i really dont know, teh answer says 'answers will vary'
#help-38
stiff scarab
tame solstice
i really dont know, teh answer says 'answers will vary'
stiff scarab
my friend suggested using cosine rule to prove the longer sides of both triangles equal ED and AB
i feel like thats a last resort if i cant find any proofs
wraith hinge
ASA congruency
vertically opposite angles
they marked de=ab
right angles
ac=cd
sin d = ec/cd
replace cd by ac
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opaque acorn
Could you check my work in part a and help me find the optimal toy production in part b?
opaque acorn
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opaque acorn
Could you check my work in part a and help me find the optimal toy production in part b?
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Could you check my work in part a and help me find the optimal toy production in...
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frozen plover
For any integer $n \geq 5$ let there be two $n \times n$ invertible matrices with real entries, $A$ and $B$, that satisfy $A^{-1} + B^{-1} = (A + B)^{-1}$. If $\det(A) = 3$, find $\det(B)$.
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NEONPerseus
frozen plover
Honestly I don't even know how to start this one
frozen plover
$A + B = A(A + B)^{-1}B$
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NEONPerseus
frozen plover
merry bluff
A = (A^-1)^-1
frozen plover
But how does that help
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full oxide
How is the domain of this function? Is not R?
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How is the domain of this function? Is not R?
split chasm
who's saying it isn't
full oxide
the answer of the exercise xd
stark bison
What is it saying?
split chasm
what are they saying the domain is
full oxide
stark bison
Dom(g) = R
full oxide
but the 0 doesn't affect no?
frozen plover
I've never seen that notation before
split chasm
that clearly states the domain is R
stark bison
I've never seen that notation before
A - B is the same as A \ B
split chasm
and isn't not R
fresh pecan
Domain is R
frozen plover
but the 0 doesn't affect no?
The function is xe^x at 0
full oxide
The function is xe^x at 0
that's it
stark bison
The case x = 0 is perfectly fine
frozen plover
A - B is the same as A \ B
I've seen that
stark bison
Oh you mean Dom( )?
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wraith hinge
Hey
feral ocean
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wraith hinge
$test$
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COLOURS
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crude blaze
wtf why it work here
true..
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tender grail
Is this right?
And how am I suppose to know if it’s perpendicular or not?
tender grail
when manipulating your equation
Yeah I took it out
split chasm
why
2x is not the same as 2
y is not 3 different things
2x-4
2-4
2
are all different
tender grail
So something like this you mean
split chasm
yes
tender grail
I thought it won’t make the difference
tender grail
Alright this looks better now
Since I had already figure out the answer how would I know if it’s perpendicular or not
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edgy abyss
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edgy abyss
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bold yacht
did you select 3 vectors whose components add to 100?
edgy abyss
did you select 3 vectors whose components add to 100?
Yes
bold yacht
what is M?
edgy abyss
Thats whAt I am confused about
Does that mean take the power of the matrix to 100
?
edgy abyss
Alr so what do for M^30 V1, V2, V3
edgy abyss
Does wolfram alpha do that
solar sequoia
<@&286206848099549185>
holy fucking shit so many pings
edgy abyss
idk if it is diagonalizable
Lemme find M
Heres the matrix M
Mb
The ppt didn’t have it so I had to get the pdf
keen ember
hello
keen ember
i was hoping for help on part b
mainly on how to take the second derivative.
I think the answer should be two but according to the answer sheet the answer is 2-2x+y
this is what it says the answer is but I’m rather lost
stoic saddle
You first derive both sides, the rhs implicitly for y also
You would have 2 - 1(y’)
You have an expression for y’ as is.
keen ember
oh wait is this the thing where you would like multiply by dx and stuff to get them on seperate sides?
i’m not sure how to do that if they’re being subtracted
keen ember
isn’t the derivative of 2x-y 2-1
stoic saddle
No, maybe revisit notes of implicit differentiation?
keen ember
i gotcha
keen ember
You first derive both sides, the rhs implicitly for y also
what does rhs stand for?
stoic saddle
Right hand side
keen ember
oh gotcha
stoic saddle
Of the equation
stoic saddle
The moment when it clicks is by far the best
stoic saddle
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scenic drift
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scenic drift
i need help w this whole page😭
tidal sky
i need help w this whole page😭
Personally I'd use apothem here
the first 4 look like regular polygons
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scenic drift
Personally I'd use apothem here
okayy?
naive geyser
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naive geyser
Ok so
Where is that upward slope coming from
How is the deravative going up on the left side
The original function only decreases on the left side
There’s no increasing
Im so confused
kindred pier
Derivative is negative on the left side
always less than (or equal to) zero
Derivative is increasing because the slope is becoming "less negative"
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flat wing
Why did they use a volume of a cone
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Why did they use a volume of a cone
flat wing
Instead of just using integration? Is it because it is quicker?
trim lichen
yes
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vague quest
I need to write it as a root but im stuck again (3rd one)
vague quest
I started with writing them as this
(4^(1/3)) / 2^(1/2))
and as far as i know it's possible to subtract powers when you're dividing as in : a^m / a^n = a^(m-n)
but yeah im stuck here
vague citrus
$\sqrt[3]{x}=\sqrt[6]{x^{2}}$ and $\sqrt{x}=\sqrt[6]{x^{3}}$
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B-eard
vague citrus
or use
$\sqrt[3]{4}=4^{\left(\frac{1}{3}\right)}=2^{2\left(\frac{1}{3}\right)}=2^{\frac{2}{3}}$
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B-eard
vague citrus
@vague quest
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welcum
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ripe topaz
How to solve a linear system where the number of equations is less than the number of unknown variables
ripe topaz
Is it even possible to solve it
Using Gauss–Jordan elimination method
Or a similar one
wraith hinge
Using Gauss–Jordan elimination method
yes
ripe topaz
How?? 🤔
sick marsh
or substitution
marble wharf
row reduce it as usual
then introduce parameters
the same way you do when you have the same number of equations as variables but end up with a zero row
ripe topaz
The last equation will end up with 2 variables
I can get rid of x in E2 and E3
Then get rid of y in E3
E3 will still have z and t as unknown variables
marble wharf
have you never reduced a system and gotten a zero row somewhere?
marble wharf
an equation that you completely eliminated
ripe topaz
Like 0=0?
marble wharf
yes
ripe topaz
Yea i did
marble wharf
and what did you do then?
ripe topaz
It's true for all values of the variables?
marble wharf
well you still have other equations you have to satisfy
ripe topaz
well you still have other equations you have to satisfy
What would u do? 😅
Move to the next equation?
marble wharf
you introduce parameters
you say "let t=lambda be given by a parameter". and then you solve the rest of the system in terms of lambda
and if you still cant do that you introduce another parameter
marble wharf
well it depends on the system how many variables you will need to introduce
ripe topaz
How do I solve this sytem?
marble wharf
start by eliminating variables from the other equations like you said
ripe topaz
start by eliminating variables from the other equations like you said
Done
This is where I'm stuck at
If i substitute t i will be left with 2 equations and 3 variables
marble wharf
well what do you get after solving t and subbing it in
ripe topaz
Idk how to solve a system that has more variables than equations
marble wharf
that's not what I want from you rn
ripe topaz
x+y+z=1/2
-2y-2z=2
marble wharf
ok
lets focus on the second one
we notice that we can set z as whatever we want, as long as y plays along
yes?
ripe topaz
Yes
marble wharf
so that's what we do
we say that z=lambda for some parameter lambda
and now we solve y = something involving lambda
marble wharf
we could also have set y= lambda if that's what you are asking
ripe topaz
Why not just leave it named z
marble wharf
I personally like it more to explicitly rename it
ripe topaz
we could also have set y= lambda if that's what you are asking
Why do we have to introduce lambda
marble wharf
and I found that its often better to understand
ripe topaz
Oh ok got it
marble wharf
you could also just solve for everything in terms of z
ripe topaz
That's what u meant by introducing parameters?
marble wharf
yes
violet nest
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violet nest
since we can't have a negative exponent, it must be in the denominator
delicate jasper
u can have a negative exponent
violet nest
does the -4 and 1/2 go on top
dull brook
You can
delicate jasper
does the -4 and 1/2 go on top
what do u mean
violet nest
power rule
I thought we have to make this a fraction since it is improper to have a negative exponent
dull brook
No, just use the basic derivation table
violet nest
-4z^-5 - 1/2z^-1/2
dull brook
You dont have to turn it into a fraction, just know that ((u(x))^y )'=(u(x))` yu(x)^(y-1)
violet nest
hmm I'm not familiar with that format
dull brook
what am I thinking of where we cant have negative exponents?
Its the general form
violet nest
ill just keep in mind its ok to have negative exponents
dull brook
Yeah, when solving these derivatives very little rules apply, most common being that if theres a fraction, the bottom side must be non-null
violet nest
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valid flame
Hey there I had this question in a practice sheet and I need to graph the equation, this is what I got so far
valid flame
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proud oracle
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proud oracle
I need urgent help with this question, I've been trying to do it for a while now and I still can't
my working if anyone is interested
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last rune
In the second step
I see you tried to multiply 10 in both sides
so the right side should be 4x instead of 4x^2
or is it like simultaneous equation you trying to solve?
unreal river
There should be infinite number of x's
So you could put a dot above the x's everywhere and use one x
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kindred pier
Stick to one channel
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stoic iron
stop having more than one channel open at a time
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stark bison
So the only part missing is the radius
quaint gazelle
state what you got for each then
area of the sector?
just give result for each part
so we can find the mistake
then we'll see what you did
for this part
ok you got a wrong radius
hint
OP =AB
OPAB is a rectangle
yes
thats why i said its the easiest part
its not very clear
how you did it
no
yes actuallyt
its sqrt(3°/2
i dont need an equation i need an explanation pls
which result/theorem did you use
lol
the angle is not 120
if you are talking about AOCC
look at your own drawing
the line doesnt go to C
for the radius
you HAVE to use the fact that the two circle touch each other
so that the radius are equal
because if not they could be different circles right
then once you know that you only need the distance
between the two centers
which convieniently is easy to find
so re do everything with new radius
and tell me if you find a good result
topaz pawn
determine the exact primitive function G(x) to g(x) with the following rules:
This is what i got can somenoe help check if its right
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quaint gazelle
ok so lets find the primitives of the two terms
ffor the first one
$\frac{1}{\sqrt[3]{x}} = x^{-1/3}$
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quaint gazelle
doest that help you?
it seems you were on the right track
but its not exactly the right answer
gentle sleet
Is there a better way to prove this?
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Is there a better way to prove this?
gentle sleet
without usin specific negative integers
main sparrow
I think a simple counter example is a fine way to disprove something
main sparrow
Hey, I'm no expert on proving things though. Might want to wait for one of those if you're really worried
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wraith hinge
in a simple parabola if i take focus as (0,3) then should i always take directrix as 3 units away from origin?
subtle prairie
can you recall the definition of a parabola?
dense breach
Simple parabola means opens up and vertex at (0,0)?
wraith hinge
Simple parabola means opens up and vertex at (0,0)?
yep
wraith hinge
assume the directrix is a horizontal line anyways
ok..
ok i will change the qn i am getting focus as (0,-2.25) here so is directix always 2.25?
subtle prairie
In addition to be a parabola, the distance from focus to any point in parabola and to the directrix must be same.
certainly vertex lies in the parabola, so the distance from vertex to focus must be equal to distance from vertex to directrix
wraith hinge
certainly vertex lies in the parabola, so the distance from vertex to focus must...
ok
subtle prairie
if focus is at (0,3), directrix must be at x=-3, so as to keep intact with the definition of a parabola
try to always see things as consequence of definitions.
wraith hinge
Why is he adding 6 root(2) with 5root(2). Did the sign flip and please explain why the sign flipped please.
split chasm
because they screwed up
split chasm
yes
wraith hinge
wait he made a mistake?
split chasm
yes
wraith hinge
so the answer is 11-1root(2) over 16
split chasm
their lack of () in text is disturbing
wraith hinge
so is the answer 11-root(2) over 17
split chasm
in plain text
(11 - sqrt(2))/17
split chasm
or written
$$\frac{11-\zqrt{2}}{17}$$
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ℝamonov
split chasm
what's your issue with reading the plain text
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flat wing
What did I do wrong for 17a?
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What did I do wrong for 17a?
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flat wing
.reopen
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cerulean hearth
if i have a plan such as : x - y + 5z = 0
what represent the vector (1;-1;5)?
cerulean hearth
i have trouble to image it in my head
cerulean hearth
this should be in your course
it's just a question i asked myself
quaint gazelle
ok
to see it
remember that two vectors are normal to each other
if and only if
their dot product is 0
so (x,y,z) is normal to (1,-1,5) if and only if
(x,y,z).(1,-1,5)=0
which if you developp the dot product
gets you?
quaint gazelle
so essentially a plane can also be represented by a normal vector
or a normal line
by just saying: its the vectors normal to this line/vector
knowing the line or the plane is the same
(this is actually an important concept in linear algebra if you go far this path)
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daring dragon
@quaint gazelle could I get some help in my channel?
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true coral
$$\lim_{n\rightarrow \infty} (1-\frac{1}{n})^n$$
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$$\lim_{n\rightarrow \infty} (1-\frac{1}{n})^n$$
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bigpufik
true coral
i thought i knew i dont know lmao
solid kilnBOT
true coral
yeah i take the n down
twin swan
yep
true coral
and what 1/n goes to 0 and i can just use that?
true coral
but then i have e^0
true coral
its the same book two probelms later
true coral
the textbook didnt cover that yet
like i know what that it is
but
ur supposed to solve this using normal methods
twin swan
fine hmm
quaint gazelle
is this exercise part of a particular sub topic?
wraith hinge
maybe use definition of e = lim n -> inf ( 1+1/n)^n and substitute n with something to make a similar expression. since limits preserve order or something
true coral
lim (1+1/n)^n
quaint gazelle
do you have a proof? or just a statement
solid kilnBOT
true coral
and that sequence <1
solid kilnBOT
quaint gazelle
yeah
twin swan
if you can show the bottom goes to e then its simple
prob some messing around with the fraction
turn it into 1 + something
true coral
ohhhh makes sense now
quaint gazelle
yeah prove this is the inverse of the sequence that goes to e
true coral
$$\lim_{n\rightarrow \infty} (\prod^{n+1}_{k=2} (1-\frac{1}{k^2}))$$
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bigpufik
true coral
i got 1/2 but im not sure
quaint gazelle
show your work then
sour talon
a box containing 10 balls
5 Red and have the numbers :{1;1;2;2;3}
3 white : {1;2;3}
2 Green : {2;3}
the problem is to fing the probability of getting three numbers that give 5 when added and having a defferent color
were gonna use nCr here
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a box containing 10 balls
5 Red and have the numbers :{1;1;2;2;3}
3 white : {1;...
@sour talon Has your question been resolved?
sour talon
<@&286206848099549185> c'mon
shrewd ridge
well you just count
i don't know what hint you'd like to see
almost no combination works, you manually enumerate those that do
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last fossil
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last fossil
im having trouble solving this inequality due to casses
tranquil junco
What have you tried so far?
last fossil
i get that we want to examine when x < 0
and when x > 0
but what trips me up is 0 < x < 1
!rotate
shoot is tht not a command
tranquil junco
,rotate
solid kilnBOT
last fossil
im kind of running in circles here
last fossil
that x^2 is inbetween 0 and 0.666...
tranquil junco
Correct
last fossil
so i also might want a case where x is larger than o.66... ?
tranquil junco
And when you take the sqrt?
tranquil junco
so i also might want a case where x is larger than o.66... ?
Well, not really
last fossil
x = +- 2/3
tranquil junco
Because if x > 0, then you are sure that x^2 is btwn 0 and 0.66666
last fossil
Because if x > 0, then you are sure that x^2 is btwn 0 and 0.66666
what if x was 1?
well i think then the inequality wouldnt hold
tranquil junco
what if x was 1?
Well test it in the original inequality
tranquil junco
last fossil
oh shoot
yeah that dosent work nvm
ahhh i see
so the inequality only holds as long as 0<x<0.666
tranquil junco
For the first case, yes
last fossil
so then not x = +- sqrt(2/3)
it would just be
x <= sqrt(2/3)
or wait no
im confused about sqr rooting with a inequality
casue for the inequality to be true 0 < x <2/3
but at the momment i have x^2 < 2/3
tranquil junco
You've done the first case, now consider the other one
tranquil junco
casue for the inequality to be true 0 < x <2/3
This is your final answer for the first case
Now do the other one
last fossil
for the second one if x is less then 0 we flip the ineq sign
but im not sure what to do from what i have in the picture
last fossil
sqrt(2/3) < x
so for the second inequality so be true when x is less then 0
x needs to be greater then sqrt(2/3)
i punched in into my calculator and it made sense cause if its smaller than sqrt(2/3) we get -2.85... > -2.1
tranquil junco
sqrt(2/3) < x
And what else?
solid kilnBOT
tranquil junco
This is almost true
last fossil
This is almost true
i think i got it cause i needed to add that x is less then 0 the realized for that to be true sqrt(2/3) has to be in the negative to
tranquil junco
You dropped a negative
x^2 > 2/3 means what?
x > sqrt(2/3) or x < -sqrt(2/3), right?
last fossil
You dropped a negative
the last 2 lines are just my example
like i was trying to show u my thought process
tranquil junco
Unfortunately this is incorrect
last fossil
so you suggest that it should be sqrt(2/3) < x < 0?
tranquil junco
Nope
tranquil junco
x > sqrt(2/3) or x < -sqrt(2/3), right?
I wrote it out for you
last fossil
x > sqrt(2/3) or x < -sqrt(2/3), right?
i think the ineq needs to be flipped for each
last fossil
Check this by testing values
cause if thats the case then i think i forgor to flip them in my sollution
tranquil junco
do we flip the sign?
Idt there's a hard formula for switching signs with sqrts
last fossil
x > sqrt(2/3) or x < -sqrt(2/3), right?
ok so x > sqrt(2/3)
but i thought my case one was right
where its sqrt(2/3) > x > 0
tranquil junco
ok so x > sqrt(2/3)
Consider which case this is for
tranquil junco
where its sqrt(2/3) > x > 0
This is right, because x > 0
last fossil
x > sqrt(2/3) or x < -sqrt(2/3), right?
this is both for case 2 when x < 0 ?
tranquil junco
Yes
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bright tulip
How am I supposed to guess the solution for x1 and x2
There should be two dots above x2
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amber wolf
can anyone explain how this happened
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can anyone explain how this happened
long basin
I'm assuming f(x) = e^x
amber wolf
yes
amber wolf
how did e^ln15 turn into just 15?
steep cypress
how did e^ln15 turn into just 15?
do you know what ln is?
amber wolf
log right
steep cypress
log of what
amber wolf
of 15
steep cypress
could you explain please
TLDR:
Logarithms are the inverse functions of exponentiation
turbid bough
ln means Log in base e
so you have e ^ (ln(15))
and that's why it's 15, cause ln has the same base
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eager sparrow
How would you do this?
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cunning cairn
I need help with the following probability math question:
We throw a fair dice, with eyes 1 through 6. The random variable X represents the number of eyes we got. Then we throw a fair coin with heads and tails, where a second random variable Y represents the number of heads we got after throwing the coin X times.
Now I am supposed to show the joined distribution of Y and X.
I have the table right before me and it says for example
P((1,0)) = 1/12.
Meaning, throwing a 1 with the dice and then no heads with the coin has a prob of 1/12.
What seems reasonable to me is that one would calculate
1/6 for the 1 from the dice times 1/2 for either heads or tails. But why would we calculate that, if that is correct.
I though we'd need to use bayes and then do
P(Y=0 | X=1)
Can someone explain ?
ocean mural
I need help with the following probability math question:
We throw a fair dice,...
wdym "why would we calculate that"?
cunning cairn
wdym "why would we calculate that"?
if that way is correct
why is that
why wouldnt we use bayes formula
ocean mural
Do you think the two events, getting some number on die and getting heads on a coin are dependent?
cunning cairn
Do you think the two events, getting some number on die and getting heads on a c...
I would say so
like the, the number of heads you can get is limited by what the dice gives you
hm
no but
the outcome of the coin is not dependent on the dice
oh I understand
ocean mural
like the, the number of heads you can get is limited by what the dice gives you
but does it affect if you get head or not
cunning cairn
but Y is about how many times you get heads
ocean mural
but Y is about how many times you get heads
yeah ik
cunning cairn
but i totally see your point
if I were to pick numbered balls from a box, lets say 1 through 10 and then say given the first pick , whats the lowest minimum I can get
that would be dependent , right ?
ocean mural
that would be dependent , right ?
yes
cunning cairn
so since X and Y are independent I simply multiply and its done
thank you now I understand
ocean mural
yes
cunning cairn
I have a similar question
I'll just pose it, dont feel obligated to answer
but If you want I would really appreciate it of course
so calculating P(Y=5) = 1/48
but I dont understand why that is
ocean mural
so calculating P(Y=5) = 1/48
is this given in the solution or you found it?
You have to add P(X=5,Y=5) + P(X=6,Y=5)
cunning cairn
is this given in the solution or you found it?
thats given
its the solution
the textbook Im working with gives you the answers
cunning cairn
You have to add P(X=5,Y=5) + P(X=6,Y=5)
but why ?
ocean mural
(1/6)(1/32) + (1/6)(6/64) = (1/6)(8/64) = 1/48
ocean mural
S = {(1,0),(1,1),(2,0),(2,1),(2,2)...}
cunning cairn
yes
cunning cairn
oh hold on
ocean mural
Y
cunning cairn
what was confusing me was that you would use both radnom variables
ocean mural
X is independent of taking any possible values
cunning cairn
because it was only asked about Y
but my mind was telling me we need X in order to get this
but I didnt know how
and why is it 6/64 ?
ohhh
binomial distribution ?
yes thats it
ocean mural
binomial distribution ?
yes
cunning cairn
I was close actually
but didnt see that what you provided
thanks man , I really appreciate your help !!!
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ocean mural
thanks man , I really appreciate your help !!!
you're welcome dear 
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Available help channel!
gentle sleet
Is my proof invalid?
kind ermine
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Is my proof invalid?
kind ermine
How to find D brother
gentle sleet
dam bro my bad
jaunty pebble
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jaunty pebble
<@&286206848099549185> i really want an answer for the question
i was trying to solve it from last night
jovial kiln
You can‘t get answers here.. only how to solve it
jovial kiln
Plus, what does this even relate to?
jaunty pebble
You can‘t get answers here.. only how to solve it
that's not what i meant
warm lily
@jaunty pebble what is the topic?
jaunty pebble
its an i\q question
so we have 16 and 21 what opreation we can make between them to give us 110
jaunty pebble
i gave an answer yesterday
yes thank u but it didnt work
jaunty pebble
waittt
pastel robin
36*3=108
jaunty pebble
sheeeesh
jaunty pebble
like is it luck or u know how to solve this questions
jaunty pebble
they can be fun to solve
haa
fringe glacier
E is the answer
jaunty pebble
i have another one can u give it a look
pastel robin
i played this types of puzzles
fringe glacier
I think
jaunty pebble
E is the answer
yes
fringe glacier
Blind guess
jaunty pebble
pastel robin
1 for A 2 for D 7 for Y 4 for Z 6 for K
fringe glacier
D is the answer
pastel robin
that is the logic just think simple
fringe glacier
Credit goes to me
pastel robin
i didnt say you were wrong
fringe glacier
Thanks
pastel robin
i am explaining the logic behind the question
fringe glacier
But credit goes to me
pastel robin
for him
fringe glacier
I would also if he replies
pastel robin
okay
jaunty pebble
1 for A 2 for D 7 for Y 4 for Z 6 for K
i have a whole book if u want it pdf
fringe glacier
Why not?
jaunty pebble
Çağdaş yos i\q
pastel robin
there are plenty of apps which you can download for free
for these types of puzzles
fringe glacier
Some names
pastel robin
i strongly recommend, as i said yesterday
fringe glacier
Would be appreciated
jaunty pebble
there are plenty of apps which you can download for free
fyi im studying this book cuz i have University entrance exam
pastel robin
all these types of questions heavily based on numbers
pastel robin
fyi im studying this book cuz i have University entrance exam
yes i know
jaunty pebble
thats why im asking too many question of it
pastel robin
good luck at your exam
fringe glacier
I think along with logic, number theory should also be used
jaunty pebble
thank u guys i send the book to u in case u want it
for two of u
in privet chat
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subtle shard
Let k be a positive real parameter. Prove that the following integral is independent of k and find its value.
subtle shard
f(x) = x^3 - 6x^2 + 11x - 6
I was thinking about replacing 2+k and 2-k in the integral
potent dome
just solve the integral
if its independent of k then the k should disappear once you solve it and then its proven
pastel robin
isnt it a bit long
but yeah you could do it
it seems like the way
i thought about 2 being symmetry axis
it can be
2+k is k distance from x=2 and 2-k is k distance from x=2 too
just right and left
subtle shard
The result is 0 so I would try to use only the additivity property of integrals, because otherwise the calculations are too long
In fact, if I replace 2+ k in the integral, I get something like (2+k)^4 which is difficilt to solve…that’s why I’m looking for an alternative
pastel robin
x=2 is the symmetry axis
subtle shard
Is this correct?
timid fulcrum
you can simplify the polynomial to f(x) = (x-1)(x-2)(x-3), then you can let y = x - 2 so that f(y) = (y-1)(y)(y+1) which is the same function just moved to be centered about the origin, allowing you to shift the bounds of the integrals as well since then it would just be from -k to k
to prove its an odd function you can just show that f(-y) = -f(y)
but henceforth since its an odd function, the integral (taken over a symmetric interval) is just always 0 (symmetry property over an odd function)
therefore you have both proved the value of the integral (always 0) and the fact that no matter what k is, the value is constant (not dependant on k)
timid fulcrum
you can simplify the polynomial to f(x) = (x-1)(x-2)(x-3), then you can let y =...
you can think of this step as doing a u substitution, then shifting the bounds accordingly
if thats easier to see while studying calculus
@subtle shard
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@subtle shard Has your question been resolved?
subtle shard
therefore you have both proved the value of the integral (always 0) and the fact...
Thank you so much!
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Available help channel!
smoky kite
can anyone explain why they subtracted 0.8918 in the penultimate line
whole coral
Presumably that's their P(Y<50.5) - P(Y<49.5) i would guess?
whole coral
Well if i remember right i think you have to do it like that because that's what the calculator would return you(?)
Think they only do up to a point so you have to do those probabilities separately
left oriole
often if you have a table or equivalent software function for standard normal probabilities, it's defined in terms of P(Y < val), so if you want something like P(val1 < Y < val2), you can get it using P(Y < val2) - P(Y < val1)
smoky kite
yep what on earth am i saying
left oriole
think of it as the probability that Y is less than val2, but not less than val1, so you subtract the P(Y < val1) part
left oriole
sure, gl
smoky kite
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chilly birch
Hey so, I have a question about graph theory (it's related to today's daily leetcode! actually) If i have an acyclic directed graph, and i know that there exists a unique dominating set. How do I know that the dominating set for this graph is exactly all vertices with indegree 0?
chilly birch
like i think it's not too hard to show that all acyclic directed graphs must have at least one vertex
of indegree 0
kindred pier
I don't have an answer for you, but my experience with graph proofs is always induction
chilly birch
mmm yeah lots of em are induction
kindred pier
Prove for your base case (empty graph or graph of one node), and then assume true for some DAG, and add/insert a node somewhere
chilly birch
sadly idk much about directed graphs :(
guh :(
okay we give up for now
will ask me friends irl
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cerulean hearth
i don't understand the logical behind this formula if someone can explain it
marble wharf
what kind of logic are you looking for
cerulean hearth
geometric logic
marble wharf
x = sum a_i e_i because x is in the span of the e_i
and then you just compute <x,e_j> and see that you get a_j = <x, e_j>
well essentially you are projecting down onto every axis
cyan zinc
The sum of orthogonal projections yields the original vector
olive shale
yup you can think of it as all the portions of x as a vector you get when you project it down onto each of the basis vectors e_i, if you add up all those projected vectors you get the original x.
cerulean hearth
ok so <x,e_i>e_i
is the projection of x on e_i
so the dot product is like the proportion of a vector over another one
cyan zinc
Basically (1,1,1) = 100 + 010 + 001
stoic iron
it works particularly nice here precisely because the e_i are orthonormal
cerulean hearth
it works particularly nice here precisely because the e_i are orthonormal
ye because the norm is one it's really a proportion i gess
stoic iron
exactly, otherwise you'd have to do some annoying scaling
olive shale
Basically (1,1,1) = 100 + 010 + 001
^ following this in general for any n, <x,e_j> as a dot product is just x_1(0)+x_2(0)+...x_j(1)+...+x_n(0), so just the jth coordinate of x, then you multiply that by the vector with just 1 in the jth's coordinate and 0 everywhere else, e_j and you should just get the projection of x onto the jth coordinate. If you add up a vector that is 0 everywhere but just contains the jth coordinate of x, and you do this for every coordinate you just get x.
In general I think you can represent any projection onto a vector as a matrix which is defined by the outer product of that vector with itself? (I don't quite remember but I think this is correct) so if you had randomly skewed vectors this is what you would want to sum up all these outer products and multiply them by x. Somebody check this my outer products is a bit rusty
if you had an arbitrary orthogonal set of vectors you made these outer products out of