Mr. Gamer π΅πΈ
#help-0
ocean sealBOT
molten pivot
can we factor again?
patent moat
yes because they both have A and B
molten pivot
so what do we write
patent moat
-16x^2 (a +b)
molten pivot
almost!
maybe if i write it a different way then you'll see it
try factoring this:
$$x^2y - 16y$$
ocean sealBOT
Mr. Gamer π΅πΈ
patent moat
y (x^2 - 16)
patent moat
a + b (x^2 - 16)
patent moat
okay, thank you!
molten pivot
so let's recap:
factor the first two terms
factor the last two terms
then factor everything
sometimes you might have to rearrange the terms but that's ok
molten pivot
π
patent moat
Is the answer to question 13
$$x + 1 (x^2 - 1)$$
ocean sealBOT
Grey
lone heartBOT
@patent moat Has your question been resolved?
carmine rapids
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carmine rapids
How would I figure out maximum number of real zeros
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leaden creek
can somebody just confirm if this is right or wrong for me
lone heartBOT
can somebody just confirm if this is right or wrong for me
leaden creek
im like 80% sure thisi s correct
because
108*3 = 324, and it follows the correct form of 4(3)^n-1
alpine sable
π
leaden creek
but then i get confused because when you put that information into a calculator it somehow says it's 971
and 971 isnt even any of the answers
modern bone
π
alpine sable
Substituting x =1 eliminates,options 1 2 3
leaden creek
im just confused why the calculator says its 971..
modern bone
4 * (3)^(5-1) = 4 * (3)^4 = 4 * 81 = 324
alpine sable
leaden creek
thank you guys
modern bone
π you've got this.
Not sure what's up with the calculator answer though. Quick rules for division:
If the last 2 digits ('71' part of 971) are not divisible by 4, then the number is not divisible by 4
If the sum of the digits (9+7+1 = 17) are not divisible by 3, then the number is not divisible by 3
so 971 is not divisible by 4 nor 3. O_O Be careful with that calculator π
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dusk ice
Solve sin(x + Ο/4) = β2 cos(x) for 0 β€ x β€ 2Ο
i need help with this question
lone heartBOT
Solve sin(x + Ο/4) = β2 cos(x) **for 0 β€ x β€ 2Ο**
i need help with this questi...
pearl hare
Solve sin(x + Ο/4) = β2 cos(x) **for 0 β€ x β€ 2Ο**
i need help with this questi...
hi did you try half angle formula
hybrid comet
sin(Ο/4 + x ) = sin(Ο/4)cos(x) + cos(Ο/4)sin(x) = (cos(x) + sin(x))/β2
Now ,
(cos(x) + sin(x))/β2 = β2 cos(x)
Or,
cos(x) + sin(x) = 2cos(x)
cos(x) = sin(x )
tan(x) = 1
x = Ο/4
dusk ice
hi did you try half angle formula
idk what htat is
pearl hare
tan(x) = sin(x)/cos(x) so if cos(x)=sin(x) then tan(x)=1
lone heartBOT
@dusk ice Has your question been resolved?
cunning trout
Isnβt that a 3? and not a 9
lone heartBOT
Isnβt that a 3? and not a 9
sharp gate
yeah
sharp gate
np
cunning trout
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arctic crystal
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arctic crystal
are these 2 the same formulas
im looking for formula of covariance between 2 assets
and i got these 2
and idk which one is correct
lone heartBOT
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arctic crystal
<@&286206848099549185>
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@arctic crystal Has your question been resolved?
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autumn matrix
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autumn matrix
0 is the the angle A is the arc length R is the radius of the circle
is this correct ?
reef pier
it is correct
if you're familiar with the arc length formula being l = r*theta, then this is just that rearranged
the problem wording is kinda strange to me but it seems like all they want you to do is plug in 2pi*r (circumference) for the arc length since your arc is one full rotation of the circle
lone heartBOT
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autumn matrix
.reopen
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β
hollow wren
do you know how arc lengths work
hollow wren
ohhh
ok so arc length in a way corresponds to the angle in the center
arc length is like
you have a section of the circumference
with a specific length
and it corresponds to the angle that creates it
dull hare
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small shard
might be a silly question, but why does f(x+h) = 2(x+h) - 5
lone heartBOT
might be a silly question, but why does f(x+h) = 2(x+h) - 5
dull swift
might be a silly question, but why does f(x+h) = 2(x+h) - 5
Youβre replacing the βxβ in f(x) with βx+hβ
lone heartBOT
wraith stratus
can you demonstrate 
long axle
ahhhh
small shard
this is my channel... wtf
long axle
u closed ur channel
wraith stratus
small shard
Get outta hereee
wraith stratus
small shard
It's mine until the bot removes my name from it
wraith stratus
small shard
long axle
ok i shd delete that msg
ornate ginkgo
It's mine until the bot removes my name from it
That's not very wise
wraith stratus
should we inform
ornate ginkgo
For a goodyoda
wraith stratus
or is this our (public) secret
small shard
I'm sorry...
small shard
I should be less possessive of things
You are all free to have this channel
I own nothing,
long axle
127
ornate ginkgo
That was fast
wraith stratus
it was blue nt
long axle
bruh
wraith stratus
,w hex color 0x000127
ocean sealBOT
wraith stratus
you're a fuckin genius
long axle
huh
small shard
what does this mean
long axle
is that a dark blue or light black
wraith stratus
it looks blue to me
small shard
i am thinking of a color between 0 and 1
I thought this was a joke
long axle
I thought this was a joke
u think chebyshev jokes?
you are a foolish yoda
i miss the old yoda
small shard
That is not a number between 0 and 1
long axle
oh apparently if u search color #000127 on google its pretty blue
wraith stratus
guess we gotta open a modmail
long axle
wraith stratus
@fierce herald
ocean sealBOT
Result:
true
@small shard
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autumn matrix
Heyt
narrow hedge
lone heartBOT
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
@autumn matrix Has your question been resolved?
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autumn matrix
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β
nimble latch
i have the points im just having trouble drawing it
draw? why do u need to draw it
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silver orbit
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west cipher
show work
silver orbit
i have none
west cipher
where do u think u should start
silver orbit
idk how to evaluate the 9x into the parenthesis
west cipher
$y=1600(0.56)^{9x}$
ocean sealBOT
hrdxpqurcxktdbanql
west cipher
can you get 9x by itself?
silver orbit
no
west cipher
lets try
west cipher
$y/1600=.56^{9x}$
ocean sealBOT
hrdxpqurcxktdbanql
silver orbit
ok
ocean sealBOT
ocean sealBOT
silver orbit
growth is 1st one
ocean sealBOT
hrdxpqurcxktdbanql
ocean sealBOT
west cipher
correct
silver orbit
ok
west cipher
it wants you to find
how much percent y increases by
every unit of x
in other words
it wants you to find by what factor y increases for each unit of x
silver orbit
how do i do that
west cipher
well
to find how much y increases for each unit of x
what do you think you have to do?
silver orbit
idk
west cipher
guess
silver orbit
i feel like
if i had no clue
and i was taking a test
i would plug in 2 into x and see
ocean sealBOT
west cipher
how much is a unit of x?
silver orbit
9?
silver orbit
1
silver orbit
ok yea
west cipher
so how much us a unit of a number line
silver orbit
1
silver orbit
ok
west cipher
it wants us to find the factor that y increases by for every unit of x
ie
by what factor does y increase every time x increases by 1?
we know
silver orbit
4.7%?
ocean sealBOT
west cipher
i dont know
silver orbit
should i try it lol
paper thistle
You can juste evaluate x=1 than x=2
ocean sealBOT
west cipher
so how much does it increase from x = 1 to x = 2
it just increases by y(2)-y(1), right?
silver orbit
i mean thats what i was thinking
west cipher
ok
ocean sealBOT
ocean sealBOT
west cipher
it wants us to find
the percent change of the rate of change
per unit of x
we know y(2)-y(1) is
,w calc y=1600*.56^(92) - 1600.56^(9*1
ocean sealBOT
west cipher
so what is the slope from x = 1 -> 2?
silver orbit
bruh i accidently submitted the answer trying to text in disc'
west cipher
ok is this a growth or decay
silver orbit
decay
ocean sealBOT
paper thistle
Answer is 99,5% yup
west cipher
$[1200/2]^{x/12} = [{[1200/2]^{(\frac{1}{12})}}]^x$
silver orbit
are you just implementing the 1/2 straight into 1200
ocean sealBOT
hrdxpqurcxktdbanql
silver orbit
bro im so confused
west cipher
is this correct
silver orbit
i have no clue
west cipher
ok
silver orbit
like u lost me im ngl
ocean sealBOT
hrdxpqurcxktdbanql
silver orbit
multiply
ocean sealBOT
hrdxpqurcxktdbanql
west cipher
Can you then undo the multiplication here: ?
silver orbit
Can you then undo the multiplication here: ?
yes
west cipher
$1200*.5^{x/12}$
silver orbit
ok i kinda get what ur doing
ocean sealBOT
hrdxpqurcxktdbanql
west cipher
$= 1200*[.5^{1/12}]^x$
ocean sealBOT
hrdxpqurcxktdbanql
ocean sealBOT
hrdxpqurcxktdbanql
silver orbit
less
west cipher
,w calc .5^(1/12)
ocean sealBOT
silver orbit
decay
silver orbit
yeah idk how to do that
ocean sealBOT
hrdxpqurcxktdbanql
west cipher
the website says to break apart the .5 into 1-r and solve for r
and r is the increase factor
silver orbit
ok
west cipher
1-r tells you the magnitude of increase / decrase
ocean sealBOT
hrdxpqurcxktdbanql
silver orbit
oh
west cipher
sorry i told you wrong
silver orbit
ok
west cipher
i meant whatever is to the power if x you replace with 1-r
so what is to the power of x here
silver orbit
so do i solve that
west cipher
yes
silver orbit
west cipher
,w calc 1-r=.5^(1/12) percent
west cipher
ok goto bed
west cipher
ok
ocean sealBOT
Requested by hrdxpqurcxktdbanql
west cipher
can you get it in the form of a number * another number ^ x ?
silver orbit
let me try
west cipher
ok
silver orbit
2000 x [2^1/11]^x
silver orbit
less
ocean sealBOT
west cipher
rip
silver orbit
ok
ocean sealBOT
hrdxpqurcxktdbanql
silver orbit
yes
west cipher
what number is to the power of x here?
silver orbit
x?
west cipher
ocean sealBOT
west cipher
,w 1-2^(1/11)
ocean sealBOT
ocean sealBOT
west cipher
,w 2^(1/11) -1
ocean sealBOT
west cipher
did you get it?
west cipher
ok next
west cipher
np
silver orbit
thakn you so much
silver orbit
goodnight lol
silver orbit
its midnight for me
still crazy
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terse meteor
When applying functions/transformations/operations to equations, sometimes we lose or gain solutions, these are known to be the extraneous solutions and missing solutions. Does this also pertain to inequalities? Do we also lose or gain solutions when applying functions/transformations/operations to inequalities?
hearty quartz
When applying functions/transformations/operations to equations, sometimes we lo...
I think your onto somethng but not asking the right question. Let say we do some transformation like multiplication by 5x. The issue is we don't know if 5x is positive or negative
I guess the answer is it will depend on the transformation
alpine sable
I guess the answer is it will depend on the transformation
We can't cancel variables on inequalities
Unless the question says x is always positive or negative
terse meteor
I think your onto somethng but not asking the right question. Let say we do some...
I'm not really asking that, ofc when we multiply by a variable such as 5x, we have to go through the possibilities of x being positive, negative, and zero. Let me give an example to demonstrate what I'm asking: Suppose x = 2, we square both sides, x^2 = 4, but this quadratic equation has two solutions, +2 and -2, so there's an extraneous solution. Similarly, for an equation such as x^2 + 2x = 0, we can write it as x^2 = -2x, divide x on both sides, and we get x = -2, which is indeed a valid solution, but by dividing by x (a variable), we lost another solution x=0. So, does this also occur for inequalities? Do we gain or lose the range of possible values of x that satisfy the inequality?
Maybe a smaller range of values get added to the solution set or maybe a part of the solution set gets lost? Or maybe the solution set is completely replaced by another? Does this happen?
hearty quartz
square both sides can cause trouble. Think of the case x<5
severe portal
I'm not really asking that, ofc when we multiply by a variable such as 5x, we ha...
x^2 = 4 might have two solutions but the constraint x = 2 still exists
I guess the way I think about it is that by squaring both sides to get x^2 = 4, a new equation is created
to solve for x, we must satisfy all the equations we have
x^2 = 4 (Eq 2)
x = 2 (Eq 1)
Solution is 2
terse meteor
square both sides can cause trouble. Think of the case x<5
Yeah, it's monotonically decreasing for negative inputs and monotonically increasing for positive inputs and stationary for zero input. So depending on the value of x, the inequality could get flipped.
We'll have to consider different possibilities, so ig as long as we consider the various possibilities, we won't lose or gain a solution to a particular possibility.
is that it? So the answer is, "We won't lose or gain solutions to inequalities as long as we consider different possibilities that could occur for the transformation done."
mortal trellis
well if you divide by x you lose a solution because the process of dividing by x in the first place is not defined if x=0. thats conceptually different from the other examples
in general whenever we have some equation and we apply something to it, we only gain solutions. a=b implies f(a)=f(b). the problem is that f(a)=f(b) does not need to imply a=b
lone heartBOT
@terse meteor Has your question been resolved?
terse meteor
in general whenever we have some equation and we apply something to it, we only ...
ok but what about inequalities? If a (fill in some inequality) b, then f(a) (fill in some inequality) f(b), if f is monotonically increasing, the inequality stays the same and flips if it's decreasing or if it's increasing and decreasing within different intervals of its domain, then the values of a and b determine whether the inequality stays the same or flips. And, if a or b, or both are variables, then we'll have to consider different possibilities of the variable's values. So, as long as these rules are followed, the solution set should not change, right? Also, what happens if fis stationary? Does the inequality change into equality?
mortal trellis
well the solution set can still get bigger just like with the equation example. cause knowledge about a,b gives knowledge about f(a),f(b) but not necessarily vice versa
with stationary do you mean constant?
constant functions are monotone
terse meteor
wait, let me think of a function that decreases/increases first and then stays stationary for a while and then again increases/decreases.
mortal trellis
well the same rules of "just consider every possibility of what a,b can be" still apply
terse meteor
So, this means that if suppose x < y, and we apply a function f: X ---> Y which follows the kind of description I gave above, and supposef is stationary on the interval I, then if x,y belong to I, then it should mean that applying the function to both sides of the inequality gives us x = y, correct?
so x < y transforms into x = y
terse meteor
well the solution set can still get bigger just like with the equation example. ...
so that means we can gain solutions to inequalities too?
mortal trellis
as a stupid example, apply the constant zero function on both sides
x<=y implies 0<=0. the second has a much larger solution set
terse meteor
x<=y implies 0<=0. the second has a much larger solution set
shouldn't it be x < = y implies x = y? Because it's a constant zero function, any two different values of the domain result in the same value, changing the inequality to equality.
mortal trellis
the zero function applied to both sides results in both sides being 0
terse meteor
Oh sorry, I mean x < =y implies 0 = 0
mortal trellis
well 0<=0 or 0=0, same thing
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zenith olive
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zinc haven
,,
(a+b)^3=a^3+3a^2b+3ab^2+b^3\
(a-b)^3=a^3-3a^2b+3ab^2-b^3
ocean sealBOT
elder forge
the one where they start to bracket
he wrote 3sinxcos^2x as sinxcos^2x and 2sinxcos^2x
now he took sinx common from sin^3x and sinxcos^2x
zenith olive
he wrote 3sinxcos^2x as sinxcos^2x and 2sinxcos^2x
now he took sinx common from ...
what happened to the 3?
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strange fractal
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strange fractal
did i do dis right
im so finished for my exam on tuesday
i just started studying
lethal belfry
looks right
lethal belfry
im so finished for my exam on tuesday
good luck
strange fractal
guys..
i think im doing this wrong..
wait.
can i use sin..
oh..
do u think i will get full marks if i wrote this
lone heartBOT
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lethal belfry
do u think i will get full marks if i wrote this
how is $cos(\mu + \theta)=AM$
ocean sealBOT
Ζ(Why am. I here)=I don't Know
strange fractal
what do u mean
lethal belfry
what do u mean
didn't notice that it was a rectangle lol
strange fractal
loooll
lethal belfry
looks alright to me
lone heartBOT
@strange fractal Has your question been resolved?
safe tapir
hello, I'm a high school student in Italy. I would like to ask a question about analitic geometry (my english isnt the best)
safe tapir
the first questions asks for the line that is tangent to the circle with P(-24/25,y) and this P in the third quadrant
i tried starting with the formula of distance between line and point ,with d= r of the circle
but then i realized i could find the point by putting the x in the circle
and being y<=0 i used a negative y
but the anwser is different from the textbook one
alpine sable
the first questions asks for the line that is tangent to the circle with P(-24/2...
What's p
alpine sable
Ok I am not reading your work just giving an idea
Do you know derivative is slope of the curve ?
safe tapir
i dont think so
safe tapir
β?
alpine sable
yes
safe tapir
so b^2-4ac
alpine sable
b^2 - 4ac and what it means ?
yes
so you know that if equation has only one solution then discriminant is 0 right ?
safe tapir
yes
alpine sable
okay so let's consider a line y = mx + c
and let equation of some general circle be x^2 + y^2 = r^2
if you solve those equations with respect to any of x or y
it will be a quadratic
which should have only one solution because tangent must only touch the curve
so it can't cut it twice
hence it can't have two solutions
so it will have only one solution
safe tapir
so i have to do this?
alpine sable
solving this you should get c^2 = r^2(1+m^2)
alpine sable
so i have to do this?
Given that you don't know calculus yes you would have to do this
alpine sable
No I mean just put y = mx + c in the circle equation
safe tapir
oh i see
alpine sable
Nah I mean x^2 + (mx +c)^2 = 64/25
safe tapir
and the mx+c i get from y-yp=m(x-xp)?
safe tapir
ok
alpine sable
by putting discriminant = 0 you will get an equation in c and m
safe tapir
alpine sable
Yes
safe tapir
so since i have yp, and it says its on the third quadrante with y<=0
i give put a negative in front of my anwser?
alpine sable
By putting xp in circle equation you Will get 2 values for y
So since we want it to be in third quadratic
Yp has to be negative
Now you have complete point p (xp,yp)
This should also satisfy the line equation y = mx + c
Because it is tangent
Do you have one more equation in C and M
safe tapir
sorry, i dont think i quite follow
alpine sable
Did you get yp
safe tapir
yes
safe tapir
yes
alpine sable
So we got one more equation in c and m right ?
alpine sable
Note that c and m both should be negative if you look at you figure because we want tangent to be in 3rd quadrant so slope must be negative and y intercept aswell
safe tapir
sorry can i ask a question
alpine sable
Yes
safe tapir
is it possible to write in the place of x^2=x(xp) and vice versa with y?
in the circle equasion
our teacher told us that the math behind it is quite hard but it works
to find our a tanget line knowing the point p(xp,yp)
alpine sable
yes and no actually math behind it is really easy you just need to learn some calculus for it
but I don't recommend using something you don't know logically
safe tapir
ok i understand
safe tapir
its just that i find the calculations quite tedious
my teacher would have me find a faster way of solving this
alpine sable
is it possible to write in the place of x^2=x(xp) and vice versa with y?
This is faster way
But you shouldn't use it now without knowledge of some basic calculus
Doing this you get equation in seconds
Because you directly find the slope using something called "derivative" which is super easy to calculate
and you already have point p so we directly use y-yp = m(x-xp)
That's all that is
Nothing rocket science
safe tapir
ok i got it i think
alpine sable
you sure ?
safe tapir
yes, i just that i have to find c and m from the sistem
and i found an error in my calculations
alpine sable
good
safe tapir
after that it asks for point A and B where the line encounters the axys
i just have to find for which value y=0 and x=0
safe tapir
in the end ,question c asks to find the elipse that has vertices in A and B
knowing that A is on x axys and B on y axis, i can find a^2 and b^2 of the elipse equasion
them being the
alpine sable
Yes
safe tapir
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lone heartBOT
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rare cove
what is theta if sin theta =1/2 and theta belongs to pi/2 ,pi both open
lethal belfry
,rotate
ocean sealBOT
rare cove
So?
left isle
do you know what the answer is
fallen oar
Itβs c but Iβm not the one who answered it and idk how itβs c
oh okay
think of unit circle
kindred saffron
Itβs c but Iβm not the one who answered it and idk how itβs c
Firstly
Sin(theta) = 1/2 means that theta is 30Β° or Pi/6, between Pi/2 and Pi, which means that the angle is Pi-30 = 5pi/6 or 150Β°
(A) must be wrong because 2pi means that the angle is Pi/6 between 0 and Pi/2
(B) Must be wrong as well because the angle in it is pi/3 (60Β°) and the 2pi will make it between 0 and pi/2
(D) is also wrong because the 2pi will make the angle -pi/6 which is 330Β°
(C) is right because it would be
Pi - Pi/6
Which will be (6pi-pi)/6 that will be 5pi/6 which is the answer we want
In other terms (-30 + 180) = 150
Or basically rotate 330Β° by 180 anti clockwise
I am not sure how to explain it by the terms of the source you are studying
But this is how I would explain it
As @fallen oar said, you should think of the unit circle
fallen oar
sin(Ο-x) = sinx
here is the graph for it
you can check this too
so it's enough you to know sin(30Β°) = sin(Ο/6) = 1/2
then you can easily say from graph that sin(Ο-Ο/6) also equals 1/2
which is between (Ο/2,Ο) intervals
Ο/2 < 5Ο/6<Ο
lone heartBOT
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lavish holly
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lavish holly
For the last part's P(M+T=12), would it follow Po(4) as its two independent Poisson distibution.
(i) treat it as binomial since the probability of n is independent of r, only taking n to be the sample size for the binomial distribution
(ii) algebra to get e^{-2}2^r/r!, so Po(2)
(iii) conditional probability, P(M=8|M+T=12), with each day being Po(2) and independent, so I think M+T is modelled as Po(4)
long axle
P(N(1) = 8 | N(2) = 12)
are u familiar with the concepts of independent and stationary increments
long axle
well in the context of this problem, the number of casualties that need surgery
like for example, the increments from 4 to 6 and 6 to 8 would be indepedent and stationary
lavish holly
idk that
just in terms of the distribution is the last part right?
i just have vague understandings of statistics distributions, mainly just using their expectation and formulae to do questions, not needing to understand more
someone else got back to me
.close
lone heartBOT
lone heartBOT
Available help channel!
undone bison
38 International Women's Day is coming, Minh was assigned by the class to prepare ribbons to decorate gift boxes for the homeroom teacher and 14 female classmates. You need to cut a 36 dm long string into 1 piece of string with a length of 4 dm, 4 pieces of string with a length of 3 dm and 10 pieces of string with a length of 2 dm. How many ways do you have to cut the red wire according to the target? help me !
lone heartBOT
38 International Women's Day is coming, Minh was assigned by the class to prepar...
@undone bison Has your question been resolved?
crimson jetty
38 International Women's Day is coming, Minh was assigned by the class to prepar...
Can you tell me what kind of class this question is for, as well as anything you've tried to do to solve it?
restive swan
lone heartBOT
restive swan
how do you find these? this topic confuses me alot
also for b) why is the mod of w half root 3? surely its just root three??
lone heartBOT
@restive swan Has your question been resolved?
old kettle
if cos^2(x) + sin^(x) = 1
and
cos^2(x) = 1/(tan^(x) + 1)
wouldn't it follow that sin(x) = 1 - sqrt(1/(tan^2(x) + 1))?
royal grail
if ```cos^2(x) + sin^(x) = 1```
and
```cos^2(x) = 1/(tan^(x) + 1)```
wouldn't i...
$\sin(x) = \sqrt{1 - \frac{1}{\tan^2(x) +1}$
ocean sealBOT
lgkoo
Compile Error! Click the 
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royal grail
you should be taking the square root over the entire expression, not just the fraction
lone heartBOT
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regal ridge
i need someone to explain this
lone heartBOT
i need someone to explain this
regal ridge
how did we go from f' to f'' because they didn't show the steps
ocean hawk
looks like they used the quotient rule
ocean hawk
regal ridge
yeah as i said i already know about the quotient rule, but I dont get how they got there
ocean hawk
work it out step by step then
regal ridge
it didn't work lmao
regal ridge
wait whats ((x-3)^2)'
ocean hawk
you tell me
valid veldt
Quotient rule can be hard in the beginning. What have you tried? Do you know the formula?
valid veldt
Tip: Write down every step, d/dx and so on. It's really easy to mess it up if you do it all in your head.
valid veldt
valid veldt
Great!
valid veldt
It doesnt matter
lone heartBOT
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Available help channel!
spare ocean
lone heartBOT
spare ocean
is the x = ... valid?
hollow wren
Ok so again here
spare ocean
rather than calculating the range when i substitute for u
oh
ye
oeps
i split them up
spare ocean
euhm V1 here
then below I do V2 and then I add them together to get V which is shown in the graph basically
what u see above is f(x)
hollow wren
mhm
hollow wren
what x =
also you didnt need to chain rule
you already did the power rule when you u-subbed
WIT
NVM
i misread it
yeah ur correct
spare ocean
I vaguely remember that when I substitute u
that the integral range
isnt -1 to 3 but smth else
hollow wren
OHH
hollow wren
ur talking about changing the limits of integration
spare ocean
yes
hollow wren
well okay so the way it works
if you change the limits
you dont have to re substitute the original equation
back into u
you can just plug the new limits straight into u
but if you dont change the limits
it still works
you just have to plug back in your values for u
and the second way is what you did
so its completely fine to leave the limits of integration unchanged with u-sub
as long as you remember to plug back in whatever expression is equal to u
which you did perfectly
spare ocean
okie makes sense, thanksπ
hollow wren
np
spare ocean
Ill make sure to split them again next time and apply what uve taught me. Means a lot
hollow wren
lol np
i learned about half this stuff last week lol
i cram self studied for the AP calc AB exam that happened this past monday in like 3 weeks
lel
spare ocean
I've been studying maths for 2 months now, basically knew nothing
ive got my exam next thursday, hopefully ill make it
prime sluice
i am not sure how you solve this type of question
lone heartBOT
i am not sure how you solve this type of question
ocean hawk
go by the definitions of one-to-one and onto
prime sluice
this is essentially cryptic language to me but what i got is for every value i give the transform it should give me at least one output?
remote heron
this is essentially cryptic language to me but what i got is for every value i g...
it will give you one output thats given
onto is interpreted as like
covering the codomain
so say something like...
prime sluice
covering the codomain
so i should be able to get every value in the codomain?
remote heron
so i should be able to get every value in the codomain?
yes, by at least one input
prime sluice
R to R3 as x1 to (x1, 0, 0)
ok i see
ah woops the question is asking for one to one
ocean hawk
it's asking for both
prime sluice
well eitherway this should be one to one no?
remote heron
is it?
prime sluice
it's asking for both
ooohh yeah the mapping of R2 to R3
remote heron
to be sure, one to one is onto and injective
prime sluice
yeah i am not sure how to prove this, iam drowning in the 80 page lecture on the topic
remote heron
yeah i am not sure how to prove this, iam drowning in the 80 page lecture on the...
you need to see if its one to one right
prime sluice
and onto
remote heron
one to one is two things
its onto
and its injective
nooo
@ocean hawk are you on a pc
ocean hawk
no, one-to-one is just injective. onto is surjective
prime sluice
no, one-to-one is just injective. onto is surjective
what is going on
prime sluice
dont mind me
alr dw i will, take your time
ocean hawk
is this for linear algebra?
ocean hawk
ok, so $T: \mathbb{R}^2 \to \mathbb{R}^3$, right? what does that tell you about the size of the matrix
ocean sealBOT
cwatson
ocean hawk
no
prime sluice
yep no clue
ocean hawk
because T(x_1, x_2) is multiplication of T by a 2x1 vector
prime sluice
1
ocean hawk
no
prime sluice
ok nice
ocean hawk
there are several ways in linear algebra to show that it's one-to-one
the easiest probably is that the columns are linearly independent
ocean hawk
well there are only 2 columns, so it's actually really easy. what's the only way a set of 2 vectors can be linearly dependent?
prime sluice
they are parallel?
ocean hawk
yes, meaning?
prime sluice
not sure, they are a multiple of the other?
prime sluice
but isnt T still blank to me
ocean hawk
yes, but you can find it from the function definition
prime sluice
oh wait i think i need to stack the equations?
ocean hawk
the columns of T are formed from what T does to each standard basis vector
yes you can do that
prime sluice
well they dont look parallel
so its not one to one?
i also tried to find RREF because i dont trust myself, and only one column is a pivot
ocean hawk
no, since the columns are linearly independent, it is one-to-one
the RREF should have 2 pivots
prime sluice
i guess i suck, got it
prime sluice
thank you
ocean hawk
one condition for it to be onto is that the columns span R3
ocean hawk
no, meaning any vector in R3 can be formed by a linear combination of the columns
prime sluice
oh wait no
prime sluice
non parallel
im thinking in 2d oops
prime sluice
no, meaning any vector in R3 can be formed by a linear combination of the column...
the span would be only a plane in R3 so ig no
prime sluice
damn
ocean hawk
another way to show it's onto is if the rows are linearly independent. but that's false as well
prime sluice
i see
prime sluice
so it is one-to-one, but not onto
that was quite a journey to reach that conclusion, thanks!
ocean hawk
yw