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#help-38
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split chasm
why can't r(x) be 0
gleaming oxide
ah, 9th grade ncert
gleaming oxide
wait really
potent pewter
Yea
split chasm
r(x) being 0 is an indication that the divisor is a factor, divides perfectly
potent pewter
So tell me does g(x) is not equal to 0 means constant 0 or polynomial can't be zero if first one then how does r be equal to 0
But it says that for every 2 polynomials
And not every polynomial divides the other perfectly
fair forge
if i want to swap C_2 and C_3, how will that affect the det() calculation?
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if i want to swap C_2 and C_3, how will that affect the det() calculation?
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pulsar dust
if i want to swap C_2 and C_3, how will that affect the det() calculation?
theres a sign change
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quiet turret
what is the 2nd derivative test and how do we use it
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what is the 2nd derivative test and how do we use it
sharp yarrow
what is the 2nd derivative test and how do we use it
the 2nd derivative test is a way to find out whether a stationary point is a local minimum or a local maximum
quiet turret
the 2nd derivative test is a way to find out whether a stationary point is a loc...
mmm and how do i do it
sharp yarrow
consider function f(x)
f'(a) = 0 so there is a stationary point at x = a
if f''(a) > 0, the stationary point is a minimum
if f''(a) < 0, the stationary point is a maximum
the 2nd derivative tells you the concavity of a function
clear cloud
what about solving f'x >= 0 instead ?
sharp yarrow
what about solving f'x >= 0 instead ?
he's asking about the 2nd derivative test?
clear cloud
he's asking about the 2nd derivative test?
oh yeah nvm
quiet turret
if f''(a) > 0, the stationary point is a minimum
whats f"(a)
wait
can i give you an example function
and you guide me through it?
this is my function
sharp yarrow
that's a slightly complex function to give as an example, should we try a simpler one?
sharp yarrow
let's try f(x) = x^3
mild oxide
Yo
quiet turret
let's try f(x) = x^3
okok so what do we do
sharp yarrow
find f'(x)
quiet turret
f'(x) = 2x^2
sharp yarrow
multiply by the power then subtract 1 from the power
sharp yarrow
the power is 3
quiet turret
3x^2 = 0
sharp yarrow
solve it
quiet turret
x = 0?
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chilly raven
Ok, so I am currently calculating a weighted "mean" of sorts and I want to normalize the data so it outputs only a specific range (1-5) but retains the "information" of the data set. Here is how I have it setup
Factor 1: weight = 1
Factor 2: weight = 1
Factor 3: weight = 99
Total weight = 101
Formula: computed mean = Factor 1 * (weight / total weight) + Factor 2 * (weight / total weight)... you get the point
I want that formula to output from 1-5, instead of say -60 or 150 or whatever values may arise from it. Is this possible? Sorry for asking the question so terribly, I can't even wrap my own head around what I should do.
chilly raven
For some more context, I want it to be from 1-5 because I want specific cutoffs from ranges 1-2, 2-3, 3-4, 4-5 exclusive which will act as decision points or "if" branches.
shrewd ridge
chilly raven
ok one sec
This is what I have
The way it works is that it takes the largest of the two values (10 in this case) and divides it by the smallest (5) and then multiplies it by its weight (0.1111) for the first one
Then adds them all up, from a starting point of 0
But I want it to be normalized so the data makes sense and is useful, which is why I thought a range of 1-5 would be ideal
Do you have any ideas of what I might need to do?
oh, I forgot to mention, in this case zScore is the "weighted mean"
<@&286206848099549185>
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cobalt cloak
when asked to prove a system of 3 equations is consistent, i only have to show the determinant =/= zero, but asked to prove that it has a unique solution, i need det =/= 0 and to express the solutions?
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when asked to prove a system of 3 equations is consistent, i only have to show t...
@cobalt cloak Has your question been resolved?
sharp yarrow
when asked to prove a system of 3 equations is consistent, i only have to show t...
for a system of equations to have a unique solution, you need to prove they are consistent and that they are linearly dependent
cobalt cloak
for a system of equations to have a unique solution, you need to prove they are ...
alr ty
umbral abyss
Say we have 3^(1/3) then there exists a k^(1/k) = 3^(1/3) where k does not equal 3 and is a positive real number. What is k?
umbral abyss
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void jacinth
Hello
void jacinth
Is there an easier way to do this question? Within the context of the textbook I’m not even supposed to know how to use log yet
Since the answer is a decimal value I couldn’t have trial and error tested it
split chasm
if you have a calc you can trial and error
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craggy narwhal
how would i get started with this?
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how would i get started with this?
nova spire
how would i get started with this?
hi, use a polynomial division algorithm to find gcd(n+2008,n^2+2008) = gcd(n+2008, ?) with ? that doesn't depend on n.
or rather if (n+2008) divides n^2+2008, then n+2008 has to divide another number that doesn't depend on n
you'll be able to do the same with n+2009.
worldly wing
hint: $\frac{n^2+2008}{n+2008}=\frac{n^2-2008^2+2008+2008^2}{n+2008}=n-2008+\frac{2008\cdot 2009}{n+2008}$
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kheerii
craggy narwhal
hi, use a polynomial division algorithm to find gcd(n+2008,n^2+2008) = gcd(n+200...
i don't know what a polynomial division algorithm is
craggy narwhal
hint: $\frac{n^2+2008}{n+2008}=\frac{n^2-2008^2+2008+2008^2}{n+2008}=n-2008+\fra...
- how did you come up with this?
- what would be my next steps from here?
worldly wing
what I wrote is an example of a poly division algorithm
worldly wing
1. how did you come up with this?
2. what would be my next steps from here?
I came up with this because having n in both the numerator and denominator isn't ideal
and your next step should be that for $n^2+2008$ to be divisible by $n+2008$, $\frac{n^2+2008}{n+2008}=n-2008+\frac{2008\cdot 2009}{n+2008}$ must be an integer, so $\frac{2008\cdot 2009}{n+2008}$ must be an integer
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kheerii
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fair wren
how does one start this question? multiply by the reciprocal of the graph idk
left oriole
you can simplify the fraction to maybe make it easier
fair wren
you can simplify the fraction to maybe make it easier
how would you simplify it?
left oriole
do the division
left oriole
(x+3)/x = 1 + 3/x right?
fair wren
oh yeah
left oriole
yes and x/x is 1 (for any nonzero x)
fair wren
you still have the issue of 3/0
fair wren
trying to think
im not sure
i know there is way
to get rid of the x
from the demon at least
fair wren
well on the function itself there wouldnt be a value
but i dont know if that means there isn't a limit
oh
it doesnt exist
because
the left and right limits as x -> 0 are going to opposite infinities
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fair wren
<@&286206848099549185>
winged totem
<@&286206848099549185>
what is the matter?
fair wren
how does one start this question? multiply by the reciprocal of the graph idk
@winged totem
winged totem
<@1194593681588629538>
result: 1
mild oxide
,w limit x tends to 0 (x+3)/(x)
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fair wren
Yeah I also got DNE
mild oxide
Bcz if it tends 0- denominator is -ve and numerator is +ve so -ve inf. And if it tends 0+ whole thing positive so + inf
fair wren
yeah true
winged totem
Bcz if it tends 0- denominator is -ve and numerator is +ve so -ve inf. And if it...
yeah, right
fair wren
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vague orbit
I don't even know what to say about this problem. I'm supposed to be finding the definite integral of a function, but somewhere my math got all messed up
vague orbit
this is what the original problem looks like btw
and I got 82.14
and my calculator got 307.06
I need to find the definite integral of this function at different points, likely using substitution at some point to do that
or wait, no, I need the average
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vague orbit
I tried again, but I got the same answer as last time
<@&286206848099549185>
I'm going to fucking explode
vernal briar
You changed variables incorrectly
u = pi/6 * t
du = pi/6 * dt
That's correct
And your integrand is correct, but when you replace dt with du, you added the constant pi/6 in the front
But you need to add 6/pi in the front, not pi/6
dt = 6/pi * du
@vague orbit
vague orbit
thank you for your help. I did eventually figure out that that was what I was doing wrong. Took me forever to, though. It conflicts with what I had previously thought I'd been taught. It's been very frustrating
vernal briar
so you're good now?
vague orbit
as far as this problem goes, yes
now I got however many more ahead of me, somewhere in the dozens
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sterile isle
how can i convert 1000 random observations from a Unif(90,170) distribution into Uni(0,1)
knotty locust
how can i convert 1000 random observations from a Unif(90,170) distribution into...
What is the size of the range that the observations from the Unif(90, 170) can take on?
sterile isle
What is the size of the range that the observations from the Unif(90, 170) can t...
60?
knotty locust
No
sterile isle
instead of U(0,1), use U(0,0.8)?
knotty locust
No
We are trying to convert our observations from Unif(90,170) to Unif(0,1)
Imagine just transforming the interval (90,170) to try to make it match (0,1)
sterile isle
yeh, theres a hint in the solution sheet, saying to use CDF inversion
but that just gets me the same numbers
knotty locust
Its 80 times to big, so we divide its length by 80, but then it still doesnt work since the interval becomes
(90/80, 170/80)
which isn't (0,1)
But, they both have length 1
so what can we do now?
sterile isle
devide by 8?
knotty locust
we already divided by 80
the length of the interval is 1 now, which is correct. its just in the wrong place
sterile isle
0,1
knotty locust
So our observations were distributed across (90,170)
but if we divide each observation by 80, then subtract 90/80
they're now distributed across (0,1)
sterile isle
right i see i see, thank you very much
strange they asked me to do cdf inversion tho
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hardy jolt
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sharp yarrow
which part?
hardy jolt
a
sharp yarrow
what is the equation? it's cut off
sharp yarrow
do you know what part a) is asking you?
hardy jolt
no
sharp yarrow
it's basically saying to rewrite that equation (x = 4000 - 10p) into p = ...
so instead of x = 4000 - 10p, we want p = something in terms of x
hardy jolt
ok
sharp yarrow
do you have an idea of where to start now?
hardy jolt
I'm lost ngl
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wraith hinge
how do i solve this type of matrices question?
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how do i solve this type of matrices question?
vernal briar
$(AB)^t = ?$
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RedstonePlayz09
vernal briar
How would you write this differently?
wraith hinge
Are you using a book?
No, i used an youtube video
vernal briar
You're only using youtube videos?
wraith hinge
You're only using youtube videos?
Yeah
vernal briar
$(AB)^t = B^tA^t$
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RedstonePlayz09
vernal briar
This is a known equality
wraith hinge
So, i know T = is Tranpose
vernal briar
**RedstonePlayz09**
Use this
vernal briar
Multiply what
vernal briar
(Sorry, if i’m not smart enough 😭)
Don't apologize
Everything is fine
I'm trying to help
wraith hinge
Okay, so for my knowledge i know T we swirch the row with the columns then we multiply the matrices
(Unless, there’s another way to solve this)
vernal briar
Transpose is not switching rows and columns
It's kind of like "reflecting" along the main diagonal
Oh
I see what you mean
Yeah I guess you can say that
vernal briar
Okay, so for my knowledge i know T we swirch the row with the columns then we mu...
You are not given the matrices
You are given what AB is
You need to find B^tA^t
vernal briar
**RedstonePlayz09**
And here is the relation between the two. Use this
wraith hinge
And here is the relation between the two. Use this
okay gimme a sec
vernal briar
No
wraith hinge
huh 🥲
vernal briar
You don't know what A is, and you don't know what B is.
What you DO know is that A * B is this:
A times B
And you want to use this to find B^t times A^t
AB is A * B. Just like how 2x is 2 * x.
We don't write the multiplication sign every time
vernal briar
Yes
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vernal briar
Np
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dawn nimbus
Number 6
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blissful geyser
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tulip violet
Honestly, that's really hard to follow
wind cloak
is (C) correct?
Yes it is indeed correct'
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blissful geyser
Honestly, that's really hard to follow
you dont have to follow i just need smone to confirm that c is correct
blissful geyser
Yes it is indeed correct'
thx
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shell zodiac
When is open again?
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When is open again?
broken pilot
When is open again?
When is what open?
broken pilot
The bot mat?
This channel is now open
shell zodiac
ok thanks ^^
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shell zodiac
ooohh..
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somber sparrow
I don't understand
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I don't understand
mighty canyon
Hint: think about the way to get 504 using 12 and 36
mighty canyon
for example, 30 pans of 12 bars and 4 pan of 36 bars yields 504 bars
mighty canyon
yes. and that should give you a line or relation for b)
somber sparrow
is there another way to do that without using table of values
mighty canyon
I mean, you can just put it into the linear equation straightaway
you know that you need some amount of 12 bar pans and some amount of 36 bar pans
somber sparrow
yes\
mighty canyon
and you know total amount of bar you need
somber sparrow
12x + 36y = 504
mighty canyon
yes
somber sparrow
and then i can put random numbers in
mighty canyon
sure
somber sparrow
i mean uh
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proper sandal
I'm not exactly sure what it wants me to find here, Im not sure what Z of alpha 0.057 is
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I'm not exactly sure what it wants me to find here, Im not sure what Z of alpha ...
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proper sandal
<@&286206848099549185> 
I wish these questions had any context for me, I just need to know what they want, like.. the area under the rest of the curve... the critical z value...
clear cloud
For me this writing is related to complex numbers, like z_a are the coordinates in complex plane express like :
a+ib.
It is called "affixe" in my language but idk how to say it in english
Oh wait nvm
You could probably read it on the table, or calculate it with given stuff in the table linked
proper sandal
I've done this type of question for this program before and I know its the most simple math ever but the way they word it makes u go "huh?" and I just cant remember what they wanted
sacred sapphire
I'm not exactly sure what it wants me to find here, Im not sure what Z of alpha ...
i think its the area
proper sandal
Its either the area like 1-0.057 or its like the Z score that correlates to 0.057..
I don't think its the area bc that would go to 4 decimals and theyd ask to round to 4 if that was the case
and z scores usually go to 2 decimals
😔 The math is easy peasy its just finding out what the heck these guys want from me is hard
proper sandal
ohhh! so it wants me to basically do
1-0.057
and then take that 0.943 and find the zscore for it?
south latch
ohhh! so it wants me to basically do
1-0.057
and then take that 0.943 and find...
yes, that left tail area.
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whole kernel
🙏 Thank you oh my gosh
np
wraith hinge
can i multiply 2 inequalities?
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can i multiply 2 inequalities?
sudden mist
Suppose we have that -1 > -2
and -1 > -2, here a=c=-1 and b=d=-2
What happens?
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sudden mist
Let’s start loose, so you have that a > b
merry scaffold
ye u can multiply and divide both side in an inequality by a posituve number. but * or : by negative one flips the sign
sudden mist
Then you can multiply both sides by c and not flip the inequality if c > 0
I.e we have that ac > bc
Notice that we can do the same procedure for the inequality c>d but with b>0
wraith hinge
and d>0?
sudden mist
So we have have that bc > bd if b>0
And so ac > bc > bd if b>0,c>0
By transitivity we can conclude that
ac > bd
But this is only one scenario, in which d here can be either negative or positive
We could have started differently to let b be that number for example
Makes sense?
So all in all, you need atleast 3 numbers here to be positive for this to work
wraith hinge
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wraith hinge
someone help
wind cloak
I think you can proceed by multiplying both of these equations
maybe you'll find something
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wind cloak
someone help
did you tried multiplying the equations?
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subtle ether
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subtle ether
why do they have to use differentiation here?
I plugged in all the values from 0 to 6
and found the lowest value was -6
shouldn't that be the minimum point too?
ah fuck
I shoulda just completed the square
nvm
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subtle ether
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marble wharf
who says you cant?
subtle ether
I got no real roots when I did it 😭
oh fuck
I wrote plus 2
instead of minus
but then again
how is it neg
double neg should be positive 2 no?
oh
fak
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wraith hinge
wraith hinge
Studying limits on KA and have learned through proof and use case that lim x -> 0 of sin x / x = 1 but when I put sin(0)/0 into my calculator I get a math error, any advice on what I'm doing wrong/missing please?
split chasm
limit isn't necessarily the same the function value
as x gets closer to x=0, sin(x)/x gets closer to x=1
sin(x)/x itself is undefined at x=0, because of divisiionby 0
main sigil
This is the graph of sin(x) / x, the only point where it's undefined is at x = 0
wraith hinge
I see but when I try to do even 0.1 or 0.0001 for x to have it closer and closer to 0 without being 0, I still get like a strange number being 0.017~
It's supposed to be getting closer to 1 no?
main sigil
I see but when I try to do even 0.1 or 0.0001 for x to have it closer and closer...
you are entering it in degrees
split chasm
makle sure your calc is set to radians
main sigil
switch to radian mode
wraith hinge
I'll experiment with that then, thanks. It's gonna take a minute to figure out how to set calculator to radian mode but i'll free up this channel in the mean time, thank you! 🙏
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wraith hinge
ah i understand now, ty
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last falcon
can someone help me with polar coordinates - the graph of four leaved rose?
the thing is that we have a graph and then we have the four leaved rose graph but i think it doesnt match the graph
last falcon
for example
here theta = π check out that part of the graph on the left
well that equals to the 1 on the x
i am very confused of this graph
r represents the distance from origin to the point
and theta represents the angle
i know that part
but like when theta = π it says r = 1 but point is P(-r, π) ins't it?
im confused on the part where
if (+) r is upside and (-) downside it is (+) to the right and (-) to the left right?
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spring current
im confused on the part where
if (+) r is upside and (-) downside it is (+) to...
yeah
positive is up or to the right
and negative is down or to the left
last falcon
yeah
positive is up or to the right
and negative is down or to the left
then why does it say positive
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then why does it say positive
last falcon
at π
its left
<@&286206848099549185> my channel got closed and opened again and im using the same channel**
this is why i pinged u now
opaque basin
Emmm
generally r>=0
it is marked at the left graph
12345678
②
r<0
so drawing ②(pi/4,pi/2)
seems like (5pi/4,3pi/2)
spring current
then why does it say positive
what part are you talking about
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keen void
Assume that G is a simple graph with ten vertices and is also disjoint. Show that G has at most 36 edges
wraith hinge
Solve the following integration
Integrate sqrt(sec^2x +1) dx
Can someone help me out with this
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brisk scroll
I have a question about the distributive property.
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I have a question about the distributive property.
brisk scroll
This is the problem I’m working on
And this is my progress
I thought I was done but when I typed my answer it I was told it was wrong
Do I need to simply
3y^3 + 2y^2 -4y
Or did I multiply wrong
sand condor
you are distributing the y into the polynomial on the right and then distributing the 1 after, which is essentially just 1(y(3y^2+2y-4)).
winged totem
(3y^3 +2y^2-4y)+(3y^2-2y-4) = 3y^3+5y^2-2y-4
sand condor
It looks like you are using the FOIL method, try "foil"ing the y term, then the +1 term separately, then add these individual results together into a sum, and then combine like terms
brisk scroll
(3y^3 +2y^2-4y)+(3y^2-2y-4) = 3y^3+5y^2-2y-4
So that basically
Okay
Thanks for the help
strong token
help
winged totem
a= (1/27)^(1/2) and -(1/27)^(1/2)
strong token
a= (1/27)^(1/2) and -(1/27)^(1/2)
can you help by writing it on a paper?
winged totem
ok, just a moment
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winged totem
can you help by writing it on a paper?
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winged totem
can you help by writing it on a paper?
trim joltBOT
As a helper, please do not give out answers that could be copied as a homework solution. Have the student work through the problem themselves and guide them along the way.
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delicate pagoda
A maths course I'm doing right now usually has a simple question of the form "determine all complex eigenvalues of this matrix" on the exam. The matrices used are almost always 4×4 or 5×5 and consist of real numbers only.
So this boils down to finding all complex roots of a 4th or 5th degree polynomial with real coefficients... How would one do that, in the general case?
delicate pagoda
(Assuming that I can't just blindly guess enough roots to eventually reduce it to a quadratic and absolutely won't be learning the generic formulas for 4th degree polynomials... Also there is none for degree 5 iirc?)
marble wharf
the exam will probably be designed in a way that its entirely possible to guess enough roots
they wont require you to solve a 4 degree equation
5 degree as you said is literally impossible to solve in general
kindred pier
(Assuming that I can't just blindly guess enough roots to eventually reduce it t...
No 5th degree equation using just roots and arithmetic
delicate pagoda
the exam will probably be designed in a way that its entirely possible to guess ...
I would hope so, but historically, no.
kindred pier
But you can solve 5th degree with bring radicals or elliptic integrals iirc
But you won't need that
Just review rational root theorem
winged totem
what is the problem?
delicate pagoda
Just review rational root theorem
wait but how does that apply for equations with non-real roots?
kindred pier
wait but how does that apply for equations with non-real roots?
For 5th degree, at least one root must be real. For 4th degree, if all roots are complex, then the polynomial will either be trivial, or your teacher is a monster
delicate pagoda
what is the problem?
complex eigenvalues of a 4x4 or 5x5 real-valued matrix, ie. complex roots of 4th or 5th degree polynomials with real coefficients
delicate pagoda
For 5th degree, at least one root must be real. For 4th degree, if all roots are...
so in essence, I first try to guess a few options, then try to apply the rational root theorem and if that doesn't work I move on and accept the lost points?
kindred pier
Basically
Your teacher won't give you some arbitrary polynomial that it's incredibly hard to factor
And if that happens, take the L
delicate pagoda
Your teacher won't give you some arbitrary polynomial that it's incredibly hard ...
this professor routinely comes up with the most unfair exams you've ever seen
I think his last exam for this course had something like a 70% fail rate
kindred pier
No one would ever make you use quartic equation
merry scaffold
I think his last exam for this course had something like a 70% fail rate
at least u have a 0.33 percent chance of surviving
positive side 👍
delicate pagoda
this professor routinely comes up with the most unfair exams you've ever seen
this year, he decided that some poor first semester compSci students will now need to understand manifolds and the implicit function theorem... just to put things into perspective
anyways.. thanks for the advice!
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tidal river
how do i find the bounds from the equations of the surfaces that enclose a region of triple integral and solve it?
I'm supposed to find the volume enclosed between the two surfaces $x^2+y^2+z^2=4$ and $x^2+y^2=3z$
when i did it in cartesian coordinates i got some bounds but the integral became a mess
here're the bounds:
$$\left{ \left(x,y,z\right)\left|\begin{array}{c}
-1\leq x\leq1\
-\sqrt{3-x^{2}}\leq y\leq\sqrt{3-x^{2}}\
\frac{x^{2}+y^{2}}{3}\leq z\leq\sqrt{4-x^{2}-y^{2}}
\end{array}\right.\right} $$
i then tried to switch to cylindrical coordinates but now im not sure of my bounds, i can picture in my head the volume in cartesian which helped me found the bounds there, but in cylindrical its mostly gone.
i thought to myself that the bounds for cylindrical are as follows:
$$\left{ \left(\rho,\theta,z\right)\left|\begin{array}{c}
0\leq\rho\leq\sqrt{4-\rho^{2}\cos^{2}\theta}\
0\leq\theta\leq2\pi\
\frac{\rho^{2}}{3}\leq z\leq\sqrt{4-\rho^{2}}
\end{array}\right.\right} $$
the bounds for rho and z are simply conversions from the bounds i found in cartesian, and the theta is from 0 to 2*pi as its symetrical all around so it should do the entire theta.
is that correct? and if so how do i show it mathematically?
solid kilnBOT
horizon2.0
bleak saddle
hi
tidal river
what bothers me are the bounds for $\rho$
solid kilnBOT
horizon2.0
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cyan grail
How would i solve this?
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How would i solve this?
merry scaffold
the last time i met blud bldu said he kinda understood 🤔
OOH NVM this is area ratio
not sides ratio
= )
cyan grail
Yeahhh this is different i got the other one
merry scaffold
the ratio of area is square the ratio of the size
river willow
side^2 gives u area
cyan grail
Yeah
merry scaffold
and when i said 3/5 i am comparing the side of tri ABE TO tri ACD
-> $\frac{S ABE}{S ACD} = (\frac{3}{5})^{2}$ (S is the area)
solid kilnBOT
≅ Semicolons
cyan grail
But how would i solve b then? The same way?
merry scaffold
do u see 4 identical shapes ;)
cyan grail
OHHHH
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cyan grail
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wraith hinge
p
wraith hinge
to find the x vertex
we do -b/2a right?
or do completing the square
and the thingy next to the x is the vertex just switch the sign
is there a rule for y vertex?
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coarse talon
I'm a bit stuck on this one, I tried converting the sin²x to 1-cos²x but that didnt seem to lead anywhere. I think making the equation given sin(θ+φ+ψ)=1 could be useful.
coarse talon
Expanding $\sin(\theta+\phi+\psi)=1$ you get \
$\cos\psi(\sin\theta\cos\theta+\sin\phi\cos\theta)+\sin\psi(\cos\theta\cos\phi-\sin\theta\sin\phi)=1$
solid kilnBOT
noah
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coarse talon
<@&286206848099549185>
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hoary moss
@coarse talon give me a sec
hoary moss
use the identitie 1/2(1-cos2x)
gusty radish
I'm a bit stuck on this one, I tried converting the sin²x to 1-cos²x but that di...
let θ = x , ϕ = y and ψ = z
- sinz=sin( pie/2−(x+y))=cos(x+y) (use the indentity)
- expand cos(x+y) by Angle addition formula
and u will get
sinz=cosxcosy−sinxsiny
3)Substitute sinz in the Identity to Prove
(sinx)^2 +(siny)^2+(cosxcosy−sinxsiny) ^2+2sinxsiny(cosxcosy−sinxsiny)=1 - Simplify the Expression:
Using the Pythagorean identity u will get the ans
molten comet
Hello
molten comet
Is \vec_a x \vec_a always equal to 0 vector?
wind cloak
no
wind cloak
yeah it is always I guess
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manic creek
heu
manic creek
hey
Shouldnt it not be D ( -infinity, 3) since the highest X value is 4, not -4? Or is the x value for Domain cacluated by (4 - x) = 0 -> -4? Square root here instead of brackets
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manic creek
Should be (-oo, 4]
sure? if u check the right side, he put -x, -y on top of the answer
manic creek
does this have other constraints, or hes wrong
solid kilnBOT
𝔸dωn𝓲²s
limpid dawn
does this have other constraints, or hes wrong
uhm not that I knew
manic creek
Ah ok, do you know why he put the -x, -y
since if it was an accident, he wouldnt have put there im guessing?
limpid dawn
time stamp?
manic creek
35:25
I was guessing that its similar to (4 - x) = 0, so that would be -x = --4, but no idea
limpid dawn
I was guessing that its similar to (4 - x) = 0, so that would be -x = --4, but ...
The domain is what I wrote $x \leq 4 \Leftrightarrow (-\infty, 4]$. The reason he wrote $-x,-y$ is because there is I think a rule of thumb of how the function behaves depending on the sign of $x$ and $y$, but it doesn't have to do with the domain here, just to explain how he got the idea of how the graph looks like.
solid kilnBOT
𝔸dωn𝓲²s
manic creek
The domain is what I wrote $x \leq 4 \Leftrightarrow (-\infty, 4]$. The reason h...
ah okay, I think he overcomplicated it lol
manic creek
Yea, but
since its -x in the square roots
does the domain have to end at the -x = -4
or it doesnt matter
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manic creek
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manic creek
or it doesnt matter
@limpid dawn
limpid dawn
Ok look at it this way
you have some random square root function
you know the inner part must not become negative
manic creek
yea
limpid dawn
not negative is the same as positive or 0
manic creek
Yes
limpid dawn
the inner part here is (4-x)
so you check for which values of x is 4-x >= 0 true
because then the function is defined
if you solved the inequality you have your domain
manic creek
so you check for which values of x is 4-x >= 0 true
that would be 4
so x = 4 and we put 4 in the domain
limpid dawn
x <= 4*
manic creek
D(-infinity, 4]
limpid dawn
ye
manic creek
Thx
limpid dawn
manic creek
@limpid dawn also makes sense since you cant have a negative number inside a square root or its imaginery
manic creek
Ahhh damn lol
limpid dawn
<@587715738484211754> also makes sense since you cant have a negative number ins...
yes exactly
manic creek
I was thinking if there was some advanced algebra formula i had to learn here lol
but just a mistake
limpid dawn
ye
manic creek
ye
I just realised the -x and -y, was talking about the quadrants lol
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urban copper
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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
0
urban copper
,, f(x) = x + \frac{3}{x}
solid kilnBOT
938c2cc0dcc05f2b68c4287040cfcf71
urban copper
$\textbf{increasing interval is when } x + \frac{3}{x} > 0$
urban copper
wdym
clear cloud
To get increasing interval
clear cloud
I need derivative for increasing interval?
Yes you study the sign
That was the table last time
urban copper
,, f'(x) = 1 + 3\frac{d}{dx} \frac{1}{x}
urban copper
That was the table last time
I thought that was for the image
solid kilnBOT
938c2cc0dcc05f2b68c4287040cfcf71
urban copper
,, f'(x) = 1 + 3 \cdot -1 \cdot \frac{1}{x^2}
solid kilnBOT
938c2cc0dcc05f2b68c4287040cfcf71
urban copper
,, f'(x) = 1 - \frac{3}{x^2}
solid kilnBOT
938c2cc0dcc05f2b68c4287040cfcf71
clear cloud
Ok
urban copper
correct or no
clear cloud
Yeah correct
urban copper
$\textbf{ now we need }1 - \frac{3}{x^2} > 0 \ \textbf{ for increasing interval}$
solid kilnBOT
938c2cc0dcc05f2b68c4287040cfcf71
urban copper
correct?
clear cloud
Yeah
urban copper
,, 1 > \frac{3}{x^2}
solid kilnBOT
938c2cc0dcc05f2b68c4287040cfcf71
urban copper
,, x^2 > 3
solid kilnBOT
938c2cc0dcc05f2b68c4287040cfcf71
urban copper
,, |x| > \sqrt{3}
solid kilnBOT
938c2cc0dcc05f2b68c4287040cfcf71
clear cloud
Ok and now for x ?
urban copper
idk
maybe like this
,, -\sqrt{3}> x > \sqrt{3} \implies x \in (-\sqrt{3}, \sqrt{3})
clear cloud
Indeed
solid kilnBOT
938c2cc0dcc05f2b68c4287040cfcf71
clear cloud
Yeah
clear cloud
**938c2cc0dcc05f2b68c4287040cfcf71**
About the interval answer
urban copper
what we need to check the endpoints
clear cloud
Nah
urban copper
or is it $x \in (\sqrt{3}, -\sqrt{3})$
solid kilnBOT
938c2cc0dcc05f2b68c4287040cfcf71
urban copper
impossible
clear cloud
One number in the interval is causing trouble
clear cloud
Be careful to this
clear cloud
Its not a question of sign, you wrote that if x is in (-sqrt3, sqrt3) it increased but f(0) is not defined so you cant let 0 in the interval
urban copper
ok
urban copper
yeah
my bad
,, f(x) = x + \frac{3}{x} \ Dom(f) = \mathbb{R} \setminus {0}
this?
clear cloud
Yeah
solid kilnBOT
938c2cc0dcc05f2b68c4287040cfcf71
clear cloud
D_f as writing is fine too
clear cloud
or like this
This is better
Ok so
Do you really think you need for decreasing interval or you can already deduce them ?
urban copper
or is it $x \in (-\sqrt{3}, 0) \cup (0, +\sqrt{3})$
solid kilnBOT
938c2cc0dcc05f2b68c4287040cfcf71
clear cloud
**938c2cc0dcc05f2b68c4287040cfcf71**
This is for the intervals where it increased
clear cloud
how do I remove 0 from increasing inteval
Two possibility
clear cloud
**938c2cc0dcc05f2b68c4287040cfcf71**
You can do this its great
Or (-sqrt3, sqrt3) \ {0}
As you want
clear cloud
how do I find the decreasing one
Its the opposite from the increasing one
So (-inf,-sqrt3]U[sqrt3,+inf)
Oh wait
clear cloud
**938c2cc0dcc05f2b68c4287040cfcf71**
It include the sqrt
Since its where its 0
Its positive and negative at the same time
urban copper
sure
clear cloud
Or maybe you check apart the f' = 0 ?
Idk, i used to solve f'x >= 0
And include the zero
And study them as minimum maximum
urban copper
ok
clear cloud
**938c2cc0dcc05f2b68c4287040cfcf71**
This is not the solution of |x| > sqrt3
solid kilnBOT
938c2cc0dcc05f2b68c4287040cfcf71
urban copper
not true
clear cloud
Well |x| > sqrt3, the solution must be greater than sqrt 3 or less than -sqrt3
clear cloud
**938c2cc0dcc05f2b68c4287040cfcf71**
So the opposite of the written here
urban copper
,, |x| > \sqrt{3}
solid kilnBOT
938c2cc0dcc05f2b68c4287040cfcf71
urban copper
,, |x| > \sqrt{3} \implies -\sqrt{3} < x > \sqrt{3}
solid kilnBOT
938c2cc0dcc05f2b68c4287040cfcf71
urban copper
whatever that means
clear cloud
**938c2cc0dcc05f2b68c4287040cfcf71**
No
urban copper
usually I do it this way but this time is not working
like if we would have had |x| < sqrt(3) it would have worked 
clear cloud
So its the opposite
urban copper
okay let me draw |x| > sqrt(3) on the number line
clear cloud
So (-inf,-sqrt3]U[sqrt3,+inf)
It increased on this interval
clear cloud
**938c2cc0dcc05f2b68c4287040cfcf71**
And deacresed on this one
clear cloud
Ok so f'x >0 ==> |x| > sqrt3
clear cloud
It increased on this interval
And the solutions of the inequality is this union
clear cloud
okay let me draw |x| > sqrt(3) on the number line
I think its the best way tl figure out
,solve abs(x) > \sqrt{3}
,solve abs(x) > \sqrt{3}
urban copper
,w
clear cloud
,w abs(x) > \sqrt{3}
clear cloud
Thus
urban copper
how
clear cloud
They draw the line
urban copper
urban copper
that was hard!
noticing that if you multiply by -1 the inequality sign flips
anyways, that was pretty obvious but I didnt know that
clear cloud
Btw
Little tips to avoid
This trouble
You see x^2 > 3
You set it like x^2 -3 > 0
You search for the roots
So -sqrt3 and sqrt3
urban copper
it is not inclusive sqrt3
just wanted to add
you say that 0 is positive negative but is not positive
impossible
yeah we can find the roots of x^2 = 3 for finding the interval easier but it is not inclusive sqrt(3)
imo
clear cloud
It included both, and by finding the roots, you know that the sign of the polynomial depends on the sign of a in ax^2 +bx + c
clear cloud
why do u include sqrt(3) and -sqrt(3)
I do most of the time including them, treating them after as extreme
It avoid the confusion between f'x = 0 and a forbbiden value
clear cloud
It included both, and by finding the roots, you know that the sign of the polyno...
If a < 0 then the function is positive between the roots
Otherwise if a >0 then its negative between the roots
Anyway it was a tip aside, to not fall into absolute value trap, so what are the intervals of increasing and decresing finally ?
urban copper
absolute value trap got me locked in like 20 min
increasing is (-infty, -sqrt(3)) u (sqrt(3), +infty)
decreasing u say its (-sqrt(3), sqrt(3)) but I havent checked but Is prolly less troublesome than the increasing one
clear cloud
decreasing u say its (-sqrt(3), sqrt(3)) but I havent checked but Is prolly less...
Its neccessarly the opposite
urban copper
shouldnt we do the variadic table aswell
clear cloud
shouldnt we do the variadic table aswell
You can, its a more visual method for corrector
I mean the man who gives grades
clear cloud
Show me how you would do it
urban copper
ok let me draw it
it was something like this if I remember
let me do the drowing+
a moment
long dust
clear cloud
!occupied
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sharp owl
how i can find this limit?
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how i can find this limit?
sudden mist
well, you can't
sharp owl
why is weird
sudden mist
what's weird?
sudden mist
that's wrong
sharp owl
sudden mist
,w limit of sin(pi/x) as x approaches 0
solid kilnBOT
sudden mist
let's assume wolfram here is wrong too, since it can at times
sudden mist
could you ask it what the limit of sin(1/x) is?
sudden mist
that's very interesting!
sudden mist
let me show you why this is wrong
so first of let u = x/pi, then as x goes to 0 then so does u
so we can infact rewrite the limit sin(pi/x) as sin(1/u)
but wait, the limit sin(1/u) as u goes to 0 doesnt exist, so what happened?
alternatively we can pick a subsequence of points for x approaching 0 that always outputs 1 and resp. -1
sudden mist
so that also reinforces why this must be wrong
sharp owl
alternatively we can pick a subsequence of points for x approaching 0 that alway...
how you can get that subsequence?
In my book choose 2/2m+1 but i dont know why
sudden mist
pick for example x = 1/(1/2+2n) where n is an integer
then you get that sin(pi/x) = 1
i could also choose x = 1/(3/2 + 2n)
so that sin(pi/x) = -1
sudden mist
In my book choose 2/2m+1 but i dont know why
Plug that into sin(pi/x)
and youll see why
sharp owl
i cant see
sudden mist
So what did you get?
sharp owl
sen((2m+1) pi/2)
sharp owl
2npi + pi/2?
sudden mist
that seems more like my example, but lets go with that
yeah so if you have sin(2npi + pi/2)
then what you can rewrite that as?
sharp owl
jidk how?
sharp owl
sum angle?
sudden mist
sum angle?
?
sharp owl
sen(a+b)
sudden mist
Uh we dont really need that here
sharp owl
yes is periodict
sharp owl
2pi/}b}