N doesn't come up in the video, and neither does your formula
#help-10
limber quartz
limber quartz
Yea so why are you insisting on using it?
Because its in the confidence interval formulas, like all of them
Yours is different
keen wadi
Confidence intervals aren’t too fun
limber quartz
Would s.d. just be 1 times half the interval width?
(Standard Norman's standard deviation scaled for the interval?)
obtuse pebbleBOT
@split marsh Has your question been resolved?
brittle swan
obtuse pebbleBOT
brittle swan
how do i begin proving this
brave bramble
Consider the span of the columns
Well, wait, which direction are we proving first?
brave bramble
Let's do the forward direction. That is:
"If the i-th column is a non-pivot, then there exists a solution for any x_i"
Take [A|b] as an augmented matrix. What would happen as it is row reduced?
brittle swan
we solve the system..?
brittle swan
Take [A|b] as an augmented matrix. What would happen as it is row reduced?
not sure what answer is expected to be given lol
obtuse pebbleBOT
@brittle swan Has your question been resolved?
brittle swan
<@&286206848099549185>
obtuse pebbleBOT
@brittle swan Has your question been resolved?
obtuse pebbleBOT
@brittle swan Has your question been resolved?
obtuse pebbleBOT
@brittle swan Has your question been resolved?
marsh lake
obtuse pebbleBOT
marsh lake
I just need help on c
I know it is yes but how would I solve it mathematically
@compact parcel if ur still here please help <3
compact parcel
If they are indeed independent events, then $\operatorname{Prob}(\text{satisfied with communication and living in Saratoga}) = \operatorname{Prob}(\text{satisfied with communication}) \cdot \operatorname{Prob}(\text{living in Saratoga})$.
warm shaleBOT
pi over four
compact parcel
So, you need to check if that is true
compact parcel
.288= .25*.365
It should be either .077 = .25 * .365 or .288 = .75 * .365
compact parcel
It should be either .077 = .25 * .365 or .288 = .75 * .365
If either of these equations are right, then they are independent events
obtuse pebbleBOT
@marsh lake Has your question been resolved?
toxic harbor
f(x) = {3 if x ≤ 0; x^2 if x > 0}
obtuse pebbleBOT
`f(x) = {3 if x ≤ 0; x^2 if x > 0}`
toxic harbor
The picture is written in the same way as the text message, right? They both mean the same thing, yeah?
last osprey
`f(x) = {3 if x ≤ 0; x^2 if x > 0}`
f(x) = 3 if x <= 0
f(x) = x^2 if x > 0
they describe the same
toxic harbor
f(x) = 3 if x <= 0
f(x) = x^2 if x > 0
they describe the same
No I mean is it okay to use ;
last osprey
depend normally i use ,
main cargo
obtuse pebbleBOT
main cargo
i get a bit confused how to simplify logs sometimes
so your meant to go left to right???
but you gotta factor out the -
so its -(2ln3 - ln8) -ln4
-ln(9/8) -ln4
-(ln(9/8) + ln4))
-ln(9/2)
ln(2/9)
but you cant change the order of things right?
so you cant do
-2ln3 + ln8 = ln8 - 2ln3
ln(8/9)
ln(8/9) - ln4 = ln(2/9)
oh
it works when i type it but not when i write it on paper lol
im prolly good then, thanks server.
.close
obtuse pebbleBOT
strong vale
i get a bit confused how to simplify logs sometimes
$\ln(\frac{a}{b}) = \ln(a) - \ln(b)$\\ $\ln(ab) = \ln(a) + \ln(b)$
royal basin
\\
strong vale
lol
warm shaleBOT
azeem321
obtuse pebbleBOT
Available help channel!
wanton kraken
obtuse pebbleBOT
fervent crag
Should be -15 at the end not -10
wanton kraken
o
but
36 squared = 4
4*5=20
no?
when I multiply 6 squared by 5 * 6 squared
I get 5 * 4
fervent crag
Sqrt36 is 6
fervent crag
Np
wanton kraken
.close
queen sand
A triangle has sides lengths of 6 cm, 8 cm and 10 cm. Use the values to show if the triangle is at right angles
obtuse pebbleBOT
A triangle has sides lengths of 6 cm, 8 cm and 10 cm. Use the values to show i...
midnight topaz
A triangle has sides lengths of 6 cm, 8 cm and 10 cm. Use the values to show i...
use Pythagoras theorem
timid silo
A triangle has sides lengths of 6 cm, 8 cm and 10 cm. Use the values to show i...
show that it forms a pythagorean triplet
queen sand
.close
uncut citrus
obtuse pebbleBOT
uncut citrus
the first differential equation is homogenus as f(x,y) = lambda^0 f(x,y) and this acheived by subsituting x=lambda x and y= lambda y
x is equal to lamba multiplied by x
so how would this play out if i were to do the same in the second differential equation (the one with the trig functions)
how do you prove its homogenity?
uncut citrus
the differntial equation in question
obtuse pebbleBOT
@uncut citrus Has your question been resolved?
obtuse pebbleBOT
@uncut citrus Has your question been resolved?
uncut citrus
.close
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
timid silo
Anyone know how to find the area of the pathway?
obtuse pebbleBOT
Anyone know how to find the area of the pathway?
obtuse pebbleBOT
@timid silo Has your question been resolved?
warm canopy
Same way you merge normal fractions, make them have a common denominator
Let's check your work quick
,w diff (x-4)/(2x+1)
warm shaleBOT
warm canopy
Nice
wanton kraken
*this is a solved problem; check the last messages for the newer ones"
obtuse pebbleBOT
*this is a solved problem; check the last messages for the newer ones"
wanton kraken
why tf
does
= (sqrt(3)+1)^2
and not
(1+sqrt(3)^2
because that's the same thing with
(sqrt(3)-1)^2
and not (1-sqrt(3))^2
answers aren't the same
warm shaleBOT
civic zealot
,w (sqrt(3)-1)^2=(1-sqrt(3))^2
warm shaleBOT
civic zealot
(1-sqrt(3))^2
(1-sqrt(3))^2 = (1-sqrt(3))(1-sqrt(3))
are you distributing (foiling) this?
civic zealot
no
civic zealot
I get 2(sqrt(3))
that's the final answer, yes
wanton kraken
that's the final answer, yes
well the answer is 2 🤣
and not 2(sqrt(3))
oh well
I used a^2-b^2
and got 2
civic zealot
..... hmm
warm shaleBOT
Zybikron
civic zealot
so
$\sqrt{(1-\sqrt3)^2} = \sqrt{4 -2\sqrt3}$\
$\sqrt{(1+\sqrt3)^2} = \sqrt{4+2\sqrt3}$
warm shaleBOT
Zybikron
civic zealot
But in general $\sqrt{x^2} = |x|$. So since $1-\sqrt{3}$ is negative, you switch the order to $\sqrt3-1$
warm shaleBOT
Zybikron
wanton kraken
what more complex formula should I use here, but it's more simple ffs
wanton kraken
But in general $\sqrt{x^2} = |x|$. So since $1-\sqrt{3}$ is negative, you switch...
wait why?
if you don't and you have:
oh
I see it
ty @civic zealot
.close
obtuse pebbleBOT
obtuse pebbleBOT
✅
wanton kraken
I need help with factoring some binomials using (a+b)^2 and (a-b)^2 etc formulas
for e.g
I can do trial & error, but I don't want to give blind answers
how do I factor this into (a-b)^2?
ill be away for a lil
ping me when responding
obtuse pebbleBOT
@wanton kraken Has your question been resolved?
wanton kraken
.close
obtuse pebbleBOT
Available help channel!
wanton horizon
@dawn meteor
obtuse pebbleBOT
<@216861413157765120>
dawn meteor
??
wanton horizon
yo sori i wasnt here earlier to respond hehe
dawn meteor
yo sori i wasnt here earlier to respond hehe
respond to?
wanton horizon
you told me to send my work but i ignored u :(
dawn meteor
you told me to send my work but i ignored u :(
ahh I see
wanton horizon
but ye this is what i tried doing butt it seems too long + i ended up getting stuck
this is the question, find the eclipse with 2 dots being known
dawn meteor
also, having a hard time reading your calculations 
wanton horizon
oof :(
dawn meteor
but I think youre on the right track
so lets try to get through it here
$$\left(\frac{21}{5}\right)^2 \frac{1}{a^2} + \frac{16}{b^2} = 1$$
warm shaleBOT
FlynnXD
dawn meteor
$$\left(\frac{28}{5}\right)^2 \frac{1}{a^2} + \frac{9}{b^2} = 1$$
warm shaleBOT
FlynnXD
dawn meteor
okay, just let 1/a^2 = x, and 1/b^2 = y
$$\left(\frac{21}{5}\right)^2 x + 16y = 1$$
$$\left(\frac{28}{5}\right)^2 x + 9y= 1$$
warm shaleBOT
FlynnXD
dawn meteor
can you try from here?
wanton horizon
leme see
would this work?
and then just put that in as y into the other equation and that solves it?
dawn meteor
and then just put that in as y into the other equation and that solves it?
yes
and then get a, b from x, y
wanton horizon
eh and now i end up with this O.o
and its rly weird to be getting so ugly numbers with my proffesor soo i dono :/
dawn meteor
,w solve (21/5)^2 x + 16y = 1 and (28/5)^2 x + 9y = 1
warm shaleBOT
dawn meteor
@wanton horizon looks like the final numbers are good
you should be getting a = 7 and b = 5, so any error is just calculation mistakes on your part
obtuse pebbleBOT
@wanton horizon Has your question been resolved?
wanton horizon
aight thank you
viral orchid
obtuse pebbleBOT
viral orchid
help
cedar lichen
What have you tried
cedar lichen
What did you get?
viral orchid
3 x 3 + 3x root3 - 3root2
cedar lichen
Looks right to me
shadow phoenix
obtuse pebbleBOT
keen wadi
Do the direction vector?
shadow phoenix
how do I do that?
timid silo
do you know what parallel means?
it means two lines have the same slope
so calculate the slope. if the slopes are equal, then they are parallel
tardy epoch
Yes
shadow phoenix
im a little confused how would I get a number from that
because i got 1 from the first one
like how would I simplify it down into a whole
or is that something you dont do
obtuse pebbleBOT
@shadow phoenix Has your question been resolved?
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
lean spear
obtuse pebbleBOT
lean spear
did i set this up wrong?
i dont exactly understand how to convert percentages to a number
warm lynx
fervent crag
70% of a number is multiplying by 0.70
signal frost
What you set up would be if the computer gained 70% value each year
lean spear
so we need 0.70^t
fervent crag
Ya
fervent crag
Think of the word "of" meaning multiplication
70% of a is a*0.70
When you are thinking of more and less than 1, that's talking about when they tell you % increase or decrease
Decrease by 20% would mean multiply by 0.80
Increase by 15% would mean multiply by 1.15
Your problem here is neither of those, it's just saying "70% of the value" so we multiply directly by 0.70
timid silo
when I want to integrate 1/3x+2x^2, why do i have to split it in partial fractions? why doesn't ln|3x+2x^2| work?
obtuse pebbleBOT
when I want to integrate 1/3x+2x^2, why do i have to split it in partial fractio...
restive pilot
do you have $\frac{1}{3x+2x^2}$ or do you have $\frac{1}{3x}+2x^2$
warm shaleBOT
Eichhorst
warm shaleBOT
hx
solar trellis
Differentiate what you think the answer is to check if you're right
timid silo
i would get $\frac{3+4x}{3x+2x^2}$
warm shaleBOT
hx
timid silo
but why does splitting it into partial fractions work
timid silo
so it's impossible to integrate it without splitting it into partial fractions?
solar trellis
I can't say, maybe there is a way I can't think of at the moment. There are often many ways to solve things
But partial fractions is probably the most straightforward
timid silo
i just want to know why it works rlly
because 3x+2x^2 is still a function in itself and the power rule can't be used because it can be treated as (3x+2x^2)^1, and my mind took the approach to simply integrate it to ln|3x+2x^2|
does it not work doing it that way because the integral of 1/x rule doesn't apply to something containing a variable that isn't x on its own?
like the +2x^2 here
solar trellis
does it not work doing it that way because the integral of 1/x rule doesn't appl...
Yes because of the chain rule
lean spear
if something is losing 75% of its value of the year before
obtuse pebbleBOT
if something is losing 75% of its value of the year before
lean spear
am i just multiplying by .75?
fervent crag
0.25
(1 - 0.75) = 0.25
remember,
decrease by 20%: (1 - 0.20) = 0.80
increase by 15%: (1 + 0.15) = 1.15
merry trench
what would the equation for f(x) be? i tried f(x) = e^-x-1 but apparently its incorrect
obtuse pebbleBOT
what would the equation for f(x) be? i tried f(x) = e^-x-1 but apparently its in...
compact parcel
If the graph is the only given information, it is extremely hard to find the original function.
Is it given that this is an exponential function?
fierce lagoon
I mean it looks asymptomatic to y = -1
fervent crag
super likely this is just find the exponential function
fierce lagoon
Yeah I think it's just $e^{-x} - 1$
warm shaleBOT
Umbraleviathan
merry trench
If the graph is the only given information, it is extremely hard to find the ori...
we were only given an hint to transform an exponential function with an integer base.
compact parcel
Maybe it’s $3^{-x} -1$
fervent crag
use the two given points to find a and b
warm shaleBOT
Umbraleviathan
compact parcel
what would the equation for f(x) be? i tried f(x) = e^-x-1 but apparently its in...
Because it looks like f(-1) = 2
merry trench
alright, thanks
obtuse pebbleBOT
@merry trench Has your question been resolved?
nocturne minnow
Don't do the work for people. Even if you are providing it for someone to check. You don't know if someone was waiting for you to give the answers and then they were going to take it and run
It's better if you help them through the parts they are stuck on
dim rock
obtuse pebbleBOT
dim rock
guys someone explain me how is g = ax+by?
haughty coyote
Since g is in S, there exists 2 integers, which we'll name x0 and y0, such that g = ax0 + by0. That's just because every element in S can be written in this way
pallid flame
a
dim rock
as well as in it isint it supposed to be d | a-b but how did it become ax+by instead?
dim rock
Since g is in S, there exists 2 integers, which we'll name x0 and y0, such that ...
u mean since we dont know the exact value yet so we can write it down in this way?
haughty coyote
d divides any linear combination of the numbers it divides. Since it divides a and b, it divides any number writable in the form ax + by
It's the minimum, it exists, and it can be written this way. Until you've seen Euclid's algorithm, the gcd only exists anyways
Because you have no explicit formula to obtain it
dim rock
d divides any linear combination of the numbers it divides. Since it divides a a...
can u gimme an example for this?
<@&286206848099549185>
obtuse pebbleBOT
@dim rock Has your question been resolved?
haughty coyote
can u gimme an example for this?
They're both factorisable by d
obtuse pebbleBOT
They're both factorisable by d
haughty coyote
So you have p, q such that a = dp, b = dq. Then for any x, y in Z, ax + by = dpx + dqy = d(px + qy), hence it's divisible by d. For any x and y, hence that includes g
.close
obtuse pebbleBOT
haughty coyote
For example d = 2, a = 4, b = 10, any x and y you want
lucid falcon
This is in French, (sorry). All it says is : This is an addition of rational expressions where the denominators are non-zero. What is the expression you get when simplifying it?
obtuse pebbleBOT
Available help channel!
toxic star
obtuse pebbleBOT
fierce lagoon
Plug x back into the original equation
Basically plug -4 back into the original equation
toxic star
So I tried that and I have
-4^2+8(-4)+9
-16-32+9
Did I do something with the 16 incorrectly?
Because if I go further then I don’t get -7 unless the -16 is a positive
toxic star
Oh
fierce lagoon
$$ -4^2 = -16$$
$$(-4)^2 = 16$$
warm shaleBOT
Umbraleviathan
fierce lagoon
It's very important that x^2 is basically (x)^2
toxic star
Thank you
fierce lagoon
Np
obtuse pebbleBOT
@toxic star Has your question been resolved?
sleek raptor
help on this question, calculus
obtuse pebbleBOT
@sleek raptor Has your question been resolved?
cedar lichen
What have you tried
compact shadow
z=rcos(θ),x=1+rsin(θ),y=y, use dxdydz=rdrdθdy
obtuse pebbleBOT
@sleek raptor Has your question been resolved?
sudden otter
obtuse pebbleBOT
obtuse pebbleBOT
@sudden otter Has your question been resolved?
sudden otter
I am still here
compact shadow
Just let f(x)=a(x+t)(x^2-6x+13)(x-1+sqrt(2))(x+1)
And use the coefficient of x^2 being 9 to solve t
compact shadow
Basic fact
compact shadow
For the convenience
You can calculate coefficient of x^2 by (1/2)f”(0)
f”(0)=18
sudden otter
Just let f(x)=a(x+t)(x^2-6x+13)(x-1+sqrt(2))(x+1)
yeah so if x is 3, then just solve?
compact shadow
? Where did 3 come from
sudden otter
sqrt9
compact shadow
Can’t get it
sudden otter
wdym cant get it?
compact shadow
I can’t understand what you said
Anyway, f”(x)=a[2(x+t)(x-1+sqrt(2))(x+1)+(x^2-6x+13)(x+t+x-1+sqrt(2)+x+1)+(2x-6)((x+t)(x-1+sqrt(2))+(x+t)(x+1)+(x-1+sqrt(2))(x+1))], so f”(0)=
sudden otter
what is the "
compact shadow
a[2t(sqrt(2)-1)+13(t+sqrt(2))-6(sqrt(2)t+sqrt(2)-1)]=18
compact shadow
what is the "
Second order derivative
sudden otter
Ah
compact shadow
Never mind seems like we still need to expand f(x)
compact shadow
Just let f(x)=a(x+t)(x^2-6x+13)(x-1+sqrt(2))(x+1)
I need to restart from this step again, sorry
sudden otter
your good
fierce schooner
Never mind seems like we still need to expand f(x)
It is messy. I must made a calculation error somewhere because I didn't get 9 for x^2 term.
compact shadow
I got
$a,\sqrt{2}-6+t,12-5\sqrt{2}+t(\sqrt{2}-6),13\sqrt{2}-13+t(7\sqrt{2}+6),t(13\sqrt{2}-13)$ need to be rational and $a(7\sqrt{2}+6+t(12-5\sqrt{2}))=9$
warm shaleBOT
Cogwheels of the mind
sudden otter
think he meant not claiming the helper role
compact shadow
I don’t want to be a helper, I just answer questions I feel interesting or when I am in the mood
sudden otter
a math vigilante
sudden otter
$a,\sqrt{2}-6+t,12-5\sqrt{2}+t(\sqrt{2}-6),13\sqrt{2}-13+t(7\sqrt{2}+6),t(13\sqr...
But yeah it still =9
compact shadow
Yeah so you can solve a=9/4
Because those coefficients being rational give you t=-1-sqrt(2), so 4a=9
sudden otter
Just let f(x)=a(x+t)(x^2-6x+13)(x-1+sqrt(2))(x+1)
just plug in a=9/4 to this equation?
compact shadow
? I mean f(x)=(9/4)(x^2-6x+13)(x-1+sqrt(2))(x+1)(x-1-sqrt(2))
Isn’t this what you want to solve?
sudden otter
right
sudden otter
? I mean f(x)=(9/4)(x^2-6x+13)(x-1+sqrt(2))(x+1)(x-1-sqrt(2))
yeah so just solve that
compact shadow
(9/4)(x^5-9x^4+16x^3+4x^2-33x-13)
compact shadow
You're profile doesn't say helper.
I am not a helper, I don’t answer tedious or too easy or application or probability/PDE/any other subjects I am not interested in questions
Yeah if you want
sudden otter
Yeah if you want
oh is it not necessary?
fierce schooner
I am not a helper, I don’t answer tedious or too easy or application or probabil...
You seems to be a good helper. I was just wondering that is all.
compact shadow
Idk, you can if you want to be certain
sudden otter
.close
tardy epoch
What are the two answers and show your work
valid badger
idk what I did wrong
obtuse pebbleBOT
idk what I did wrong
valid badger
Ive went over it a dozen times and I still think its right 0_0
valid badger
nvm, my tierd self put the y intercept on the x axis, ty for the help anyways though
ive seriously been looking at this question for 20 minutes to lol
took my posting it to figure it out
.close
timid silo
obtuse pebbleBOT
brittle blaze
What is d
timid silo
All is states is “use the properties of logarithms to write the expression as a single logarithm”
brittle blaze
Do you know the properties of logs
timid silo
No I do not, that is what I’m trying to learn tbh
brittle blaze
Use the middle
timid silo
Alright thank u
obtuse pebbleBOT
Available help channel!
light sand
Hey
static sun
Well the first one is just the coordinates
Since they both intersect at a exact value on the graph
light sand
My bad I should have siad
static sun
What's the last equation
light sand
I type it one sec
light sand
X squared -4 =-3
static sun
Is it asking for the intercepts?
static sun
Well algebraic x= 1 and -1
static sun
Yeah one second
light sand
Kk
static sun
Just drawing a diagram
static sun
All good
Basically
What it's asking you to do
Where does the point -3 lay on y=x^2-4
Sorry really crappy at making accurate drawings
light sand
Ya no touch
static sun
Yep so x=1 or -1
light sand
That it
static sun
Yeah basically
But just draw those lines on the graph
To prove you used it
Hope that helped!
light sand
Like just draw a lien at 3
warm canopy
light sand
Ok
light sand
So there is this question that is like explain why this pattern is linear
But they give lines
So I can’t graph it and say oh look it a straight line
static sun
Can you show me a photo of it
static sun
So b?
light sand
We have to prove with our a graph
light sand
Yes
light sand
By 0.6
static sun
Look at value 2 and 3
light sand
It goes up by 0.6
static sun
You're missing a point
It goes 22.6
Then it goes down
22.2
Then goes back to 22.8
Is that a linear line?
Here's a graph of your data
light sand
No it goes up only
static sun
light sand
23.2
static sun
Do you see that?
light sand
It is
static sun
Do you see the graph I put up?
light sand
Hehe
static sun
That was my bad
light sand
Algoods
light sand
So is the gradient 0.6
static sun
Yep
light sand
And it goes up by 0.6 each time
static sun
So for your answer
static sun
The data is linear, as the gradient is 0.6, causing the data to be linear
Something along those lines
And sure
light sand
Thanks
static sun
You can probably close this then
light sand
Have a nice day
static sun
You too
light sand
.cloes
static sun
Did you add for further question?
static sun
Oh haha
light sand
.cloes
static sun
.close?
warm canopy
.close
lime scarab
how
tranquil arch
like for x = 2, you can find 2 y that satisfied the equation
tranquil arch
yes, so it's not a function
lime scarab
alr thanks
obtuse pebbleBOT
@lime scarab Has your question been resolved?
stark mural
What is this called?
obtuse pebbleBOT
What is this called?
obtuse pebbleBOT
@stark mural Has your question been resolved?
toxic harbor
Why is it undefined when a negative number is under a square root?
obtuse pebbleBOT
Why is it undefined when a negative number is under a square root?
warm shaleBOT
ホタル
stark ether
what real number multiplied by itself is -4
@toxic harbor if you try this you'll see that it isnt possible
toxic harbor
because negatives aren't real numbers?
stark ether
no no
timid silo
Why is it undefined when a negative number is under a square root?
It's something called complex number, you gonna learn about it later
stark ether
its because lets say you take positive number, if you multiply by itself you'll get a positive number
lets say you take negative number, if you multiply by itself you'll STILL get a positive number
warm shaleBOT
stark ether
x such that x greater than 0
toxic harbor
so x>0
stark ether
yes thats just set builder notation
toxic harbor
Why is x such that important?
stark ether
its not, you can use any variable name
toxic harbor
hm
How do I write this in interval notation?
(0, infinity]
But what about x=/=2?
stark ether
(0, infinity]
infinity is never included in interval notation
toxic harbor
Ah okay ty
If I wanted to say that it's never x+b, can I say (0, inf) - {x+b}? @stark ether
toxic harbor
oh does that make no since since on part says all positive infinity numbers are in the domain but then the second part says that some positive infinity numbers are not?
Since you can define x as like 5
or something
But - {x} means "exclude x", right?
stark ether
yes it does mean that
toxic harbor
Okay tysm
toxic harbor
obtuse pebbleBOT
toxic harbor
Is this answer incorrect?
royal basin
it has a typo
stark ether
yeah
royal basin
actually hold on
stark ether
-2 isn't greater than 2
royal basin
no it has more than that
toxic harbor
Since -2 isn't the only number that would put a negative number under the sqrt, right?
royal basin
the domain specification should be x < -2 or x > 2
as-is it looks like they wanted to write -2 < x < 2 but typo'd, when in fact doing so would have given us almost the complement of the real answer
stark ether
toxic harbor
Wait but we can't put -1 under the square root, right?
stark ether
yes we cant, for the real functions
royal basin
"the real domain"?
stark ether
mb
I meant for real functions
sqrt(-1) is undefined in the real world but is defined in the complex world
toxic harbor
But it can't be part of our domain then
stark ether
if you had
$\sqrt(x)$ then the domain is $x \in [0, \infty)$
warm shaleBOT
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stark ether
for $\sqrt{x^2}$ the domain is $x \in (\infty, \infty)$
warm shaleBOT
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toxic harbor
-1^2 is 1
stark ether
yes
stark ether
(not really put brackets (-1)^2 = 1)
toxic harbor
1/sqrt(-3)
toxic harbor
as-is it looks like they wanted to write -2 < x < 2 but typo'd, when in fact doi...
But according to this it is
toxic harbor
-2 < x < 2 means that x can be -1, right?
stark ether
what they wrote is incorrect
royal basin
for $\sqrt{x^2}$ the domain is $x \in (\infty, \infty)$
you mean $(-\infty, \infty)$
warm shaleBOT
Ann
stark ether
the picture
stark ether
you mean $(-\infty, \infty)$
yes mb another typo
stark ether
yes
stark ether
stark ether
they wanted to write -2 < x < 2 which in itself is wrong
toxic harbor
Ah okay
stark ether
What does complement of the real answer mean?
R - {}
toxic harbor
AHHHH
stark ether
assuming the universal set is R
royal basin
Isn't -2 > x > 2 correct though
when you chain two inequalities together they're connected by an AND and never by an OR
royal basin
R - {}
no
i should've just said "true answer" instead of "real answer" so as not to cause confusion here
stark ether
no
wdym now I'm confused too
complement of an interval would be R - that interval right
royal basin
yes
what im saying is
the actual answer is (-∞, -2) ∪ (2, +∞)
but what they wrote looks like they meant to write (-2, 2) but typo'd
hence why i said their answer is almost the complement of the true answer (the complement of (-∞, -2) ∪ (2, +∞) is [-2, 2])
toxic harbor
Wait one sec, so the correct domain of ^ is all numbers other excluding 2, -2, and -1?
royal basin
no
royal basin
the domain of 1/sqrt(x^2 - 4) is the set of all real numbers excluding everything between -2 and 2 inclusive.
stark ether
there are real numbers between -2 and 2 too
toxic harbor
Ahhh okay
How do you write that in interval notation?
(-infinity, infinity) - {-2 and 2}?
- {-2 and 2} this is wrong but idk what is right
stark ether
I understand nothing
see lets go over it again real quick
for $\frac{1}{\sqrt{x^2-4}}$
we get 2 conditions for the domain
$\sqrt{x^2-4} \ne 0$ as denominator can't be 0
and $x^2-4 \ge 0$ for the radical
combining the inequalities we get $x^2-4 > 0 \implies x^2 > 4 \implies x \in \mathbb{R}-[-2, 2]$
royal basin
{-2, 2} would mean the set consisting of just the points 2 and -2 and nothing else
it's >, not \gt.
stark ether
oh
warm shaleBOT
ホタル
toxic harbor
{-2, 2} would mean the set consisting of just the points 2 and -2 and nothing el...
Oh okay
I see
(-infinity, infinity) - [-2,2]
stark ether
yes that's correct
royal basin
is that genuine or sarcastic?
stark ether
💀
toxic harbor
is that genuine or sarcastic?
LOL, nono genuine
It was my fault that I didn't understand
But the explanation was really good lol
Honestly
tysm
stark ether
toxic harbor
For this one
If we didn't simplify it to (1/x) (- 4), would we have not been able to use negative numbers as x?
If so, it's kind of strange that we can at one point and then can't at another.
stark ether
tbh I'm not sure about this one @royal basin could probably help, from what I've done before we do look at the original function without simplification
toxic harbor
tbh I'm not sure about this one <@183668144404037632> could probably help, from ...
Okay, thank you!
royal basin
and here we come to the ambiguity inherent to any attempts to infer the domain of a function from its formula alone
presumably we have m: R -> R defined by m(t) = t^2 - 4, and p: (0, +∞) -> R defined by p(x) = 1/sqrt(x)
and if we don't engage in these mindgames where the functions' signatures are hidden from us, their composition has the same domain as the inner function
i.e. (0, +∞)
so their composition has the formula x ↦ 1/x - 4 (which, by the way, CANNOT be written with parentheses around the -4) but its domain is not R \ {0} as this answer key claims
stark ether
so lets say you had f(x) = (sqrt(x))^2, does that mean the domain is R+ and not R?
royal basin
as written, the inferring-from-formula domain is indeed [0, +∞)
the problem is that domain-as-inferred-from-formula is not invariant under simplification
and this leads to all sorts of mysticisms about how you need to find the domain before simplification or shit like that
toxic harbor
Okay so when you simplify you can get a different domain that when we don’t simplify in these isolated cases?
royal basin
...
toxic harbor
I’m sorry then I didn’t understand
You said ambiguity
So I figured that this was the case
toxic harbor
the problem is that domain-as-inferred-from-formula is not invariant under simpl...
This too. Doesn’t this literally mean that when you simplify the domain can change?
royal basin
yes, but at no point during this tirade did i mean for you to interpret any of it as an instruction
toxic harbor
I didn’t take it as an instruction, I’m just wondering if that’s what you meant.
So is it recommended to find the domain before simplifying then, or after?
royal basin
i'm not going to recommend anything
toxic harbor
.close
distant moth
obtuse pebbleBOT
compact shadow
Just find {x: 5^x<=1 and 7^(-x)>7} union {x: 5^x>=1 and 7^(-x)<7}
distant moth
Just find {x: 5^x<=1 and 7^(-x)>7} union {x: 5^x>=1 and 7^(-x)<7}
i got the first part of what u said when i tried but how do you know that we have to looks for the ones where 5^x is greater 7^-x is lesser, like when i did other domain problems related to square root, i never had to think of this way, i would mostly >=0 and take intersections and i never usually took extra steps
or maybe the excercise was purposefully designed this way so i dont encounter them at the beginning
compact shadow
a/b >=0 iff a>=0,b>0 or a<=0 and b<0
timid silo
can someone help me determine f(r(t))
obtuse pebbleBOT
can someone help me determine f(r(t))
sweet flint
What is the solid line called through x+y so i can research, i don't get it. x'.y' is simple enough
distant moth
What is the solid line called through x+y so i can research, i don't get it. x'...
i think it means complement
not sure if its right word tho, complemt of A is basically what is not in A
sweet flint
Your right i think, wasn't sure why it was joined.
distant moth
its saying like (x union y) complement as in complement over the entire thing
distant moth
Opposite of x + y = opposite of x*y
im a bit confused how did this come💀
i think u mean opp of x * opp of y on the rhs
obtuse pebbleBOT
@timid silo Has your question been resolved?
compact parcel
can someone help me determine f(r(t))
$F(x, y, z) = (y^2, y^2, zx)$. This means when the function $F$ changes $(x, y, z)$ to $(y^2, y^2, zx)$. So, $F(r(t)) = F(\cos t, \sin t, t)$ by directly substituting. By the same way $F$ changes $(x, y, z)$ to $(y^2, y^2, zx)$, $F(\cos t, \sin t, t)$ will give $(\sin^2 t, \sin^2 t, t\cos t)$.
obtuse pebbleBOT
warm shaleBOT
pi over four
obtuse pebbleBOT
Available help channel!
raven plume
can someone help me calculate determinant of
obtuse pebbleBOT
can someone help me calculate determinant of
warm shaleBOT
alpine raven
raven plume
I think there must be some way to simplify the matrix
but idk how
bruh do I need to calculate this the traditional way
.close
obtuse pebbleBOT
obtuse pebbleBOT
Available help channel!
grave halo
Does anyone know where to find a proof / intuitive explanation as to why
grave halo
when calculating oblique asymptotes
m = lim x-> inf [f(x)/x]
and h = lim x->inf (f[x] - mx)
haughty coyote
Intuitively, if you measure the slope from (0,0) to (close to infinity), then the offset compared to the true slope is caused by the vertical offset you had before before sufficiently straight. But when you divide by x, that offset is sent to 0. You can imagine more simply if f does whatever it wants and then abruptly switches to a straight line. The final slope will be the slope from (0,0) as the initial conditions are made irrelevant by infinity
grave halo
Why does the offset dissapear when you divide by x @haughty coyote?
haughty coyote
If we take the simpler example where it's a straight line after some time, we can write (for x big enough) f(x) = ax+b. Then f(x) / x -> a
grave halo
yes
haughty coyote
Then it's about rigorously making sure that if you approach that slope, this is still going to work.
grave halo
Then it's about rigorously making sure that if you approach that slope, this is ...
So you are saying that dividing by x is akin to taking the derivative of the function and therefore finding the slope? If so, why do we take the limit of the function divided by x and not the derivative, as in some cases I don't believe it would give the same answer.
haughty coyote
Bold of you to assume f is differentiable
grave halo
Thanks so much for your help btw, I just am looking to fully grasp the concept as I need to present it
grave halo
Bold of you to assume f is differentiable
Ah I haven't learnt about non-differentiable function