np
#help-36
storm storm
steep forum
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dull matrix
i need help with logistic growth
marsh temple
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static oak
Is this correct
loud acorn
when doing the root test on a power-series (p-series) (hopefully it translate to p-series) is there a way to find out if it is absolute convergent and divergent for the result.
final saddleBOT
when doing the root test on a power-series (p-series) (hopefully it translate to...
loud acorn
As an example. let say the root test gives 1, therefore the convergence radius R=1. My teacher, always writes directly after that for example, x<1 is convergent and x>1 is divergent, how does he come to that conclusion? I assume he must be doing some test that is not written but I am not sure.
- Is there any better chanel to be asking these kinds of questions about series?
yes, hold on
question
translate to find what x converge the series
and to use the conjugat and seperate to two different term
full solutions
loud sundial
Ig he used the ratio test
loud acorn
yes, sorry. But still doesnt understand
loud sundial
https://tutorial.math.lamar.edu/classes/calcii/ratiotest.aspx
does this look familiar
loud acorn
yes, but checking again did he not use the root-test?
loud sundial
Oh true
loud acorn
(a_k)^1/k is root test right?
loud sundial
Oh I see
loud acorn
does this apply to all series that has a convergence radius of 1
loud sundial
Radius is 1 -> it converges for all x that are up to 1 away from 0
So yeah if you had just x and R=1, you can say it converges for |x|<1
Ignore what I said esrlier
I have 2 brain cells rn
loud acorn
okay, alright.
bro i appreciate all the help i can get, i am very thankful!
thank you so much i got it now!
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glad sandal
How can I simplify this further?
8.5 * (7.5 * 10^8) / (6.6710^-11)
final saddleBOT
How can I simplify this further?
8.5 * (7.5 * 10^8) / (6.6710^-11)
sharp shadow
How can I simplify this further?
8.5 * (7.5 * 10^8) / (6.6710^-11)
well you could do 8.5 * (7.5 * 10^8) and get
result / (6.6710^-11)
glad sandal
The number is too massive, @sharp shadow . And i need to compare this to earth's mass to see which is larger
I cant easily compare that massive whole number to 5.9722*10^22
sharp shadow
ahh i see
bright epoch
How can I simplify this further?
8.5 * (7.5 * 10^8) / (6.6710^-11)
Is there a missing *?
i feel like that should be 6.67*10^-11
glad sandal
i feel like that should be 6.67*10^-11
Should be the gravitational constant. Now that I see it I feel like there should be a * there as well, but I'd have to check what is is again.
bright epoch
if it's G yes there should be a *
glad sandal
if it's G yes there should be a *
Good catch.
bright epoch
in which case u can move the 10^-11 up top and get 10^11
to get (8.5*7.5/6.67)*10^19
and comparing that to the earth's mass is way easier
glad sandal
Exaclty what I need! Thanks @bright epoch.
But if I may ask:
How exactly did you "move" 10^-11 to the "top"?
I feel quite silly asking.
bright epoch
well, 1/(a^-b) is identical to a^b
sorry by move to the top i mean converting the fraction such that that part of the denominator becomes part of the numerator in such a way as to give something numerically identical
bright epoch
e.g. if I have $\frac{5}{2^{-2}}$, that's the same as $\frac{5 \cdot 2^{2}}{1}$
soft zealotBOT
Out Of Nosh
glad sandal
Oh ok. Interesting. I never knew that.
Thank you, @bright epoch . You were incredibly helpful.
This answers my question.
Now how do I close the help channel?
glad sandal
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sacred lynx
in a garden there is apple and orange trees. bob picked 300kg from one apple tree andd 800kg from orange tree . from one tree he picked 600kg average what percent of trees are the apple trees
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in a garden there is apple and orange trees. bob picked 300kg from one apple tre...
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sacred lynx
<@&286206848099549185>
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crisp carbon
Find a basis, B, of $R^3$ with the first vector of which is $\vec{b_1} = \begin{bmatrix}1\0\0\end{bmatrix}$ and so that the coordinates of $\begin{bmatrix}1\2\3\end{bmatrix}$ with respect to B are $\begin{bmatrix}0\1\1\end{bmatrix}$ and the coordinates of $\begin{bmatrix}3\2\1\end{bmatrix}$ are $\begin{bmatrix}1\0\1\end{bmatrix}$
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Find a basis, B, of $R^3$ with the first vector of which is $\vec{b_1} = \begin{...
soft zealotBOT
š Is Toast Modern? š
crisp carbon
I tried doing an algebra type system where b_1 + 2b_2 + 3b_3 = 0,1,1 and so on, but that didn't get me very far
or is it saying that b2 + b3 = 1, 2, 3?
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@crisp carbon Has your question been resolved?
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@crisp carbon Has your question been resolved?
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@crisp carbon Has your question been resolved?
lime ginkgo
@versed meteor
lament geode
hey @crisp carbon im taking a peek at this now, ill let you know if i come up with anything
You are correct, you need a system of equations. Here is how I started
lime ginkgo
@unverified
modest pollen
Hi i need help with naming carbon compounds, ik this is a maths server but i rly need help im sure there is people who can help me, all the other chem servers im in rn are super ded nobody online
final saddleBOT
Hi i need help with naming carbon compounds, ik this is a maths server but i rly...
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stray lark
how do i do this?
final saddleBOT
how do i do this?
west berry
Substitute the equation of line in the parabola
And it should intersect at one point, so after you get the quadratic in x set D=0
stray lark
And it should intersect at one point, so after you get the quadratic in x set D=...
sp (mx+2)^2 = 5x?
west berry
Yes
stray lark
And it should intersect at one point, so after you get the quadratic in x set D=...
i use discriminant?
west berry
Yes
west berry
Cuz it's a tangent
west berry
Np
stray lark
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paper talon
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paper talon
i get the 9-2x part
but then at the bottom left
we multiply by x again
why?
<@&286206848099549185>
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woven valley
Volume=lengthĆwidthĆheight
Height=x
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paper talon
Volume=lengthĆwidthĆheight
Height=x
but we used x as the width for like the squares we cut out
final saddleBOT
but we used x as the width for like the squares we cut out
woven valley
The rectangles x*(9-2x) are folded up,so x becomes height
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steel tinsel
help with this please
final saddleBOT
help with this please
versed crater
Whatās an orthogonal matrix
steel tinsel
R^T dotted with R is equal to the identity
basically a matrix inverse is just its transpose
steel tinsel
im trying that but stuck
versed crater
How did your first vv^t become v^tv
versed crater
(vw)^t = w^tv^t no?
versed crater
So you have (vv^t)^t
steel tinsel
yes
versed crater
But (vw)^t is w^tv^t
steel tinsel
i get it now it flips
steel tinsel
which means R^t is just R
versed crater
It does seem so!
steel tinsel
so now i want RR =In
versed crater
Wdym dot product
steel tinsel
i find it tricky to do it when i dont have values
versed crater
We have no concept of a dot product
steel tinsel
is it not R dotted with R equal to identity
versed crater
They are matrices we do matrix multiplication to them
Dot products are defined for vectors
āNot matricesā (this is technically false but not the point)
steel tinsel
but matrix multiplication is just dot the first row with the columns of the other etc
i will try and just multiply by distrubtuion it dosenet feel correct
steel tinsel
This where Iām air
At not air
omg i see it
i think
it just simplify to In which i wanted so R is orthogonal correct or no?
no wait maybe not
versed crater
Look at your wack term
The last one
Pull the constants out
You get vv^tvv^t
But v^tv is clearly just v ā¢ v ok the dot product is kinda useful I guess
But thatās also just ||v||Ā²
Which is a scalar so just pull it to the front
steel tinsel
why can i pull the constants outfrom the first one?
versed crater
aAbB = abAB where lower case is scalars and upper case is matrices
Since matrices are linear
steel tinsel
Like this
versed crater
,rotate
soft zealotBOT
steel tinsel
Ahhh wait I see what your saying
versed crater
Yo what happened to the square on the norm of v on the last line
It just disappeared into this air
versed crater
Also (-2)Ā² is most definitely not -2
steel tinsel
I though you said pull out constant
versed crater
Also (||v||Ā²)Ā² is also not ||v||Ā²
Yeah but thereās still 2 copies of the constants
steel tinsel
Ahh wait so square it first then pull it that makes more sense to me
steel tinsel
Ok give me a few min and Iāll give you nest solution of where im at if thatās ok
versed crater
aAbB = abAB where lower case is scalars and upper case is matrices
Thatās why I said this
steel tinsel
Thank you
versed crater
Except a and b were your constants
soft zealotBOT
steel tinsel
Just want now to understand why v^tv is v dotted v
thats the only bit i donet get
and also does R being symmetric have anything to do with it
versed crater
Well the way they constructed R makes it symmetric
Or do you mean does being orthogonal imply symmetry
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gleaming bluff
can someone help me with this?
final saddleBOT
can someone help me with this?
gleaming bluff
this is the sketch btw
i havent included multiplying by -2 here
but im solving question according to desmos and according to desmos f(x) = 1/x and f(x-2)'s graph is not this
so im doin smthn wrong but idk what im doing wrong
this thing u see is my graph for f(x-2)
This is the one according to desmos
it doesnt even intercept with +1
gleaming bluff
can someone help me with this?
va = vertical asymptote
ha = horizontal asymptote
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ocean robin
Guys
A = 115Ā° b = 15 c = 10
Find a B C
Help me solve this guys please
Im having a hard time
final saddleBOT
Guys
A = 115Ā° b = 15 c = 10
Find a B C
Help me solve this guys please
Im having...
ornate knot
use laws of sines and cosines to solve
actually u can only use law of sine in here to solve the entire question
pliant shore
actually u can only use law of sine in here to solve the entire question
Nah you need law of cosines first to find side a
Then law of sines
final tangle
cos will be required
ocean robin
yeah a / sin A b / sin B c / sin C?
ornate knot
Nah you need law of cosines first to find side a
cant u find angle C first and then by property of triangle find angle B and then side b?
oh wait thats angle A and not B , my bad
pliant shore
cant u find angle C first and then by property of triangle find angle B and then...
You don't know angle B though just from the info in the question
Okok
ocean robin
Ok which angle do i start on
pliant shore
Ok which angle do i start on
Not an angle
Start by finding side c using the cosine rule
ocean robin
c^2=a^2+b^2-2ab cos(C)
pliant shore
c^2=a^2+b^2-2ab cos(C)
Yes
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frank nebula
How to integrate from -pi/2 to pi/2; (sin^2(x))/(1+sinxcosx)
final saddleBOT
How to integrate from -pi/2 to pi/2; (sin^2(x))/(1+sinxcosx)
warm python
try using king's rule
warm python
$2I=\int \frac{dx}{1+\sin\left(x\right)\cos\left(x\right)}$
soft zealotBOT
Ę(Why am. I here)=I don't know
pliant shore
numerator became one
Wait so what integral do you get now?
pliant shore
$2I=\int \frac{dx}{1+\sin\left(x\right)\cos\left(x\right)}$
Damn
frank nebula
yes
warm python
now integrate this
frank nebula
will dividing by cos^2(x) work?
warm python
probably
final saddleBOT
pliant shore
Alternatively, $I = \int \frac{\sin^2 x}{1 - \sin x \cos x} \ dx$
soft zealotBOT
south
pliant shore
So $2I = \int \frac{2 \sin^2 x}{1 - \sin^2 x \cos^2 x} \ dx$
soft zealotBOT
south
pliant shore
Ah that integral isn't as nice
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gleaming bluff
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gleaming bluff
this is the sketch btw
.
I havent included multiplying by -2 here
but im solving question according to desmos and according to fedmod f(x) = 1/c and f(x-2)ās graph is not this. So im doing smthn wting but idk what im doing wrong.
Desmos shows this
it doesnt even intercept with +1
va = vertical asymptote
ha = horizontal asymptote
i asked this question here before but the channel got closed due to timeout after a short time so i guess i can tag helpers
this happened 3 hours ago
<@&286206848099549185>
tired galleon
Desmos shows this
it doesnt even intercept with +1
The minus part of -2 is very important
have a look at -f(x) and see if that makes things a little clearer
If the part confusing you is why the y-int isn't where you expect it?
inland kettle
!15m
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pliant shore
Shape isn't correct for -2f(x - 2)
Recall that y = 1/x fills only the 1st and 3rd quadrants: the 2nd and 4th quadrants are completely empty
So y = -1/x, reflecting 1/x across the y-axis, fills the 2nd and 4th quadrants and leaves the 1st and 3rd quadrants empty
pliant shore
I havent included multiplying by -2 here
but im solving question according to ...
So you definitely need to take the -2 into account
-2 * -2 is 4 > 0 so the shape will look similar to 1/x
Except that it is translated right a few units
Also it's not that hard to find $-2f(x - 2) = -2 \frac{-2}{(x - 2) - 2}$ explicitly
soft zealotBOT
south
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gleaming bluff
The minus part of -2 is very important
idk what happens when i multiply
gleaming bluff
!15m
i already explained the reason I tagged them. I asked this question 3 hours ago. Please read my texts before writing this š¤¦š»āāļø
tired galleon
So y = -1/x, reflecting 1/x across the y-axis, fills the 2nd and 4th quadrants a...
This happens
-f(x) is a reflection in the y-axis
The 2 is just a stretch scale factor 2, also in the y axis (main thing being it doubles the value of your y-int)
Anything outside the brackets in f(x) is in relation to the y-axis and anything inside is in relation to the x axis (and also sorta does the opposite of what one might assume, hence f(x-2) is a shift to the right by 2 units)
wintry cipher
How many routes exist from A to B? The two cubes touch at the bottom front righttop left rear corners respectively. Please explain which method and which principle you used to get your answer.
final saddleBOT
How many routes exist from A to B? The two cubes touch at the bottom front right...
zenith pollen
How many routes exist from A to B? The two cubes touch at the bottom front right...
is there any restriction, like can I just loop around a bunch on the A cube? usually these problems force you to travel down/right the entire time
wintry cipher
I donāt think you can visit the same vertice more than one
You just get from route A to route B using pascals method or the recursive method
zenith pollen
hmm I haven't seen pascal's method for paths that let you essentially go backwards before
zenith pollen
the way I'd go about it is counting the paths from A to the first corner, starting with a rightward move. you'd probably need to just draw out the 4 or 5 paths, and then with symmetry you can figure out all the paths to the first corner and then all the paths to B
zenith pollen
hmm for both of these methods you're assuming you can only take certain moves, like this weird looping path isn't allowed, and I'm wondering if it's allowed in your initial question
wintry cipher
It should be allowed
Iām sure as long as you go from route a to b without going over the same line your fine
zenith pollen
ok, I just don't think you can use pascal's method then, you sort of just have to draw all the paths for a simple case and then multiply it out to the full amount
like all the pascals' methods I see online have a certain movement restriction
zenith pollen
assuming any path at all, I'd start with this and find all the paths to get to the corner
wintry cipher
Ok lemme try
zenith pollen
if the problem said you can only travel down/right/frontward then you'd be able to use pascal/recursion for something like this, but your question would have to mention that??
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wintry cipher
if the problem said you can only travel down/right/frontward then you'd be able ...
Yeah ur correct
wintry cipher
if the problem said you can only travel down/right/frontward then you'd be able ...
This looks right thanks so much!!
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nocturne ravine
How can we prove that a power series is decreasing if we dont know its x value
nocturne ravine
Like in this question
You would assume the x value is in the IOC (0,1/10)?
And i k the formula for the power series is x^n/n
dapper lodge
I have question about combinatorics
final saddleBOT
I have question about combinatorics
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eager stirrup
how would i do this? im not sure where to start
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how would i do this? im not sure where to start
mellow axle
try a simpler version of this question, there are 3 stones and you can split any 2 stones into 2 groups of 1 stone each with equal weight
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eager stirrup
try a simpler version of this question, there are 3 stones and you can split any...
How does this scale up though?
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sturdy cypress
i don't get it but it's a cool question
eager stirrup
lol
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steep tendon
if any 12 stones can be put into 2 groups of 6 then bothe sides are = right? if you randomly change the arrangement of the 12 stones it stays the same
eager stirrup
interesting
steep tendon
so does that logically mean the 12 stones are the same
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sturdy cypress
i googled the solution
sturdy cypress
- the stones can't have both odds and evens, then it would always be possible to choose 12 stones that add up to something odd
- out of all possible solutions, for the weight of 13 stones, lets choose any solution where the max weight is smallest possible, and they aren't all equal. Is it all evens or all odds?
- if all stones are odd, we can subtract 1 from each weight, and it will still work, so it must be all evens
- but then we can divide all weights by 2 and it will still work
- so this solution can't exist, except if dividing doesn't do anything to the weights. That can only be 0,0,0,0,0,0... but we were choosing a solution where they aren't equal
very confusing
distant ridge
prob by contradiction
there is a stone that doesn't weight the same and for the 13 stones, you can pick 12 of them and split them into two groups of 6 of equal weight
any arrange would be weight{x,x,x,x,x,x} = weight{x,x,x,x,x,x} with y outside
swapping one x with y, would be weight{x,x,x,x,x,y} which doesnt equal weight{x,x,x,x,x,x} with x outside
ok probably i did a mistake cuz i dont think it would be that easy
sturdy cypress
yeah i get it, so any solution would necessarily let us do ā1 or /2 to it
which must end in all zeroes, therefore we started from all-equal
i'd never think of it
no wait
- a solution with all evens lets us divide it by 2 and get another solution
- this solution still has all odds or all evens, because it's a solution
- a solution with all odds lets us subtract 1 and get another solution
- this will eventually end with 0,0,0,0,0, and by reversing the process we would get to original solution. And that solution will be all equal
this seems flawless, so hard to think of though
frail mountain
Hi, what subjects am I working with here? I know it's multivariable calculus, but I don't know what keywords I should be looking up
final saddleBOT
Hi, what subjects am I working with here? I know it's multivariable calculus, b...
frail mountain
pliant shore
You need to set the partial derivatives with respect to x and y equal to 0, for questions 1 and 2
The key word is "optimisation"
desert mantle
googling for local extrema multivariable calc is probably gonna give better results
pliant shore
For 3, this should remind you of similar problems for functions in one variable
For example, the functions under the square root have to be 0 or greater
And like the denominator can't be 0 etc
frail mountain
Gotcha, there was also a specific way to describe the domain something like D:(x,y) etc is there a name for this format?
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ruby bough
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ruby bough
how whould i expand smth like this
terse folio
how whould i expand smth like this
the thing on the right?
dense crystal
notice $(x-2)(x+2)$
soft zealotBOT
Renz
dense crystal
you should recognize this is a difference of 2 squares formula
ruby bough
the thing on the right?
left is question right is answer
terse folio
you want to factorize or expand
dense crystal
oh
dense crystal
let u = x^2
yep do this
terse folio
u get u^2-3u-4, this can be factorized easily
(u-4)(u+1), resubstitute:
(x^2-4)(x^2+1)
a^2-b^2:
(x+2)(x-2)(x^2+1)
dense crystal
a good rule would be, if you notice x^4 and x^2 and a constant term
just make u=x^2
ruby bough
ok thank you
tranquil pine
a,b,c is a geometric progression. b-a c-b and c-a is a arithmetic progression. Find q. Q is (r or multiplication calue.)
final saddleBOT
a,b,c is a geometric progression. b-a c-b and c-a is a arithmetic progression. F...
tranquil pine
.
vast summit
b/a=c/b, (c-b)-(b-a)=(c-a)-(c-b)
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tranquil pine
b/a=c/b, (c-b)-(b-a)=(c-a)-(c-b)
Find q
vast summit
is q the common ratio of the geometric series? If so then try rearranging the second equation to get c as the subject, and substitute into the first to solve for b/a
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chrome prism
In a standard deck of playing cards, there are 52 different cards. Each card is one of 13 different values, and one of 4 different suits (of which there are 2 red suits and 2 black suits). How many 9-card hands contain four cards of the same value?
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In a standard deck of playing cards, there are 52 different cards. Each card is ...
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@chrome prism Has your question been resolved?
cyan kayak
@chrome prism you must have one of 13 values 4 times. So you have 13 choices, then the other 5 cards are chosen from the remaining 48, so 48 choose 5.
However, it is possible to have two sets of 4 cards using this method, so you need to subtract those because you double counted. So choose two sets, 13 choose 2, and then one card from the remaining 44.
chrome prism
sorry i forgot to reply to the bot!!! mb g
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chrome prism
<@312196139128586250> you must have one of 13 values 4 times. So you have 13 cho...
but yea essentially that is the solution thanks
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trail dove
My notes for pre-cal/trig probably need to be rewritten, but im almost done with class so just pushing through it. Will my pre-cal/trig notes be useful for future classes though?
final saddleBOT
My notes for pre-cal/trig probably need to be rewritten, but im almost done with...
dense coral
possibly
depends on how much more math you want to study, and how well youāve internalized the material already
trail dove
Im going for a comp science degree so going to be doing a lot of math
dense coral
well, if you already know the material like the back of your hand, your notes will probably not be useful because you wonāt need them
otherwise, they could help
lucid marsh
well you probably need precalc for, you know, calc, but calculus isnt that hard w/o notes from previous classes
dense coral
yeah if you havenāt already internalized the content, calc will probably force you to lol
after that, the notes will likely not be useful
but for calc specifically, perhaps they could be of use
trail dove
If I have the formulas Im usually good, thats what most my notes are.
They just need to be better organized, which I got some advice on already.
My calc and my trig notes are in the same book which is part of the problem as well lol
Alright, I may try and trim it down for reference later then still.
languid yoke
I've never really found previous notes to be helpful. Usually if the content overlaps, the prof will go over it again so you can just take new notes
trail dove
Im not understanding where the B_2 is coming from?
final saddleBOT
Im not understanding where the B_2 is coming from?
vast summit
there are usually multiple solutions to trigonometric equations
sin(theta)=sin(180-theta)
ornate knot
sinB is positive this means that angle B lies in either Quadrant 1 or 2 (in these quadrant sin(angle) must be positive) so if it lies in Quadrant 1
sinB = 0.522
B = sin inverse 0.522 (the general angle )
if it lies in Quadrant 2 , then
theta = 180 - general angle
btw cos theta is positive in quadrant 1 and 4
tan theta is positive in quadrant 1 and 3
trail dove
So is it only when its outside Q1?
Dont guess it matters for me actually, gonna be easier to check with 180-angle than decide the quad
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limpid burrow
How to find the smallest value of the function y=-xĀ² on the interval [-1;3]?
tranquil pine
are you familiar with derivatives?
limpid burrow
Yes
wise crescent
to find the answer to this while in exam, we can simply use logic
we want to find smallest value of -x^2
so that means we need the largest value of x^2
which we get at x=3
hence the smallest value is y=-9
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trail dove
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trail dove
So I just did law of cos after not being able to use law of sin
but im wondering why I get a different answer for law of sin. Isnt it 7*sin(34)/4?
then sin^-1(.9785875)
trail dove
A, hence the 34
distant ridge
with cosines?
trail dove
yeah, then attempted the other 2 with law of sins
but they were wrong
sorry realizing I may not have explained well enough.
I used law of cos for A which was correct, so then tried to use law of sin for B and C, but couldnt get the right answer so then just did law of cos for B and C
The answers for law of cos for B and C are right, but wondering why law of sin was giving me wrong answers for B and C
Since they should result in the same answer unless im misunderstanding something
distant ridge
ok i think i know why
because of the range of sin^-1
it would be from -90 to 90
and b has an angle greater than 90
sin(180-x)=sinx
because of sin^-1 you got then sin^-1(.9785875)=78.12183566197
but the angle is greater than the range of sin^-1
so it should be 180-78.12183566197
for c,.. uhm i dont think you would get a wrong answer (using sines)
trail dove
I dont think I did c actually, gave up with b now that I recall
distant ridge
cosines always work because the range of arccos is from 0 to 180
and any angle of a triangle is in that interval
ok maybe too much stuff
recap
- sin^-1(x) returns a value between -90 to 90 degrees
- because the angle you are trying to find is greater than 90 degrees, you have the use the identity sin(180-x)=sinx
- 180-78.12183566197 aprox 102
trail dove
Ok, I think I gotcha
I wasnt aware of 1.
2. & 3. make sense though
Should I just stick with cos for future reference then?
distant ridge
cosine will never fail, but using sines could be a shortcut if you have a picture of how the triangle looks like (if the angle is not acute, then use 180-angle)
ok just
yes, cosine will never fail
but you can still use sines by doing 180-arcsin for "big" angles
the ones greater than 90
trail dove
Alright, I think I got what youre saying. has to do with the range and since triangles are always 180 cos always works vs sin you need to do the identity after 90
distant ridge
yes
trail dove
Much appreciated then, usually just doing equations without fully understanding what's going on so nice to get a grasp on things lol
I sorta got the deleted part, but probably a bit out my wheelhouse for now. I got the main stuff though
bronze otter
I actually understand Sin and Cos now .. what the holy ...
trail dove
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ā
lusty pecan
so what have you tried
trail dove
I made a graph and the points make a line
thats it
ive never seen a question like this and the book doesnt show a example of one.
lusty pecan
have you learned how to find the distance between two points?
as a hint its related to the pythagorean theorem
trail dove
sqrt((x_1-x_2)^2+(y_1-y_2)^2)
Ok, I think I can work with that. Forgot about it honestly. š
lusty pecan
all good lol
trail dove
It just making a line on the graph also just confused the hell out of me
I got it, thanks for the reminder lol.
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tranquil pine
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fiery prism
how would i solve b
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how would i solve b
random rampart
put everything to left hand side
to make it into something like ax^2+bx+c=0
then solve for x
honest egret
and you have to change the signs
fiery prism
what
final saddleBOT
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signal vector
Move everything to one side
You will have a quadratic equation after doing so
Solve the quadratic equation
wise flame
Hello! I'm trying to wrap my head around double and triple integrals. It makes sense to me that Double integrals would be finding the volume under a 3-D surface. So the area of the region would just be the double integral when f(x,y) = 0
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Hello! I'm trying to wrap my head around double and triple integrals. It makes s...
wise flame
for triple integrals, it is finding the 4D volume? I'm not sure how to think of this
formal trail
well if you integrate f(x,y,z) = 1 then it would be the volume, like if you integrate f(x,y) = 1 it's the area
wise flame
additionally if w = 1 then am i finding the volume of the region?
is the region 3d?
yeah
i think it is
so then when we find mass from a mass density function whats really going on there
what is a mass density funciton
my text didnt really explian
formal trail
other than that, while there is technically a "4d volume" you could pursue, it may be easier to go with some other physical analogy, for example if f(x,y,z) is the mass density of a region, then its integral is the mass
a mass density function is a function that gives the amount of mass per unit volume in a region
formal trail
mass/volume * volume is mass, so when we integrate we are multiplying it by the volume and getting mass out
wise flame
oh
formal trail
so literally the density
yes, but you can also have densities of other things. for example, in electrostatics you can have a charge density
wise flame
right
wise flame
mass/volume * volume is mass, so when we integrate we are multiplying it by the ...
im trying to understand this but im a bit lost
aren't we summing the mass/volume
OPOOh
srry
f(xi,yj,zk) dV
sum
the differential
is that right?
formal trail
yes
wise flame
where dv. = dx dy dz
oooh
so f(x,y,z) is the mass density function = mass/volume
ok
still feel like im missing something
because the triple riemenn sum
are we just summing the mass in different directions
which gives us total mass
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dense garnet
Define a parallel rectangle P in R^n to be an n-fold Cartesian product of compact intervals, so its sides are parallel to the coordinate axes. Let an oblique rectangle O be a rectangle whose sides are not parallel to the coordinate axes; it is obtained as the image under an orthogonal transformation T of a parallel rectangle, so T(P)=O. An almost disjoint collection of sets means that the sets intersect each other at most along their boundary. v(.) is the volume of a rectangle, i.e. the product of the length of its sides (where length of say [a,b] is simply b-a). Note, that since an orthogonal transformation preserves lengths and angles, we have v(P)=v(O).
Consider then the following lemma in my lecture notes:
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Define a parallel rectangle P in R^n to be an n-fold Cartesian product of compac...
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Philip
dense garnet
How hard is it to prove this lemma? Do you know a proof or a reference where this is proved? I'm grateful for anything! Prior to this, the author has proved some results about parallel rectangles which may be useful in proving the above lemma. If you want to know more, let me know and I'll post some more results the author has already proved.
The above lemma is from here by the way.
on page 27, section 2.8 on linear transformations
It's a critical result that most of the theorems in that section rely on, hence my eagerness to see a proof of the lemma.
fading dock
How hard is it to prove this lemma? Do you know a proof or a reference where thi...
I think I remember discussing this lemma in one of the analysis channels, is that correct? iirc it boiled down to whether the volume of an oblique rectangle is equal to its Lebesgue measure. Looking at it again, I remember a result for the Lebesgue measure on R^n that generalizes the properties of translation invariance and dilation equivariance; specifically, that for any invertible linear transformation T we have that for any Lebesgue measurable set A then m(T(A)) = |det T|m(A). I think you would need to use this property to prove the lemma, which would give you that last step
because basically you obtain an oblique rectangle from a rotation and a translation, so it has the same measure as the parallel rectangle it was achieved from because rotation matrices have determinent 1 and also by translation invariance
actually you don't even need the translation invariance, just the rotation
dense garnet
I think I remember discussing this lemma in one of the analysis channels, is tha...
indeed, I posted about it in one of the analysis channels. thanks for responding there and here š I agree, it does boil down to whether the volume of an oblique rectangle is equal to the Lebesgue measure of the oblique rectangle. I think this has not yet been established in the text. But do you think that the identity m(T(A)) = |det T|m(A) can be proved without the above lemma? The identity m(T(A)) = |det T|m(A) is in that section as another proposition, but it comes later.
fading dock
indeed, I posted about it in one of the analysis channels. thanks for responding...
I think you should be able to prove it without the lemma because it suffices to just show that the identity holds true for the three types of elementary transformations
does your textbook assume the lemma in that proof? I would think not
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dense garnet
does your textbook assume the lemma in that proof? I would think not
well, it does kind of š„² the thing is, this lemma appears at the very start of that section (on linear transformations of Lebesgue measure) and then the author uses it to prove another proposition, which in turn is used in proving m(T(A))=|det T| m(A). Maybe there is an alternative way of proving this, I don't know...
fading dock
hmm that's unfortunate. what's your textbook, out of curiosity?
dense garnet
hmm that's unfortunate. what's your textbook, out of curiosity?
I'm using these lecture notes. In the document outline on the left, you can navigate to section 2.8, where you'll find the lemma (it's page 31 in the pdf). Theorem 2.31 is the theorem you are talking about, i.e. mu^star(T(A))=mu^star(A), where T is orthogonal and mu^star is Lebesgue outer measure, which is the Lebesgue measure when restricted to the Lebesgue measurable sets.
fading dock
I'm using these [lecture notes](https://www.math.ucdavis.edu/~hunter/measure_the...
Hmm yeah it seems this source goes about things in a different order than I have previously seen. Unfortunately, without assuming anything about the behavior of the Lebesgue measure under linear transformations, I can't think of any clean way to show the lemma. It probably requires a very tedious argument using inner or outer regularity or some other construction but I'm not sure. Sorry I couldn't be more help.
dense garnet
Hmm yeah it seems this source goes about things in a different order than I have...
ok š„² which book/text did you use when studying the Lebesgue measure?
fading dock
ok š„² which book/text did you use when studying the Lebesgue measure?
primarily Folland
dense garnet
primarily Folland
I don't think he ever proves rotational invariance š there is the following theorem:
i.e. invariance under translation and dilation, but I have not been able to find anything about a general linear transformation
fading dock
He does. Theorems 2.44 and 2.45 and corollary 2.46
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lime cedar
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lime cedar
The explanation is this, but I would like some help interpreting it.
I haven't taken a Stats class, but in my state's competition you're expected to know how to use combinations and permutations to find probability so oftentimes I don't know what the explanations are trying to say
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@lime cedar Has your question been resolved?
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@lime cedar Has your question been resolved?
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trail mango
damn i wanted to prove it carelessly
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@old hornet Has your question been resolved?
static fractal
the LHS is the number of ways to select a k-subset of from [n]. for each subset, we could alternatively choose to include the element 1, then pick k-1 elements from the remaining n-1, or exclude the element 1, then pick k elements from the remaining n-1
the RHS corresponds to this second way of picking subsets
try giving an explicit bijection using this construction
tranquil pine
being f and g defined as [a, +infinity[and such that lim x-> +infinity f(x)/g(x) =0, lim x->+infinity g(x)=0 and g(x)!=0, to every x >= a. Calculate , if exists, lim x->+infinity f(x).
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being f and g defined as [a, +infinity[and such that lim x-> +infinity f(x)/g(x)...
tranquil pine
someone can explain it to me
muted ore
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muted ore
not sure how to do this?
I don't know how to parametrise it
or if that's even necessary
formal trail
to parameterize a line, the simplest way is to pick a range for t from 0 to 1
then you know at t = 0 it's at the starting point and at t = 1 it's the end point
and in between then it should be a linear function of t
muted ore
right, that makes sense
a function r(t)=(x_1 * t,x_2 * t) would work? @formal trail
with t ranging from 0 to 1
formal trail
yes
muted ore
yes
Well
Is this the correct setup?
dotP being the dot product between the two vectors
formal trail
the line integral of the vector field will be V(r(t)) * dr/dt
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solemn nimbus
Open.
solemn nimbus
Number 3.
I was having trouble understanding the solution.
If anyone could explain it to me, it would be great.
Or if anyone has a different approach, feel free to let me know.
I got it on my own.
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uneven quarry
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uneven quarry
how do i find the normal vector of the plane
this is the vector equation of the plane
zenith pollen
you do the cross product of the vectors with variables in front
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plush mason
is there any technique to solve this rapidly
final saddleBOT
is there any technique to solve this rapidly
wintry kindle
is there any technique to solve this rapidly
GauĆ-Algorithm
for an arbitarily large linear system of equations
craggy plank
is there any technique to solve this rapidly
There are infinite solutions to these equations
The two equations represents a line in a 3D space
plush mason
The two equations represents a line in a 3D space
what ?
plush mason
GauĆ-Algorithm
this one ?
plush mason
The two equations represents a line in a 3D space
are you like trying to form a 3d vector
wintry kindle
this one ?
yop, not sure if GauĆ Jordan is a different variant
but GauĆ-Algorithm is the main one called
which, again, lets you solve any linear system of equations
regardless of size
craggy plank
are you like trying to form a 3d vector
2x-3y+5z=0 and 10x-4z=0 are two different plane respectively
The two non-parallel plane can form a line
plush mason
can u graph it
craggy plank
With parametric equation, you can briefly list every spots on the line
craggy plank
this one ?
This method allows you to notice that the equation has infinite solutions
plush mason
so this line is the solutions
craggy plank
so this line is the solutions
Yes
plush mason
This method allows you to notice that the equation has infinite solutions
but it's not infinite
plush mason
i think it's wrong
wintry kindle
i think it's wrong
nop
craggy plank
There is not only one parametric equation
plush mason
but from the very bigging
wintry kindle
like x = n
the first image solves it via GauĆ-Algorithm and gives you x,y,z with respect to z. In the second image I define the set of solutions to be independent of x,y,z to get the set of vectors which fulfill the given system of equations
plush mason
wait does this have any relation with the vector space lesson
wintry kindle
wait does this have any relation with the vector space lesson
the resulting set of vectors is a line, sure
its a subvector space of RĀ³ in this case
as it goes through the origin
and is closed under addition/multiplication
and Sky/Night provided the geometric interpretation
you have two equations in normal form
=> two planes
and any vector fulfilling both equations
must lie in the intersection
which is, at its largest, a plane
and at its smallest, nothing
in this case the intersection was a line
meaning the two planes are intersecting, but they aren't parallel
because if they weren't intersecting, the set of solutions would be empty
if they were parallel and intersecting, then the set of solutions would be a plane
wintry kindle
the resulting set of vectors is a line, sure
but it's a line, since we only need 1 variable.
plush mason
because if they weren't intersecting, the set of solutions would be empty
you mean they got only 0R3 as a solution
now i understand better
this equation was for getting the ker of an app
but i think that they are wrong
look
wintry kindle
it's the exact same solution I provided above
just abstracted a little using the linear combination brackets <...>
plush mason
it's the exact same solution I provided above
yeah bt look at the last line
plush mason
just abstracted a little using the linear combination brackets <...>
yeah yeah i didn't notice it my bad ...
wintry kindle
GauĆ applied:
I prefer this representation
because it makes it more clear what the steps were
and this process is better to generalize
than the way they wrote
plush mason
and this process is better to generalize
yeah and can i use that method to calculate any equation ??
wintry kindle
yeah and can i use that method to calculate any equation ??
any linear system of equations
yes
plush mason
any linear system of equations
yeah thank you alot for help !!!!
wintry kindle
yeah thank you alot for help !!!!
np :)
here a little piano piece I wrote as a reward š¦
plush mason
here a little piano piece I wrote as a reward š¦
keep the good work it's sick ngl
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tranquil pine
Can anyone use the fact that derivative of x^2 is 2x and construct a world problem so that i can understand how differentiation works in real life
I am very new to calculus
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Can anyone use the fact that derivative of x^2 is 2x and construct a world probl...
tranquil pine
how fast are you going at x = 1
Ooooooooo
Thanks thanks
1 more plz
Am slowly understanding
pliant shore
pliant shore
Can anyone use the fact that derivative of x^2 is 2x and construct a world probl...
Imagine you have a square like this, where the blue square is x by x
And your square's sides are increasing at a rate of 1 unit per hour
Then your area is actually increasing by 2x per hour
At that instant
tranquil pine
how fast do you gain weight at time x = 2
So basically derivative is difference between successive values of a function
Or slope of tangent on a point on function
Or both mean the same
tranquil pine
So basically derivative is difference between successive values of a function
Or...
@pliant shore
tranquil pine
Then your area is actually increasing by 2x per hour
Give me a bit time to comprehend this
pliant shore
So basically derivative is difference between successive values of a function
Or...
Yes, two different definitions
pliant shore
Give me a bit time to comprehend this
It's not easy to comprehend, but you can get an idea of this from (x + 1)^2 - x^2 = 2x + 1
So you need the difference to be very very tiny
mellow axle
difference between successive values of a function
yes but the 'successive values' are getting arbitrarily close together
pliant shore
If we have (x + 0.01)^2 - x^2, you get 2(0.01)x + 0.01^2
tranquil pine
> difference between successive values of a function
yes but the 'successive val...
Ohhh
Thanks mate
pliant shore
So we can write this tiny value as 2(dx) x + dx^2
mellow axle
derivative is just slope between two points, when those two poitns are getting closer and closer
pliant shore
But dx^2 goes to 0 very quickly, faster than dx goes to 0, so we can say when dx is practically 0, the dx^2 term disappears
pliant shore
derivative is just slope between two points, when those two poitns are getting c...
YES
Nice one
mellow axle
THANKS BRUV
pliant shore
derivative is just slope between two points, when those two poitns are getting c...
So the distance of these points need to approach 0
That's the really important bit
Approaching 0
Or arbitrarily close etc
tranquil pine
And your square's sides are increasing at a rate of 1 unit per hour
And what if sides are increasing at rate of 2 unit per hour
pliant shore
And what if sides are increasing at rate of 2 unit per hour
Well, you would have ((x + 2 dx)^2 - x^2)/(dx)
Or (2 * (2 dx) + (2 dx)^2)/dx
But dx goes to 0 so you would have (4 dx + 4 (dx^2))/dx = 4 + 4 dx -> 4
Or 4 units per hour
Shit you need to divide by dx
Yeah this is the proper explanation, sorry for that
tranquil pine
Yeah this is the proper explanation, sorry for that
Kindly plz don't use dx and use numbers for now
I am just 1 day old to calculus
pliant shore
Kindly plz don't use dx and use numbers for now
I am just 1 day old to calculus
Just imagine dx to be like 0.00000001 or something
pliant shore
Well, you would have ((x + 2 dx)^2 - x^2)/(dx)
(x + 2 dx)^2 - x^2 is the change in y, the change in area
dx is the change in x
So (change in y)/(change in x) = slope
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@tranquil pine Has your question been resolved?
tranquil pine
(x + 2 dx)^2 - x^2 is the change in y, the change in area
So the rate of change of area is 2x
At any instant right?
pliant shore
So the rate of change of area is 2x
At any instant right?
Yes exactly
final saddleBOT
Yes exactly
pliant shore
There you go, we've shown that d/dx x^2 = 2x
tranquil pine
Yes exactly
š«
pliant shore
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misty sequoia
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misty sequoia
could anyone explain the meaning of what's written in the parentheses?
versed cave
A union B is the set of all x in the set M such that x is in A or x is in B
it means that, given two subsets of M called A and B, all of the elements of AāŖB are elements x of M such that x is in either A or B (inclusive)
@misty sequoia theres your explanation, if thats all you needed please type .close
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misty sequoia
it means that, given two subsets of M called A and B, all of the elements of AāŖB...
Thank you for the explanation
Does 'v' sign mean 'or', and '^' sign mean 'and'?
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misty sequoia
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proud marsh
I know the answer but idk what is wrong in my working out
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I know the answer but idk what is wrong in my working out
gritty solar
$\frac{a^b}{a^c} = a^{b-c}$
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Ā²ā°Ne
gritty solar
you seemed to have taken it to be b/c instead
proud marsh
but when they all have the same base, can you not just equate the powers?
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Ā²ā°Ne
gritty solar
you need to have a set up like this
gritty solar
yes
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rich tide
you are my sunshine
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vale nova
Hello, how do I find the maximum and minimum of these? I donāt know how to do it with log and how to understand it
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Hello, how do I find the maximum and minimum of these? I donāt know how to do it...
hard lintel
find the derivative first
vale nova
find the derivative first
I did
hard lintel
then get the stationary points
vale nova
Itās
-e^-0,5x+6e^-3x
vale nova
then get the stationary points
When I do
0=fā(x) I donāt know how to continue because e is never 0 so Iām confused as how to find a solution
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glass pawn
How do you know when to use the ln
Whenever you have powers that you can simplify easily this way. Any tools are allowed and ln allows you to make it a simple linear equation
glass pawn
Maybe you may want to make it 2 over 5 as a final answer. Also, maybe check whether it is a maximum or minimum. But it seems correct anyway
glass pawn
You
Also, it is important that sometimes you may not be able to take a log of both sides
But in the case when you have anything (positive) to power x, it will be always larger than 0. Thus in practically all cases you may want to use log, You will be able to
But for the equation
$e^x = -1$
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mlysikowski
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red leaf
help?
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solid pond
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solid pond
i dont understand how i am supposed to answer this question
i feel like more than 1 answer is true
i know d is true for sure
i know a and b are wrong
but c seems true and d
solid pond
the way they layed it out was supposed to make all the variables obvious
u can see k here without factoring
<@&286206848099549185>
solid pond
wdym
using the variables given in the function deterime the transformation and which one is true or not
its a simple question my problem is i see there are 2 answers
vast summit
yeah but it doesnāt specify transformation from what
vast summit
from sin? from cos? from some not trig function? a combination of them?
solid pond
.
there are 2 answers in my eye
c and d
and i am pretty a mc question can only have 1 answer
it doesnt say select all that apply
vast summit
if itās transforming from sin then yeah c and d
vast summit
not sure what they mean
vast summit
I would just answer c and d
solid pond
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marsh hamlet
I am unsure how to do 3)
