😭😭😭😭😭😭😭😭
#help-0
wary root
wary root
i have everything besides the coordinates memorized, not the (1,0) (0,1) (-1,0) and (0,-1), ik these
jagged cobalt
knowing the special triangles and then the behaviour of the trig functions can be easier though less direct
wary root
what about the part where its like tan^2
jagged cobalt
these fellows
surreal crystal
what about the part where its like tan^2
thats just a square
wary root
these fellows
these i also memorized
wary root
thats just a square
wait
if tan = x/y, do i do (-1/2)/(sqrt(3)/2)?
and then square the answer?
surreal crystal
yes
surreal crystal
consider the definition of sine
wary root
y/r
wary root
what about for sin(150)
do i do (-sqrt(3)/2)/1?
so this is correct
surreal crystal
yes
surreal crystal
thank you, i will do it and put my answer here
this is one of those problems you can just check with a calculator
just remember to set to degrees lol
wary root
ya, but i have a ti-84 plus, and idk how to do tan^2 since its not an option
surreal crystal
$$\tan^2{x}=(\tan{x})^2$$
ocean sealBOT
Cycadellic
surreal crystal
into the calculator?
surreal crystal
oh, generally people keep it closed form like that
wary root
so thats the answer?
surreal crystal
you would have to compute it algorithmically
just factor sqrt(3) to make it as simple as possible
wary root
factor sqrt(3)?
surreal crystal
itll be something*sqrt(3)+something
wary root
what if i just turn the sqrt(3)/2 into 3sqrt(3)/6 and the sqrt(3)/3 into 2sqrt(3)/6
surreal crystal
by factor i refer to the distributive property
that a(b+c)=ab+ac
going right to left is called factoring
wary root
turn the fractions into improper fractions?
ik what factoring is, i just never factored a sqrt of a number
surreal crystal
that is actually just the distributive property again, language aside i just meant simplify your answer
surreal crystal
then, if it takes closed form answers, give the simplified answer, if it takes numbers, try 5.75
wary root
it just says "evaluate each of the following"
lone heartBOT
Please show the original problem, exactly as it was stated to you, with the entire original context. A picture or screenshot is best. If the original problem is not in English, then post it anyway! The additional context might still be helpful. Do your best to provide a translation.
wary root
well i need to show my work
wary root
how do i do -2(sqrt(3)/3) - 3(sqrt(3)/2) + 1
but im stuck with this
surreal crystal
how do i do -2(sqrt(3)/3) - 3(sqrt(3)/2) + 1
this one isnt right
so we are trying to find 2tan^2(120deg) + 3sin^2(150deg) - cos^2(180deg)
surreal crystal
lets find out, what is
tan(120deg)
sin(150deg)
cos(180deg)
wary root
i used this
surreal crystal
tan(120) is not -1/2 * 2/sqrt(3)
surreal crystal
yes
wary root
🤑 im lossing my mind
surreal crystal
same
wary root
is that the only thing that i did wrong
surreal crystal
i think you got them all backwards
wary root
sin is -1/2
surreal crystal
yes
wary root
this is how i feel rn
wary root
ok, now i will try and hopefully get the right answer
i just started trig, dont scare me with words like, arctan
wary root
surreal crystal
step 2 to 3 in pic is actually unclear to me what you did
wary root
step 2 is turning the tan, sin, and cos into numbers
and step 3 is multiplying inside the parenthesis
based on the unit circle and what tan, sin, and cos equal
surreal crystal
this doesnt change anything, but sin(150)=1/2 actually oops not -1/2
1/2*
im losing my mind, sorry
wary root
but it's squared
surreal crystal
yeah, it doesnt change anything i just needed to correct myself
wary root
ima just go back in time and make sure trig never becomes a thing
but the tan part is correct yes
surreal crystal
lol no im just running tired always
everything is good you just have sign issues
note that squaring reals is always non negative
wary root
.
surreal crystal
like (-sqrt(3))^2 is not -3, (-1/2)^2 is not -1/4, and (-1)^2 is not -1
surreal crystal
like (-sqrt(3))^2 is not -3, (-1/2)^2 is not -1/4, and (-1)^2 is not -1
thats your issue, everything else is good
wary root
oh, the 2's cancel
surreal crystal
yeah
surreal crystal
yes
wary root
and (-1/2)^2 is 1/4
surreal crystal
yes
wary root
and (-1)^2 is 1
wary root
tysm and sorry for making 1000 minor mistakes
wary root
all good
surreal crystal
its good
wary root
well, thanks again, and sorry for making this take longer than it should have
surreal crystal
well, thanks again, and sorry for making this take longer than it should have
thats my fault probably lol
np
wary root
.close
weak monolith
Help
wary root
i think not since the plane has to be moving to take off?
and if the conveyor belt is matching the speed of the plane the entire time, its technically not moving?
weak monolith
and if the conveyor belt is matching the speed of the plane the entire time, its...
its not like a car
wary root
but doesn't wind have to like go over and under the wings for a plane to take off, thrust i think it is, and if it's "stationary" and moving in place, then no wind is moving over or under the wings?
naive valley
if you were somehow able to lift the airplane slightly off the ground and spin its wheels, that wouldn't make it take off, right? this is basically the same thing
lone heartBOT
@weak monolith Has your question been resolved?
surreal crystal
it cant, there is no motion relative to air, so we do not generate lift force
if we put it in a wind tunnel, then it will generate lift and take off, without moving
weak monolith
it cant, there is no motion relative to air, so we do not generate lift force
How will there be no motion relative to the air?
surreal crystal
its not stated, but im assuming the air is approximately stationary like it usually is
heres an image i shamelessly ripped off the internet
we essentially use terminal velocity to our advantage
we need the air to move faster than the terminal velocity of our wings, and then even more to counteract the weight of the plane
surreal crystal
we need the air to move faster than the terminal velocity of our wings, and then...
its wayy deeper than this, but this is the gist
and the force of air resistance is F=\rho v, where v is our velocity through a fluid, and \rho is the drag coefficient
weak monolith
its not stated, but im assuming the air is approximately stationary like it usua...
No but you're assuming that the plane won't move forward
under those conditions
why?
surreal crystal
it matches the speed of the plane
weak monolith
No
surreal crystal
superposition principle, +v-v=0, so there is no speed
surreal crystal
right
and for the plane's wings to function, we need to actually move through a fluid
weak monolith
and for the plane's wings to function, we need to actually move through a fluid
no one denies this
but you're claiming that the plane won't be able to move forward under those conditions
surreal crystal
no, by definition the speed of the belt cancels the speed of the plane
weak monolith
thinking that the treadmill can provide fiction against the plane
surreal crystal
its given in the problem
weak monolith
no, by definition the speed of the belt cancels the speed of the plane
No
It never says that the plane is stationnary in the question
It only says that the treadmill matches the speed of the wheels
but in the opposite direction
surreal crystal
it specifically says that the speed of the belt matches the speed of the plane, but in the opposite direction
surreal crystal
so if the plane is velocity v, then the treadmill is velocity -v
i meant relative to us
the ground, so to speak
weak monolith
yes, but the treadmill can't tie the plane to it in an inertial frame
it can't cause frictions on the wheels to make them slow down the plane
surreal crystal
inertia is relative
surreal crystal
thats pretty rude
weak monolith
thats pretty rude
srry
surreal crystal
yes, but the treadmill can't tie the plane to it in an inertial frame
okay, so what do you mean by this, exactly?
marsh fulcrum
weak monolith
okay, so what do you mean by this, exactly?
The plane can always go faster than the treadmill, and thus the treadmill will have to accelerate ad infinitum
As such, the imparted draft generated by the belt moving at a high speed would generate a strong head-wind, thereby assisting the lifting force of the wings
surreal crystal
so we need to assume the wind is going some velocity to actually try this problem, lets say, relative to the ground the wind isnt moving
otherwise we cant answer the problem, also i think this is implied
v_{ground}=v_{air}
weak monolith
so we need to assume the wind is going some velocity to actually try this proble...
No no, the enviromental wind is 0
but the treadmill wiill generate wind
surreal crystal
why would the treadmill generate wind?
weak monolith
because the two will be moving at basically ultra high speeds
weak monolith
why would the treadmill *generate* wind?
Well, the conveyor belt will have to be moving extremely fast, because it has to try and match the speed of the wheels. The plane can always surpass that however, since the friction of the belt has no impact on its thrust. Thus, the wheels will keep going faster and faster, just like the treadmill. Because of these extremely high speeds, the belt will generate a strong head-wind
surreal crystal
v_{ground}=v\_{air}
the plane is moving v_{plane-belt} relative to the belt, and the belt is moving v_{belt-ground} relative to the ground, but it is given that v_{plane-belt}=-v_{belt-ground}, so
v_{plane-ground}=v_{plane-belt}+v_{belt-ground}=0
then, v_{plane-air} = v_{plane-ground} +v_{ground-air}=0+0.
but v_{ground-ground}=v_{ground-air}=0
so, v_{plane-air}=0
drag is given by F=\rho v, so the drag of the plane through the air is F=\rho *0=0
so there is no drag force, assuming the air is motionless relative to the ground
weak monolith
the plane is moving v\_{plane-belt} relative to the belt, and the belt is moving...
cool, but you forgot about the wind generated by the belt
surreal crystal
the belt wont generate wind
if the belt generated wind and blew it up at the plane, then we would have lift, but that is not the case, its not stated
the problem doesnt make much sense if the belt contributed, of course blowing wind at the plane will generate lift, the problem is focused on whether this plane could be stationary and undergo lift
surreal crystal
the problem doesnt make much sense if the belt contributed, of course blowing wi...
and the answer to this is the plane must travel through a fluid to undergo lift, from moving through the air or the belt generating air somehow, the answer they want is that the plane has to actually move through a fluid
lone heartBOT
@weak monolith Has your question been resolved?
weak monolith
the problem doesnt make much sense if the belt contributed, of course blowing wi...
It can be stationnary and fly indeed
if you have enough airflow
surreal crystal
yes
if you put it in a wind tunnel for instance
it just has to move through a fluid
indigo anchor
lone heartBOT
indigo anchor
hey i dont get how to do this question
inner haven
Average rate of change formula is f(b)-f(a)/b-a
fresh field
$$ rate_{avg} = \frac{(f(b) - f(a))}{(b-a)} $$
indigo anchor
just asking for derivative
?
indigo anchor
im just confused what to do with only one x point and one y point
hollow vapor
use this
ocean sealBOT
Anurag.[UdayS]
inner haven
im just confused what to do with only one x point and one y point
No they’re a and b values
fresh field
I'm seeing two points in question 1
indigo anchor
where
fresh field
im just confused what to do with only one x point and one y point
For question two, you have to know either limits or derivatives
indigo anchor
where are the two points
fresh field
Correct
hollow vapor
theyre x values not points
indigo anchor
sorry I only see [0,1]
indigo anchor
i get how to find averagerate of change but wheres the b values
fresh field
sorry I only see [0,1]
Yes, there are end points. "Terminal X values" would be a better phrase as Sho is suggesting , but we just call me points
hollow vapor
i get how to find averagerate of change but wheres the b values
0 and 1 are your a and b
alpine sable
trying to compute the chance that for 3 randomly placed points on the perimeter of a circle, the triangle formed will contain the center but my answer seems to be way off and I'm not sure why
alpine sable
I figured I could fix the first point, then calculate the avg chance for all possible placements of the 2nd point given that the 3rd point is randomly placed. i.e we have points a,b,c. a goes at the top of the circle. we then see that for any placement of b the probability that a randomly placed c will form such a triangle is:
length(a-b arc) / (circumference - length(a-b arc))
so I tried to integrate over this with:
$\int _0^{\pi }\frac{x}{2\pi -x}dx$
ocean sealBOT
Valor
alpine sable
since I'm using a circle of radius 1 as reference
and divide the result by pi
but this is giving me the wrong answer and I'm not 100% sure why
lethal pelican
trying to compute the chance that for 3 randomly placed points on the perimeter ...
there’s a 3b1b video on this exact question I think 🤯
lone heartBOT
@alpine sable Has your question been resolved?
alpine sable
there’s a 3b1b video on this exact question I think 🤯
checked out the video and he seems to use expected value for the placement of point b instead of integrating over all possible placements of b which is a really smart way to go about it
but I don't really get why my formula is giving me an incorrect value
looks like I was setting up the integral wrong. it's
$\frac{\left(\int _0^{\pi }\frac{x}{2\pi }dx\right)}{\pi }$
ocean sealBOT
Valor
lone heartBOT
lone heartBOT
Available help channel!
hollow vapor
lone heartBOT
hollow vapor
how do i do part b
empty moth
SHOOOO
hollow vapor
AYOOOO
empty moth
you know I always got ur back
hollow vapor
twas my a
hollow vapor
💀 what
hollow vapor
:,)
hollow vapor
its allg
empty moth
I don't have a better explanation
hollow vapor
this q is weird af
dusk pawn
What is the coefficient of x^1 in the expansion of (3+x)^14. Can anyone tell me what this is supposed to be I honestly thought it was 3+x^14
dusk pawn
I also tried putting 126 but it doesnt take that either
hollow vapor
binomial expansion?
nCk x (a)^k(b)^(n-k)
or you can just
yknow
pascal’s triangle
empty moth
binomial expansion?
do you even need that here cuz we are selecting only one x
hollow vapor
do you even need that here cuz we are selecting only one x
true
patent fractal
Hello. I'm doing a stats course on coursera. it's self paced and all. I came across this problem that I can't seem to work out, so I would like pieces of advice on how to tackle it
The problem goes like this
A student takes a multiple choice test with 20
questions. Each question has 5 possible answers, only one of which is correct.
b) If a student answers each question at random, what is the probability that they will answer at least 14
questions correctly? Round your answer to seven decimal places. Store your answer in p2.b.
My reasoning: Since I have to calculate the probability of students answering at least 14 questions or more at random, I can just use a sum of combinatorics from k = 14 to n = 20 divided by the sum of combinatorics from k = 0 to n = 20. I thought this was a valid answer because I can interpret the problem as the questions being chosen are the ones answered correctly. So, it'd just be a matter of adding up sums of combinatorics, but coursera doesn't accept my answer as valid
By the way, we have only covered combinatorics, permutations and probability axioms.
serene junco
Hello. I'm doing a stats course on coursera. it's self paced and all. I came acr...
this sounds vaguely correct, but I'm not exactly sure what you mean by "the sum of combinatorics from k=14 to n=20"
are you referring to a binomial probability distribution?
patent fractal
like adding up nCr(20,14)+nCr(20,15)+...+nCr(20,20)
patent fractal
are you referring to a binomial probability distribution?
yeah I think so? Solutions on the internet mention it but we havent covered it yet on the course
since this is the first week
serene junco
hm, I'm not sure of any way to do this without binomial probability, unless they're trying to get you to intuit that for yourself, but that seems unfair
patent fractal
like my reasoning, without knowing how that distribution works, is that the probability could be calculated as: (nCr(20,14)+nCr(20,15)+...+nCr(20,20))/(nCr(20,0)+nCr(20,15)+...+nCr(20,20))
patent fractal
hm, I'm not sure of any way to do this without binomial probability, unless they...
yeah. I thought the same thing haha
well
I think I'll just come back to it later when I learn how that distribution works
serene junco
I can show you the basic idea, if you understand how combinations work it isn't too tough
serene junco
so when you do nCr(20,14), that is not a probability, but rather it is the number of groups of 14 problems out of 20 that you might get correct
in particular, nCr(20,14) = 38760, meaning there are 38760 different possible groups of 14 problems out of a total of 20
But it says nothing of the probability of getting 14 questions right
The probability of getting exactly 14 questions right would be nCr(20,14) * (0.2)^14 * (0.8)^6
0.2 is the probability of getting any single question right, and 0.8 is the probability of getting any single question wrong
If you get 14 questions right, you have gotten 20-14=6 questions wrong
which is where the exponents come from
Does this make sense so far?
patent fractal
yeah, but doesnt nCr(20,14) already take care of the wrong questions?
serene junco
No, nCr(20,14) is just the number of possible groups of 14 questions you might have gotten correct
Think of it in the opposite order:
What's the probability you get (specifically) the first 14 questions right and the last 6 questions wrong?
This would be (0.2)^14 * (0.8)^6
But you could have gotten any 14 questions right, not just necessarily the first 14
So we need to multiply by the number of possible groups of 14 questions you could have gotten right
which is nCr(20,14)
patent fractal
I think I get it, so the (0.2)^14 * (0.8)^6 is a conditional probability that takes into account the (14 questions right and the rest ebing wrong), and we add all the possible combinations this couldve happened with nCr(20,14)
serene junco
yes, there are nCr(20,14) different orders in which you could have gotten 14 questions right and 6 wrong
patent fractal
ahh ok
serene junco
(0.2)^14 * (0.8)^6 only accounts for one possibility, we need to multiply by nCr(20,14) to count all of them
patent fractal
.close
lone heartBOT
lone heartBOT
Available help channel!
cloud geyser
isn't having the same dimension enough for two vector spaces to be isomorphic . Bcoz the other condition is that there must be a invertible linear map between them , but i feel like if they have the same dimensions then they should always be a way for us to define a linear invertible map between them . Am i mistaken , if i am pls provide examples to clarify my confusion . My example is P m(x) and F (m+1)
tardy stag
idk what those symbols mean but an easy example of two vector spaces that have the same dimension but aren't isomorphic are R and Z/2Z
both are one dimensional but they're very different
cloud geyser
idk what those symbols mean but an easy example of two vector spaces that have t...
Z/2Z ??
cloud geyser
idk what those symbols mean but an easy example of two vector spaces that have t...
polynomial in one variable with atmost m degree
cloud geyser
Z/2Z being {0, 1}
wait , how is that a vector space ?? if 1 exists then -1 will also exist due to additive identity or scalar multiplication .
tardy stag
it's something called a finite field. but if you're only dealing with real vector spaces, then yes all real vector spaces of the same dimensions are isomorphic to each other
surreal crystal
Z/2Z isnt a vector space, we specifically count Z as a field
scalars have to draw from R or C
i think we say its a module over a ring or field in these cases
rose sigil
scalars *have* to draw from R or C
hm?
cloud geyser
it's something called a finite field. but if you're only dealing with real vecto...
is R2 and C isomorphic ? wait can even vector space and field be isomorphic ?
hushed locust
scalars *have* to draw from R or C
vector spaces can use scalars from any field
surreal crystal
vector spaces can use scalars from any field
can they? i had a big debate with my lin al prof and he told me they werent called vector fields, he was adament they were modules instead
surreal crystal
is R2 and C isomorphic ? wait can even vector space and field be isomorphic ?
yes
tardy stag
is R2 and C isomorphic ? wait can even vector space and field be isomorphic ?
as vector spaces, yes. C also comes equipped with multiplication which is neat
rose sigil
Z/2Z isnt a vector space, we specifically count Z as a field
any field is a vector space over itself
hushed locust
can they? i had a big debate with my lin al prof and he told me they werent call...
a module can be over any ring, as opposed to vector spaces which are over fields
tardy stag
can they? i had a big debate with my lin al prof and he told me they werent call...
modules are "vector spaces" over rings
surreal crystal
"In a module, the scalars need only be a ring, so the module concept represents a significant generalization" https://en.wikipedia.org/wiki/Module_(mathematics)
wiki acknowledges that modules are generalizations, so not all modules are vector spaces
so my language is bad, but right idea
cloud geyser
they are isomorphic over real vector spaces, but not complex
so i assume that for any two vector spaces to be isomorphic , they have to the have same field ?
surreal crystal
probably
i hope i just didnt derail everything with something meaningless, sorry if so
cloud geyser
i hope i just didnt derail everything with something meaningless, sorry if so
hmmm , its fine
.close
lone heartBOT
surreal crystal
you dont need to close it
cloud geyser
.reopen
lone heartBOT
✅
cloud geyser
you dont need to close it
why ?
cloud geyser
https://en.wikipedia.org/wiki/Dimension_theorem_for_vector_spaces
I am sorry but how does this relate to isomorphism ?
hushed locust
so i assume that for any two vector spaces to be isomorphic , they have to the h...
basically the answer is yes, we require an isomorphism (like any linear transformation) to be between two vector spaces over the same field, otherwise the linearity condition doesn't make sense
surreal crystal
I am sorry but how does this relate to isomorphism ?
maybe im being crazy, but because morphisms are linear maps, this theorem gives us bijectivity, then iso is straightforward to show by linearity
so, vector spaces of the same dim are iso
cloud geyser
basically the answer is yes, we require an isomorphism (like any linear transfor...
thx for the source
cloud geyser
maybe im being crazy, but because morphisms are linear maps, this theorem gives ...
isomorphism (from my knowledge) is true if there is 'any' invertible linear map between the vector spaces , therefore here is the linear map is not defined by default , so i believe it is not straightforward like verifying bijectivity .(correct me if i am wrong , i am new to linear algebra )
surreal crystal
you are right
a bijective morphism by definition has an inverse, i believe
yes it has to
cloud geyser
a bijective morphism by definition has an inverse, i believe
yes i am aware that bijectivity <=> inverse , my point was the linear map is not defined so it is not straightforward to prove it
surreal crystal
we just want an invertible map, we dont need a specific one, we just need to know there is one
oh i see what you mean
just map the bases to the other bases
there will always be a map by the dim theorem
lone heartBOT
lone heartBOT
Available help channel!
swift raptor
can i have some revision on the cartesian plane?
lone heartBOT
can i have some revision on the cartesian plane?
swift raptor
so like i dont have an image
but like what i need help on
is like gradient
and the y intercept
i have some understanding
<@&286206848099549185>
gritty bramble
and the y intercept
y intercept is where the graph cuts y-axis
gritty bramble
is like gradient
gradient is just how steep the line is
for example here the y-intercept is 2 and the gradient is 1
lone heartBOT
@swift raptor Has your question been resolved?
mortal sandal
Hi! I have some year level 8 maths work (algebra) and i'm struggling to understand what it means. I'm not the smartest so don't judge
Moving Out
Background: Andrew, Claire, and Fiona are moving out of their homes into a shared rental property. They are working a budget to determine how much money they will require to cover their bills each month.
Electricity:
Electricity is charged at 4 cents per unit plus a $50 connection cost per month.
a) Write an expression: Using X to represent the number of units of electricity used, create an expression that will determine the monthly cost for electricity.
b) Construct: Construct a table of values for the cost from 0 to 10 000 units going up in increments of 500.
c) Graph: Graph the cost from 0 to 10 000 units using Excel or on the following page.
d)** Write an expression:** Andrew, Claire and Fiona will split the electricity cost equally. Alter the expression from (a) so that it calculates how much each person must pay.
e) Calculate the cost: If they use 5000 units of electricity, how much will each pay, correct to the nearest cent?
f) Calculate budgeted usage: They budget $118 each for electricity for the year. How many units can they use without going over their budget?
-# If there are spelling mistakes just tell me, i scanned the letters so it could be wrong.
raw jetty
have you done any of these?
lone heartBOT
@mortal sandal Has your question been resolved?
mortal sandal
have you done any of these?
No i just reached this set of questions
lone heartBOT
No i just reached this set of questions
mortal sandal
sorry for the late response i was on a run
lone heartBOT
@mortal sandal Has your question been resolved?
gritty bramble
No i just reached this set of questions
u tried anything after that?
ocean sealBOT
Aya
grave wolf
So you have problems multiplying the matrices to get the final expression?
lone heartBOT
@cerulean grove Has your question been resolved?
grave wolf
Sorry its not my first language :/
I think I have seen the formula once or twice but its not that common as far as I know
Its usally a formula thats given
hmm i see
Sorry i cant help. Sometimes you have to be really creative to prove something and think outside the box, using equations like this
kindred nexus
lone heartBOT
kindred nexus
hai guys pls helppllspsspsls
coral seal
you may find it easier if you extend all the lines
minor hearth
Bro another question help pls
small lance
lone heartBOT
@minor hearth Has your question been resolved?
minor hearth
Yes
strange fractal
lone heartBOT
strange fractal
can i have some help with part B
coral flower
can i have some help with part B
first consider the case
P(1) then P(k) and P(k+1)
basic induction stuff
strange fractal
first consider the case
P(1) then P(k) and P(k+1)
i did
but like
idk how to start like
do i start with let n = 1
1(3(1)-1) = 2
and then 6(1) - 4 also = 2
so lhs = rhs and therefore n=1 is true
lone heartBOT
@strange fractal Has your question been resolved?
strange fractal
.close
long wing
lone heartBOT
long wing
can anyone explain what question c means
i have answer for a and b already, isn't it just asking for b?
lone heartBOT
@long wing Has your question been resolved?
lone heartBOT
@long wing Has your question been resolved?
long ledge
i have answer for a and b already, isn't it just asking for b?
i think so
ocean sealBOT
l'agit
lone heartBOT
@native kraken Has your question been resolved?
formal wraith
Hey, I did ask this question yesterday, but I didn't really understood what the people was trying to tell me so, here i go again:
formal wraith
Part C
People was telling me about the vertex, function stuff etc
I'm on precalc and so far I've seen so basic stuff, not even functions
formal wraith
So, having the statement, how can I solve it, without using vertex formula?
thanks in advance
lone heartBOT
@formal wraith Has your question been resolved?
valid trout
So, having the statement, how can I solve it, without using vertex formula?
Using calculus it would be somewhat simpler but without it I think the only way to solve it would be by transforming the equation into the form of h = a(t-b)^2+c
pallid scarab
ok so a random variable is just like the measure of some random event
say for example you roll the dice three times
and you want to measure the sum of all three rolls
well for a given outcome (a,b,c)
you know that the sum of those rolls is a+b+c
that's what the random variable we chose here does
call it X
X just takes the input of an outcome, (a,b,c)
well no matter what the outcome is, X will give you its measure
if you know how the event unfolded, you know X
so X takes the outcome (a,b,c)
and outputs a+b+c
X is just a map
from the set of possible outcomes
(Omega)
yep
so X is a map from Omega to the set of possible measures
didn't we just do it?
for any outcome (a,b,c)
it outputs a+b+c
well we always do
a random variable is a map
from the set of outcomes
to the set of "possible measures"
exactly
just like f:A->B
yeah sure
ok
Say I play heads or tails until I get tails
so what are the possible outcomes of that
(T)
(H,T)
(H,H,T)
yeah ok
yes
and now I want to know how many coin flips I needed to stop
that's gonna be a random variable X
the input is any outcome
(H,H,...,H,T)
and it outputs the length of the sequence
yep
well Omega is the set of outcomes
so X takes any outcome (aka an element of Omega) as an input
usually it can map to any set you want
sometimes you will hear of REAL random variables
meaning the outputs are real numbers
any function has to be correctly defined to exist
meaning f(x) exists for any input x possible
ah I see the confusion
again X1 and X2 are just, for any given outcome, the results of first and second roll
so if you take some outcome of two dice rolls (a,b)
X1(a,b) = a
(remember X1 and X2 are still functions)
technically we don't know what the outcome is
we know the possible outcomes
it's the elements of Omega
yeah
so take one such possible outcome for the 2 dice rolls
(a,b)
in that outcome, X1 is the first dice roll, so a
and X2 is the second dice roll, so b
X1(a,b) = a, X2(a,b) = b
ok
one of the possible outcomes is (3,3) (both dice rolls are 3)
then X1(3,3) = 3
X2(3,3) = 3
X1 is the first dice roll
what's the first dice roll in (3,3)
ok
yes
like any other function
for example
f(x) = x^2
you take a real number as an input
and the rule is you square it
here we're doing the same thing with X1 for example
you take an input of two dice rolls (a,b)
and the rule is you just take the first one
X1(a,b) = a
exactly
X2(a,b) = b
well that's the mapping here
(a,b) ---> a is how X1 works
(a,b) ---> b is how X2 works
well first you have to get the set of possible outcomes
they all look like (a,b)
with 1 <= a <= 6
same for b
then an outcome looks like (a,b)
so
how can you manipulate that possible outcome (a,b)
to give you the result of the first roll
where is the information you need
how do you obtain it
exactly
so you get how we got X1?
X1 is the map that takes an outcome (a,b)
and outputs the first entry
always 2 steps:
- what are the possible outcomes
(find omega)
well
you have to "describe" them AS random variables
so you have to write them as functions
from Omega
to a certain codomain set
lone heartBOT
@native kraken Has your question been resolved?
lone heartBOT
@native kraken Has your question been resolved?
lost gazelle
hey, where can i find a rigorous proof of why the continued fraction form of a rational number is finite? (e.g.: [a0;a1,a2,...,an] meaning it does not continue on infinitely)
alpine sable
my guess is that you can reformulate this statement into saying that euclidean algorithm eventually halts (e.g. an infinite descent argument), but I'm too lazy to formulate here
lone heartBOT
@lost gazelle Has your question been resolved?
lone heartBOT
@lost gazelle Has your question been resolved?
lone heartBOT
@lost gazelle Has your question been resolved?
lyric dune
lone heartBOT
inner haven
What do you know about the FTOC
lyric dune
Derivative is inverse of integral
inner haven
so basically
f(t) = sin(t^2)
g(x) = x^2
if u find the derivatice of ur integral
u get sin((x^2)^2)*2x
lyric dune
Ok this problem is always this easy?
lyric dune
Sin x ^4 * 2x
lone heartBOT
@lyric dune Has your question been resolved?
storm marten
initially, there are 3^80 particles at the origin. At each step the particles are moved to points above the x axis as follows:
If there are n particles at any point (x,y) then floor(n/3) of them are moved to (x+1,y+1), floor(n/3) are moved to (x,y+1) and the remaining to (x-1,y+1).
After 80 steps, the number of particles at (79,80) is?
storm marten
no idea what to do or where to start
storm marten
then?
mortal trellis
that will hopefully give you some insight into the problem
kindred atlas
lone heartBOT
kindred atlas
I tried -infinity and dne but it didn't like that
But it looks like as x\to0 it will be negative infinity as slope 🤔
verbal blaze
$$\lim_{x \rightarrow 0^+} \frac{d}{dx} \frac{3}{\sqrt{x}} = \lim_{x \rightarrow 0^+} -\frac{3}{2} \frac{1}{\sqrt{x}^3} = - \infty$$
ocean sealBOT
Shuba
spiral shore
Hi
verbal blaze
I tried -infinity and dne but it didn't like that
you're right, it should be negative infinity 
kindred atlas
I also tried putting in -1.5 thinking it just wanted any possible slope but it was mad too
maybe something weird with cengage/webassign (hw provider)
verbal blaze
I hate these online tests. Even if you have the right answer, if you don't put it in the box the way it wants you to, it'll mark it wrong.
kindred atlas
yup
I'll ask tmmrw in person to people who are experts on dealing with this app XD
verbal blaze
"does not exist", "nde", "DNE", "undefined", "-infinity", "-\infty", who knows
lone heartBOT
lone heartBOT
Available help channel!
wicked vine
Hi I need help with series of numbers. Here is a example:
lone heartBOT
Hi I need help with series of numbers. Here is a example:
lone heartBOT
@wicked vine Has your question been resolved?
simple sphinx
anyone good at this part of maths?
Umm, what's are you supposed to analyze though? (What certain conditions are there, if any)
wicked vine
@simple sphinx you have to calculate the steps between every number
the sequence
and what the next one is going to be
vagrant heart
it looks like the order is +1, x3, +1, /2, so it's probably asking you to multiply by 3 again, in which case the answer would be 4 1/2
@wicked vine
upbeat wagon
Guys, give me some help here
wicked vine
it looks like the order is +1, x3, +1, /2, so it's probably asking you to multip...
let me look
@vagrant heart omg thats correct
i almost want to cry
vagrant heart
lol no worries
vagrant heart
what's more important is that you understand the logic behind what I did
wicked vine
what's more important is that you understand the logic behind what I did
not really
i wil ltell you why
vagrant heart
the only part of this question that requires thinking is un-simplifying the simplified fraction
so
lone heartBOT
@wicked vine Has your question been resolved?
frozen sentinel
what is the smallest 7 digit number from using 1, 3, 6, 6, 7, 8, 9 where adjacent digits are relatively prime?
obtuse knoll
have you tried solving it using a greedy strategy
obtuse knoll
i mean its more or less brute force depending on how many numbers you have to look at but i cant think of a better strategy
a greedy strategy is a strategy that always chooses the best-looking option at the moment
for example, the best number to put at the start is 1
frozen sentinel
yes
obtuse knoll
then the next digit you choose is the next lowest one that can go next to 1
so, 13...
when it doesnt work you backtrack
cause if you try it immediately you get 1376 and then you cant pick another number that is relatively prime, so you would delete 6 and try again
with a bigger digit
There are ways you can be smarter about it
like you see that there are the numbers 3,6,6,9, which cant be next to each other, and the only way you can space it out is to put each of them on every other digit
like 3x6x6x9 where x's are unknown digits; you can see that it won't work either cause you have to put 8 next to both 3 and 9
i think i figured out the answer. im not really supposed to just outright give you the answer but if you understand my logic then i dont think it would be too hard to figure it out
if you do think of something we can compare answers
lone heartBOT
@frozen sentinel Has your question been resolved?
delicate skiff
How do I do compute A = CR if the matrix is not 3x3?
lone heartBOT
How do I do compute A = CR if the matrix is not 3x3?
remote heron
How do I do compute A = CR if the matrix is not 3x3?
well, you know that C has to be orthogonal, right?
what does that imply about the dimensions of C
delicate skiff
well, you know that C has to be orthogonal, right?
3x3?
i think that C is 3x3 and R is 3x6
idk how to compute the values
remote heron
idk how to compute the values
check out gram schmidt
youll need a modified process
but its just padding
delicate skiff
check out gram schmidt
i seriously think this question is not supposed to be that complex this is just the 2nd week of lin algebra
remote heron
i seriously think this question is not supposed to be that complex this is just ...
its surprising that youre already doing matrix decompositions
but gram schmidt is very constructive fwiw, it tells you just what to do
lone heartBOT
@delicate skiff Has your question been resolved?
hollow bay
$\int_{-\infty}^{\infty} xe^{-x^2}dx$
lone heartBOT
$\int_{-\infty}^{\infty} xe^{-x^2}dx$
solemn juniper
infty
serene junco
$\int_{-\infty}^{\infty} xe^{-x^2}$
ocean sealBOT
tatpoj
solemn juniper
we did it reddit
hollow bay
lol
serene junco
you need \infty, not just infty
ocean sealBOT
wakamole
hollow bay
ty
serene junco
lol np
hollow bay
not sure how to begin.
split it into 2 integrals?
-inf to 0 and 0 to inf?
or -inf to x and x to inf
?
serene junco
I don't think there's any reason to do that, there's no discontinuity in this function
naive valley
you could compute $\int_a^b$ and then take the limit as $a \to -\infty$ and $b \to \infty$
ocean sealBOT
Bungo
hollow bay
nah
naive valley
or, you could show that $\int_0^\infty$ is finite and then use the fact that the integrand is odd
ocean sealBOT
Bungo
hollow bay
tru yeah im going to do $\int_a^b$ as lim goes to neg inf and pos inf
ocean sealBOT
wakamole
hollow bay
so i was aright
serene junco
u sub the -x^2?
this seems promising
hollow bay
you split it into two integrals
naive valley
you split it into two integrals
the integrand is odd, the result is gonna be zero as long as $\int_0^\infty$ is finite, so you really only need to compute that one
ocean sealBOT
Bungo
naive valley
but of course you can compute both if you want
naive valley
if $f(x) = x e^{-x^2}$, then $f(-x) = -f(x)$ (check this)
ocean sealBOT
Bungo
hollow bay
ah ok it's just a shortcut
i know
hollow bay
this seems promising
will that work or not?
serene junco
yes that substitution will work
serene junco
is this integration by parts
it's not integration by parts, just a u sub
hollow bay
it's not integration by parts, just a u sub
i got -1/2e^x^2 are my signs correct or not?
cause u is -x^2?
so when you put it back in is it e^-u?
so e ^x^2
serene junco
should be e^u if u is -x^2
hollow bay
so i got -1/2e^-x^2
serene junco
yeah
serene junco
so i got -1/2e^-x^2
hang on, if you have this
the whole thing is $\int -\frac{1}{2}e^u \dd u$
ocean sealBOT
tatpoj
serene junco
other than the bounds
hollow bay
oh right i still have t o integrat
wait
but that is the same thing
as the antiderrivative
serene junco
yes
hollow bay
so ya
i got for antderrivative -1/2e^-x^2
so what do i do wwith the t and a bounds now[
serene junco
oh okay I see what you mean now, you integrated and then converted u back to x, right?
serene junco
okay, so then $-\frac{1}{2}e^{-x^2} \vert _{-\infty}^\infty$
hollow bay
oh i thought i had to split it into two integrals
ocean sealBOT
tatpoj
serene junco
man I suck at latex
hollow bay
its fine i get it
but i had 2 integrals
$\vert_{t}^{a}$ and $\vert_{a}^{t}$ first one is neg inf second is a to inf
ocean sealBOT
wakamole
hollow bay
youre saying to just combine them..?
serene junco
Either way you're going to end up with the same expression
hollow bay
ok but i need to show work man
what do thety yeven equal
0?
i know they do bc i have the answer but how do you know if it is 0
serene junco
I don't think there's any reason to break it up, you can just say $\li{k}{\infty} [-\frac{1}{2}e^{-x^2}]\vert_{-k}^k$
ocean sealBOT
tatpoj
serene junco
when you substitute k and -k and subtract (like any other definite integral), you'll get $\li{k}{\infty} 0$, which is 0
ocean sealBOT
tatpoj
hollow bay
oh
naive valley
I don't think there's any reason to break it up, you can just say $\li{k}{\infty...
this isn't valid in general, consider $\int_{-\infty}^\infty x,dx$
ocean sealBOT
Bungo
naive valley
convergence of the symmetric integrals $\int_{-k}^k$ doesn't imply convergence of $\int_{-\infty}^\infty$ in general
ocean sealBOT
Bungo
serene junco
oh I suppose you're right
naive valley
convegence of the asymmetric integrals $\int_{a}^b$ does, though
ocean sealBOT
Bungo
hollow bay
convegence of the asymmetric integrals $\int_{a}^b$ does, though
can you tell me if I’m doing anything wrong here?
hollow bay
oh
naive valley
so it should just be u
naive valley
your integration bounds are wrong after the sub
hollow bay
the limits?
naive valley
yea, if u = x^2 + 1 then u can't be smaller than 1
hollow bay
oh
naive valley
so your integral should be from 1 to infinity, not -infinity to infinity
naive valley
the integrand is not a 1-1 function
you should probably break it into an integral from 0 to inf and an integral from -inf to 0
naive valley
yea that's generally the way you want to proceed for integrals that go from -inf to inf
hollow bay
and since the antiderrivative is ln |x^2 + 1 | it cant be anything less than 1
ok
the other guy told me not to
sry
did i do the antiderrivaitve right?
i think it is diverging right? because the -inf DNE???
naive valley
yea the antiderivative looks right
and yea it'll diverge, both integrals diverge (0 to infinity) and (-infinity to 0)
hollow bay
hmm
then what is the point of writing 2 integrals lol
you can clearly see it will diverge then
right?
naive valley
one way to see that intuitively is that for large x, the function is approximately x / x^2, which is 1/x
and the "tail" of 1/x is too heavy for convergence
naive valley
you can clearly see it will diverge then
yep that's exactly why you do it
hollow bay
ok... so integrals are -inf to 0 and 0 to inf?
naive valley
your integrals will have values +inf and -inf
hollow bay
how do you know what to make the intervals?
naive valley
and +inf - inf is undefined
hollow bay
why not like -inf to -3 and -3 to inf
naive valley
how do you know what to make the intervals?
you can make them -inf to a and a to inf for any real number a
0 is just convenient
but you'll get the same answer no matter what a you use
naive valley
it's just linearity of the integral:
$$\int_a^b = \int_a^c + \int_c^b$$
ocean sealBOT
Bungo
hollow bay
integral is linear?
naive valley
yep
hollow bay
that means it goes forever
naive valley
no it means you can compute pieces and add them up
naive valley
another flavor of linearity is $\int_a^b (f+g) = \int_a^b f + \int_a^b g$
ocean sealBOT
Bungo
naive valley
like peicewise
yep
naive valley
I have to go after this and take NyQuil I have Covid :/
😭
lone heartBOT
@hollow bay Has your question been resolved?
lone heartBOT
lone heartBOT
Available help channel!
candid shore
How would I find the percentile of this I know it is within one standard deviation but I dont know how to find the percentile
lone heartBOT
How would I find the percentile of this I know it is within one standard deviati...
@candid shore Has your question been resolved?
candid shore
<@&286206848099549185>
pulsar aspen
What is a downrange?
Are you sure the question asked for just the x axis and not the distance from origin?
Yeah
Do you have any other question?
lone heartBOT
lone heartBOT
Available help channel!
azure axle
help
proven leaf
small lance
small lance
Is this the new sticker? Haven’t seen it before
proven leaf
Is this the new sticker? Haven’t seen it before
no it's been around for a minute :)
azure axle
can someone explain why it equals 1 and -1
proven leaf
what is -1 times -1?
buoyant saddle
(-1)^2 = 1
