X X X X X X X
10 letters have 9 empty spots
Why does 10 --> 9 here ?
topaz axle
you just see that there's 9 spaces
sacred rock
but there's 10 ?
drifting sierra
10 Xs, 9 spaces in between
sacred rock
OHHHHH
Okay that makes perfect sense now that first part Except, why do we look for the spaces? Is it because we want at least 1 of each ?
topaz axle
yes, if you put a divider at the start, where there's no space the first group has no letters
sacred rock
So if we wanted at least 2 letters would it become 8 spaces ?
topaz axle
there's no specific way to do 2+ letters
you kinda think of it as subtracting 3 letters, so 7 letters and 6 spaces
or you subtract 6 letters and do the solution for d)
sacred rock
Okay so
For Qb)
I know the resulting answer will be kC2 (Using your analogy, any 2 dividers will leave 2 of each letter minimum in each group) right?
But how do I find k
topaz axle
it would be 6
you assign 1+1+1 letters to make each group hae at least 1 letter
7 letters are left, which have 6 spaces
and that again makes each group have at least 1 letter
it would make more sense if you do it with the second method
sacred rock
it would make more sense if you do it with the second method
Which method ?
sacred rock
How would i put this into the formula though to find 6C2 from just knowing I want 2 of each letter etc ?
topaz axle
okay that's not how it goes, the denominator is (3−1)!10!
drifting sierra
There is one method you can use for all five questions
topaz axle
yeah, based on d)
sacred rock
Could you explain how I'd do that? I'm pretty stuck on combinatorics
topaz axle
look at d)
we don't care if a group has 0 letters, we freely permute the dividers mixed with the letters
XXXXXXXXXX||
we get 12c2
sacred rock
look at d)
we don't care if a group has 0 letters, we freely permute the divider...
This means my formula was incorrect as mine outputs 220 whereas 12C2 = 66
sacred rock
(3−1)!10! not 3!(10−1)!
Ohh okay yeah that makes sense
topaz axle
every other question works from this, you simply reserve however many letters you need to have
at least two of each: 10 becomes 4
1 A 2B 3C, 10 becomes 4
drifting sierra
In other words you just remove the elements that must be included and you work on the remaining set just like in (d)
This may be a good read for you https://en.wikipedia.org/wiki/Stars_and_bars_(combinatorics)
sacred rock
Thank you you've both been such a huge help, I'll definitely be giving that a read too 🙂
drifting sierra
sacred rock
I have another question if you're willing
b) and e) I have to understand
But I won't lie, I don't know where I even start with this
drifting sierra
That looks painful
sacred rock
That looks painful
Feels it 😭
drifting sierra
Did you do (a) (c) and (d)?
sacred rock
No I've not done them yet, we've been told to just comprehend b and e for now so idk if they're somehow different
drifting sierra
Ok so I have to go for like 20 min but in the meantime try to express (1+x)^n as a sum
fiery sluice
I can't wrap my head around this how does 0.03 turn into 0.0300?
pseudo basin
i mean they're the same thing aren't they
they just decided to add some trailing zeros for whatever reason
fiery sluice
Kinda I've been putting 0.03 into the calculator but it gives me a different result then 0.0300
Any clue why i keep getting this error here?
fiery sluice
What calculator are you using?
fiery sluice
I guess
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woven spade
i need solution
devout snowBOT
i need solution
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i need solution
What step are you on?
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2. I have begun but got stuck midway.
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6. I have completed the problem and don't need help anymore. Thank you.
7. None of the above
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.
@woven spade Has your question been resolved?
woven spade
no
devout snowBOT
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What step are you on?
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7. None of the above
lusty sapphire
What are the properties of a natural cubic spline. Start the maybe
woven spade
They are C2 continuous, which means that the second derivatives match at the transitions between segments.
They are interpolating, which means that the curve passes through the given control points.
The second derivatives of the spline polynomials are set equal to zero at the endpoints of the interval of interpolation. This forces the spline to be a straight line outside of the interval, while not disrupting its smoothness.
lusty sapphire
They are C2 continuous, which means that the second derivatives match at the tra...
Excellent. You can use all of these facts to find your points.
What I would do is use all of these properties, plug in given points, and see what abcd should all be
I should end up with a system of equations and that would give me the answer
Okay like, the spline should go through (1,1). That is, we require S(1)=1. If you evaluate S(1), you will see that this is true no matter what
So then I'd next look at (2,1)
I require S(2)=1.
If I evaluate S(2), I need to evaluate S1(2) which comes out to 1 again.
But here's where we finally get somewhere interesting.
The spline needs to be continues, and since S0 stops just before 2, we require S0(2)=1 for continuity
If you evaluate S0(2), you get 1+a+b
So there's your first equation
1=1+a+b, or a+b=0
You can continue this process by evaluating the derivative conditions you mentioned earlier
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restive river
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restive river
why can i not take a sphere around this here?
opaque tangle
opaque tangle
-3/2*x^5/2
How u so fast
frozen aurora
stop spamming the question everywhere
opaque tangle
stop spamming the question everywhere
Myb fr
lament schooner
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lament schooner
Why doesn’t the t+1 here have -2 as exponent
Since (t+1)^2/3
\
2/3(t+1)^-1/3
Is 3/2(t+1)^-2
Nvm I’m stupid
.close
restive river
Is my answer correct?
They are asking me to calculate A_6
devout snowBOT
Is my answer correct?
They are asking me to calculate A_6
cold bough
it seems ok ....i have not checked the sum or multiplications
restive river
can someone help me understand why this is
devout snowBOT
can someone help me understand why this is
nova ibex
ln is log base e
lilac mango
log laws
nova ibex
So imagine you had 10^5 , log_10(10^5) = 5 * log_10(10) = 5 ;
Log_e(e^(5/8)) = 5/8 * log_e(e) = 5/8
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random badge
The "room for 250" people does no make sense, as there cannot be half a compartment
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The "room for 250" people does no make sense, as there cannot be half a compartm...
random badge
I chose E as the answer, right or wrong?
dense jay
,calc 142 mod 17
woven radishBOT
Result:
6
dark tundra
modulo
random badge
modulo?
dark tundra
basically remainder when you divide
dark tundra
In this case yes
brave falcon
not sure what to do at this final step
devout snowBOT
not sure what to do at this final step
brave falcon
Part J*
Isn't the slope of the tangent line at Point Q equal to t=0.852
(dy/dt)/(dx/dt)?
charred zodiac
Part J*
Isn't the slope of the tangent line at Point Q equal to t=0.852
(dy/dt...
Yes 👍🏽
charred zodiac
try a numerical value (?
🤔
it might not recognize cot, idk
oh wait
well if the velocities are the same, what's the direction of the object's movement at point Q?
what's the slope of the line in that direction?
brave falcon
n/n = 1???
charred zodiac
yess
restive river
How do I solve this when theres an absolute value
devout snowBOT
How do I solve this when theres an absolute value
restive river
dunno how id do it w that
restive river
yes
solving for x
or at least thats what im assuming since this problem does not specify
undone fractal
I would say x is 4 but I actually dont know what the "|" is for
strange arch
How do I solve this when theres an absolute value
split it into cases
from |A| = B follows
A = B or -A = B
woven radishBOT
Matt
restive river
you'd solve two equations and these give you the range of solutions
ok so I got $\frac{14}{3}$ and $-4$
woven radishBOT
Matt
strange arch
yop
woven radishBOT
Matt
strange arch
I'd guess they have some formatting rules, either comma or as set, am not familiar with the site
either what you sent or {14/3, -4}
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brave falcon
Not sure how to calculate the magnitude here.
devout snowBOT
Not sure how to calculate the magnitude here.
dense jay
you can get the vertical and horizontal velocity components at the time it hits the ground
get an angle from those
brave falcon
those horizontal velocity and vertical velocity components form lines right?
dense jay
sure? In the sense that you can find the angle between them, yeah
brave falcon
do I use arccos to do that?
dense jay
wouldnt really be helpful, you have the adjacent and opposite
brave falcon
oh i thought the vertical velocity would have given you the hypotenuse in this
ok so i just use arctan
dense jay
oh i thought the vertical velocity would have given you the hypotenuse in this
x and y are perpendular, that wouldnt make much sense
dense jay
ok so i just use arctan
yeah
brave falcon
arctan ((-32t+4)/150)?
not sure what is the opposite or adjacent here
dense jay
vertical is opposite, horizontal is adjacent, you have it fine
it would be |-32t+4| though
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median kiln
pop. mean=266 pop. std. dev.=16/sqrt15 which gave me 4.13 samp size=15 x values= 266-9=257 & 266+9=275 input all of these in a samp. distribution calc and its ? not giving me a probability
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pop. mean=266 pop. std. dev.=16/sqrt15 which gave me 4.13 samp size=15 x values=...
median kiln
wait nvm i got it actually😭 kept looking at the question above
thank youuuu
.close
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restive river
How to solve these questions using unitary method I believe?
devout snowBOT
How to solve these questions using unitary method I believe?
restive river
languid ore
!status
devout snowBOT
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
timber pebble
hmm unitary method is new to me
do you feel familiar with the method and are confused about applying it
or dont understand the unitary method
restive river
I feel confused in how to apply like the textbook has told me to
restive river
Abstract?
restive river
Ah ok yes I understand that
restive river
Hmm your right but if I were to do conversion their should some type of you know formula or something
languid ore
Hmm your right but if I were to do conversion their should some type of you know...
kind of?
i mean
ok lets work through the first one
how many seconds are in an hour
and how many meters are in a km
restive river
3600 secs in 1 hr and 1000 meters in 1 km
restive river
1000meters/h and 1000meters/s
languid ore
not quite
when converting km/h to km/s
we have 1 km/1 h = 1 km/3600 s = (1/3600) km/s
does that make sense?
restive river
Ok 1km=3600secs? Ok
restive river
we have 1 km/1 h = 1 km/3600 s = (1/3600) km/s
Oh ok read it the wrong my bad
restive river
1x1000/2x3600 secs? In a fraction?
humble ferry
ok my explanation was bad
let's start again
so how does kilometers change to meters
1 kilometer changes to 1000 meters
the question wants the numerator part to change from kilometers to meters
restive river
Yes
humble ferry
so 216km x 1000 = 216000 meters
now let's see denominator part
it's per 1 hour, that 216 km is travelled
and the question wants you to change the hour to seconds
restive river
which is 1hr=3600 secs
humble ferry
good
humble ferry
so 216km x 1000 = 216000 meters
now u have numerator
humble ferry
now when you divide it u get 60 m/s
restive river
216000/3600=60m/s
humble ferry
yes
restive river
Oh so you this method for every question?
humble ferry
yep
idk if there's a specific requirement in your school
but this method would work every time because without logic it's easy to get wrong
want to try question 2?
restive river
Ok sure
humble ferry
so how do you change 1 Litre to mililitre(numerator), 1 hour to minutes(denominator)
restive river
1L=1000mL 1hr=60mins
humble ferry
nice
now there's 30 liters
so multiply 30 to numerator and voila
you'd get the answer
restive river
30/60=0.5?
humble ferry
nooo
restive river
or is it 3000/60
humble ferry
30 x 1000 /60
restive river
30x1000/60=500
restive river
Hmm
humble ferry
step 1 is to change the unit, step 2 is multiply how many times is that unit like 30 litres so u x30 with the new unit. step 3 is to just calculate
I gtg now now
GL
urban harbor
Any help?
this is just simple proportions
you can think of it simply as multiplying by $\frac{40cm}{180 blocks}$ or $\frac{180 blocks}{40cm}$
woven radishBOT
Soosh
urban harbor
you can simply pay attention to the units and see how they cancel:
for example in (i). you start out with 495 blocks and want to end up with cm
so you want the blocks in the fraction you multiply with to cancel out with blocks in top and then you end up with cm so you keep the cm unit of the conversion fraction on top, if that makes sense:
$495 \text{blocks} \times \frac{40\text{cm}}{180\text{blocks}}$
radiant nebula
Hi
woven radishBOT
Soosh
restive river
Oh is this using the unitary method?
urban harbor
$\frac{495 \text{blocks}}{1} \times \frac{40\text{cm}}{180\text{blocks}}$
woven radishBOT
Soosh
urban harbor
lemme even write the left as a fraction so its even more clear
see the units cancel out are youre left with 495 * 40 / 180 cm as the final answer
so just paying attention to the units is an easy way to know which way you need to multiply so you cant go wrong
restive river
Hmm I got it now
urban harbor
(and that will help you all the way to college chemistry and physics ) : )
restive river
Btw is this the unitary method or just..
restive river
(and that will help you all the way to college chemistry and physics ) : )
Am still in yr 9 lol but I hopefully remember for future refrences
urban harbor
i have no idea what that refers to, i dont know names of "methods" this is a simple multiplication
restive river
Oh ok cuz it seemed similar to the method that was shown in the textbook for a different question by same topic
urban harbor
got it from here?
restive river
If you could just help me with (ii) I can do all of the questions by myself
urban harbor
well if you can't do 2 then you didnt understand my whole explanation for i) since its the exact same thing, you should be able to do this yourself now
step 1): write down 70 cm on your paper
step 2): what units are you starting with and what units do you want your answer to have?
step 3): what fraction do you need to multiply by to make that happen?
restive river
70 40 and 180
urban harbor
?
restive river
Unit I need in cm
urban harbor
part ii is asking for an answer in cm?
restive river
Yes
urban harbor
...
restive river
No wait
urban harbor
"find the number of BLOCKS"
urban harbor
not sure how i can explain it any more clearly than i did already 🤷♂️
id just be repeating what i already said for previous problem, so go back and read that
brave falcon
Yeah hasn't stated and need to find but using which method the previous one
what do u mean
brave falcon
to get the answers for both i and ii you can just find how many blocks it takes to get 1cm worth of height
restive river
to get the answers for both i and ii you can just find how many blocks it takes ...
So your telling me that 4.5 cm = 1 block?
brave falcon
that isn't the correct proportion because it takes multiple blocks to get 1 cm of height, not multiple cm for one block
restive river
Hmm
brave falcon
it would be 4.5 block for 1 cm
restive river
so 4.5x70=315 Blocks?
brave falcon
i think so
i would recommend for future reference
always writing down units and making sure that your answers are sanity checked
restive river
Sanity meaning?
brave falcon
a sanity check is seeing if your answer would make sense in the real world or in the context of the problem
a not crazy answer
restive river
Emily is paid a bonus of 78$ which is 15% of her orginal weekly salary. Find her weekly salary rate.
devout snowBOT
Emily is paid a bonus of 78$ which is 15% of her orginal weekly salary. Find her...
devout snowBOT
short forum
Emily is paid a bonus of 78$ which is 15% of her orginal weekly salary. Find her...
let original weekly salary be $x
x(15%) = 78
you should know the rest
restive river
yea 520$
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oblique seal
knowing the function f admits primitives and F is a primitive such that F(-1)=1/e, find the value of F(1).
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knowing the function f admits primitives and F is a primitive such that F(-1)=1/...
oblique seal
my answer is 3 but there is no 3
urban harbor
primitives? real numbers?
oblique seal
antiderivative
deep vortex
urban harbor
ok guess it is beyond the scope of my head : )
oblique seal
integral of e^x is e^x and integral for the second equation is x^3+x^2+ax
since the function has primitives then that means it is also continuous
frozen aurora
e^x + C 🙃
urban harbor
i'm confused, you're saying:
such that F(-1)=1/e, find the value of F(1).
but all you are giving is a point of F with no other info about it and you're expecting to find F(1)
is one of those Fs meant to be little f?
oblique seal
F is the integral
urban harbor
of f?
oblique seal
yes
urban harbor
ok sorry i finally understand what you were saying in the problem statement, i am just used to different language
oblique seal
mb I should've translated it differently
urban harbor
no it's fine i saw you are reading in romanian so probably just different lingo used
oblique seal
yeah
urban harbor
so you said for the 2nd part of the function the antiderivative is
x^3+x^2+ax + C
and also you made the observation the function must be continuous if it is integrable
oblique seal
yes
oblique seal
it should be but I dont see 3 in the options avalaible
so I wanted to check if I did something wrong
urban harbor
yes i see how how are getting that and it makes sense to me
although one thing i am not sure is how to decide what the constant of integration should be on x^3 + x^2 + ax + C
i see how you decided on the other integral that it's 0 so it fits F(-1) but not sure about for the second piece
oh i have one idea...
@oblique seal so i guess i would make the argument that if you want the antiderivative to be valid it needs to be differentiable, otherwise it wouldn't make sense for it to be an antiderivative, so F needs to be continuous first off too and so you need to match F(0) from the left and right too i suppose
so i think the antiderivative for the second part then needs to be x^3 + x^2 + x + 1
so that F(0) = 1 from both sides?
then you would get 4 for the answer
urban harbor
im a little fuzzy on how exactly these piecewise integrals work tbh but thats the best i can come up with
urban harbor
cu placere, te salut : )
sage burrow
Show that the criteria for a group are met.
glad bough
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clever gorge
WELL FIRST OF ALL, DRAW A DIAGRAM
AND FOR QUESTIONS INVOLVING COMING UP WITH YOUR OWN FUNCTIONS/EXPRESSIONS, ITS ALWAYS HANDY TO MAKE A TABLE OF THE STUFF THAT CHANGES:
TIME, HEIGHT, ANGLE, HYPOTENUSE ETC.
cursive steppe
Hi, I'm trying to prove the following fact "An ideal of a (commutative, with a unit) ring is radical iff its quotient ring is reduced" I know how to do the proof when writing elements of the quotient ring as $\bar{f}$ but I want to understand this proof on wiki which is quite clear there's only one thing I don't understand: how $(r+J)^n = 0$ implies that $r^n + J = 0$, tried to use the binomial formula but I still don't see it...
