Previous SQT thread maxed

Oh of course

Start by calculating
n*T(n) - (n-1)*T(n-1)

The sums cancel except for the T(n-1) term, leaving you with
n*T(n) - (n-1)*T(n-1) = T(n-1) + 10*n - 1
=> T(n) - T(n-1) = 10 - 1/n
=> T(n) -> 10*n

IOW: c) f(n) = n

can any geology guys tell me what's the difference between "cleavage" and 'schistosity" on rocks? I don't think the two terms can be used interchangeably, and yet I can't figure out a difference other than the fact that schistosity only occurs on (duh) schists. both terms seem to indicate that the rock's structure has a planar orientation.

I have a test tomorrow with a really annoying teacher and I just want to have a good response in case he asks this

Matlab's acos function is return ArcCos values in the range [0, pi], which is kinda expected but I need it go from [0, 2pi]

Anyone have any ideas how I could tell whether a result should be in the range 0-pi or pi-2pi

Im sorry I dont quite follow could you expand on the working more? Its a bit too simplified for me to follow..

T(n) = 1/n * (T(0) + T(1) + T(2) + . . . + T(n-1)) + 5n

nT(n) = T(0) + T(1) + T(2) + . . . + T(n-1) + 5n^2

Where am I supposed to go from here?

cos(-x)=cos(2π-x)=cos(x)
For x in the range [0,2π], every value of cos(x) occurs twice. If that's the only information you have, there's no way to distinguish between the two possible angles.

n*T(n) = T(0)+T(1)+T(2)+...+T(n-2)+T(n-1) + 5n^2
(n-1)*T(n-1) = T(0)+T(1)+T(2)+...+T(n-2) + 5(n-1)^2

n*T(n) - (n-1)*T(n-1)=T(n-1) + 5n^2 - 5(n-1)^2
=T(n-1) + 5n^2 - 5(n^2-2n+1)
=T(n-1) + 5n^2 - (5n^2-10n+5)
=T(n-1) + 5n^2 - 5n^2 + 10n - 5
=T(n-1) + 10n - 5

=> n*T(n) - (n-1)*T(n-1) - T(n-1) = 10n - 5
=> n*T(n) - n*T(n-1) = 10n - 5
=> n*(T(n) - T(n-1)) = 10n - 5
=> T(n) - T(n-1) = 10 - 5/n
=> T(n) - T(n-1) -> 10 as n->infinity
=> T(n) -> 10n - k as n->infinity

Much clearer thanks, but what happened here?

=> n*T(n) - (n-1)*T(n-1) - T(n-1) = 10n - 5
=> n*T(n) - n*T(n-1) = 10n - 5

How does (n-1)*T(n-1) - T(n-1) simplify to
n*T(n-1)?

and

=> T(n) - T(n-1) -> 10 as n->infinity
=> T(n) -> 10n - k as n->infinity

Where did -T(n-1) go? T(n) - T(n-1) = T(n-1) right? how is 10"n" back on the last line

wot is a measurable function?
I've seen two definitions
1: a function of a measurable space (X,M) into a topological space (Y,τ)
such that the inverse of an open set from Y is a measurable set.

2.a Function on a measurable space Xinto another measurable set Y such that the inverse image of a measurable set from Y is a measurable set in X.

i see that the first definition seems to imply the second if we use the Borel σ-algebra, but idk if there are pathological examples that would make it impossible to get a topology on Y if Y is a measure space or something like that.

is there any way to find the sum of the first n squares, without resorting to pyramids or cubes?

Wanting to check if I've done these correctly. I can't really remember how to do the first one - you substitute the supplied value for ψ(x) and take the second derivative of it (Will this always be (2mE/h)ψ?
I have a feeling I'm supposed to go ψ(x) = Asin(pix/a) + Bcos(pix/a) and ψ(-a) = ψ(a) but I'm not sure where to go for there

You mean [math]\sum_{k=1}^{n}k^2[/math]? That's just [math]\frac{n(n + 1)(2n + 1)}{6}[/math].

yes - thanks for the reply

Iv been searching around on this, and i have found the formula you posted on several pages and youtube videos, but i cannot understand how this is derived.

i have seen the proofs involving pyramids, its cute but i would rather find it through algebra

All of the algebra i have found just assumes we need a cubic function to solve it. but i dont understand why we need cubes to solve a function based on squares.

nor can i see how i can start with
>1^2 + 2^2 + 3^2 ... (n-1)^2 + n^2
and end up with a cubic

please forgive my stupidity and my lack of latex formatting,

If A and B are sets, is "A contains B" just another way of saying "B is a subset of A"?

Induction, my man.

I found this: proofwiki.org/wiki/Sum_of_Sequence_of_Squares

I'm sure that will satisfy you. The first proof is just proving that the formula is valid, but the other proofs provide ways of deriving it. They all involve a bit of machinery that you may or may not have yet, but they're still valid.

What's the context? Another possibility is "B is an element of A."

yes

A contains B means every element of B is an element of A

"Let K be a compact set and let U be an open set that contains K."
Pretty sure it's Thanks, guys.

> Much clearer thanks, but what happened here?
> How does (n-1)*T(n-1) - T(n-1) simplify to n*T(n-1)?
It doesn't. But /both/ terms are negative, and
- (n-1)*T(n-1) - T(n-1)
= (-n+1)*T(n-1) - T(n-1)
= -n*T(n-1)

> => T(n) - T(n-1) -> 10 as n->infinity
> => T(n) -> 10n - k as n->infinity
> Where did -T(n-1) go?
f(x)-f(x-1)=m => f(x)=m*x+c
If the difference between successive values is a constant, then the function is linear. This is the point where the recurrence relation is converted to an explicit form.

Well, it's not quite constant because of the 5/n term, but as n gets larger 5/n approaches zero, 10-5/n approaches 10, and T(n) approaches 10*n-k. Which is enough to say that it's O(n).

>= (-n+1)*T(n-1) - T(n-1)
>= -n*T(n-1)
Man... im not getting it how is this possible youre subtracting stuff against a thing thats being multiplied with something? wtf Ive never done anything like this in my life I just cant see for the life of me whats going on

>f(x)-f(x-1)=m => f(x)=m*x+c
Oh right haha

(x+1)^n = x^n+n*x^(n-1)+...
(x+1)^n-x^n = x^n+n*x^(n-1)+... - x^n
=n*x^(n-1)+...

I.e. the difference between successive terms of a degree-n polynomial is a degree-(n-1) polynomial.

And the difference between the sum to N+1 terms and the sum to N terms is just the N+1 term.

So the sum of successive values of a degree n polynomial is a degree-(n+1) polynomial.

From there, it's just a matter of working out the coefficients.

E.g. if you have f(x)=a*x^3+b*x^2+c*x+d, then
f(x+1)=a*(x+1)^3+b*(x+1)^2+c*(x+1)+d
= a*(x^3+3*x^2+3*x+1) + b*(x^2+2*x+1) + c*(x+1) + d
f(x+1)-f(x)=a*(3*x^2+3*x+1) + b*(2*x+1) + c
= (3*a)*x^2 + (3*a+2*b)*x + (a+b+c)

So if you want f(x+1)-f(x) = (x+1)^2 = x^2+2*x+1, then
3*a=1 => a=1/3
3*a+2*b=2 => b=1/2
a+b+c=1 => c=1/6
And f(0)=0 => d=0

So f(x)=(1/3)*x^3 + (1/2)*x^2 + (1/6)
= (2*x^3+3*x^2+x)/6
= x*(x+1)*(2*x+1)/6

Somewhat easy question here,

If I have a plane (for example -5x-7y+7z=35) then the normal vector is n=. Is it true that any vector on the plane x has to satisfy the condition x.n=35?

> youre subtracting stuff against a thing thats being multiplied with something?

- (n-1)*X - X
= - (n*X - 1*X) - X
= - n*X + X - X
= - n*X

E.g. if n was 8, n-1 would be 7, -7X-X=-8X.

Yes.
(dot) = 35
-5x-7y+7z = 35

Awesome, thanks, ill digest what i can, have a great day!

Holy that's obvious bless u user

Let x be a unit vector on the plane. Then [math]x \cdot n = -5 - 7 + 7 = -5 \neq 35.[/math]

This is not a proof.

TYYYY

What are the first and second derivatives of (4-x)e^-x?

My calculator and Wolfram give different answers (opposite signs) for the first derivative, and Wolfram and DerivativeCalculator give answers with opposite signs for the second derivative.

Maybe my algebra is shitty, but I can see how it logically works both ways when I work it out on paper, and I know they aren't equivalent cause the graphs different.

******VERY IMPORTANT****

I have an interview for joining a lab tomorrow. This will be my first one, what should I expect for questions?

Same poster here:
By hand, I get:

>f'(x) = (4-x)'(e^-x)+(4-x)(e^-x)'
>f'(x) = (-1)(e^-x)+(4-x)(-e^-x)
>f'(x) = -e^-x + (-4e^-x) + xe^-x
>f'(x) = -e^-x - 4e^-x + xe^-x
>f'(x) = -e^-x (1 + 4 - x)
>f'(x) = -e^-x (5 - x)

According to Wolfram and derivativecalculator, it's positive, on derivative calculator it distributed -1 back into the parenthesis, but this changes the graph. The derivative function on my TI graphs what I got.

>derivative calculator it distributed -1 back into the parenthesis, but this changes the graph
No it doesn't. That's an equivalent function.

You're correct. This is how you go from your answer to Wolfram's answer:
[math]-e^{-x}(5 - x) = e^{-x}((-5) + x) = e^{-x}(x - 5).[/math]

And this is what Wolfram gives me, for the record.

What's the purpose of accumulation points in analysis? why do we define limits, the derivative in particular, with them, as opposed to simply requiring the domain to contain an interval or neighborhood around the point(s) of interest?

Say I have an array with 16 million numbers in it and it is sorted. Then I add 10 unsorted onto the end, sorting with insertion sort would be fastest right? Since it would be Initial input * the new input which would work out to be 160m vs for example a quick sort which would be 16m*log16m=240m would that just be O(N)?

Can some retard give me a nice simple explanation of the branch of the logarithm/complex logarithm? I just can't grasp the fucking thing. Reading out of Complex Analysis by Bak and Newman, they seem to do a shit job of explaining it. I need some intuition behind it.

I want to do either embedded systems or machine learning. Would it be smarter for me to do a BS in math then CS? I got the freedom to pick quite a few classes if I do a BS in math and so I will have the prerequisites for both. I do realize that there is a difference in difficulty. But I've never understood how somthing can be too hard to learn. Not sure if I'm just a delusional brainlet or nah.

So CS or math?

Try to rephrase the recurrence relation as a recursive function. Suppose you have computed T(n), how do you compute T(n+1) from that information? T(n+1) = [[T(n) - 5n]*n + T(n)]/(n+1)] + 5(n+1) = [(n+1)*T(n) - 5n^2)/(n+1) + 5(n+1)= T(n) + 5(n+1)^2/(n+1) - 5n^2/(n+1) = T(n) + 5*[(2n + 1)/(n+1)] = T(n) + (10n + 5)/(n+1) = T(n+1). For very large n, (10n + 5)/(n+1) is approximately equal to 10, so as n increments by 1, T(n) increments by 10, so choice C best represents this relationship (up to a constant).

I know you can express any rational number as a continued fraction in two ways [a0;a1,… an−1,an] and [a0;a1,… an−1,(an−1),1]
I want to know how you prove these are the only two ways to do it?
I know the two expressions are ending with n > 1 i.e. [ ..... , n ] or ending with 1 i.e. replace the final n by (n-1) + 1/1 i.e. [ ..... , n-1 , 1 ]
Wondering how you prove there are no other expressions

Do you have the coordinates for your point?

What sort of explanation are you looking for?

The logarithm is the inverse of the exponential, i.e. y=e^x x=log(y).

If x is complex, x=a+bi. And f(x)=e^(ix) is periodic with period 2π, i.e. e^(x+2πi)=e^x. So e^(a+bi+2nπi)=e^(a+bi) for all integer n.

Which means that log(e^(a+bi)) = log(e^(a+bi+2nπi)) = {a+bi+2nπi}, i.e. the logarithm of a complex number has infinitely many values which differ by integer multiples of 2πi.

Whereas the complex logarithm is multi-valued, a branch is a single-valued function which returns a value whose imaginary part lies in a specific interval of size 2π. A branch cut is a line in the plane between branches, where the branch is discontinuous: if a+bi lies on the branch cut, then Log(a+(b-ε)i) and Log(a+(b+ε)i) will differ by approximately 2πi (more precisely, the one-sided limits lim[ε->0-]Log(a+(b+ε)i) and lim[ε->0+]Log(a+(b+ε)i) will differ by 2πi).

if n is a positive integer, then 0

What is a limit of a function?

Oops; just realised that isn't quite enough; for 1/r=1. But if r

What f(x) looks like it would be if by looking at f(values near x).

Is there a similar document for linear algebra?

damtp.cam.ac.uk/user/fq201/DEsummary1.pdf
Found it from the sticky, also like, I found a pdf for linear algebra there, but it requires a password.
>Linear Algebra (PDF) by Jim Hefferon, Saint Michael’s College
>ftp://joshua.smcvt.edu/pub/hefferon/book/book.pdf

I'm taking DE this semester but my school doesn't require linear algebra for my major for some damn reason.

How did this happen?

Variance is based upon the square of differences.
Var(x)=E[(x-m)^2]

If the mean of x is m, then the mean of x/n is m/n.
Var(x/n)=E[(x/n-m/n)^2]
=E[(x-m)^2/n^2]
=E[(x-m)^2]/n^2
=Var(x)/n^2

Thanks user. I have a feeling that you're answering a lot of these questions ITT. How do you know so much? Are you a grad student doing maths?

What calc course do I learn differentiation under the integral sign in? It looks really cool

Math background: school, then a CS degree, then spending too much time on wikipedia and Project Euler.

Do you work as a programmer or something more mathsy?

Programmer. But work programming rarely involves much math; that's more of a hobby.

Dear resident category theory experts, in what way does a span represent a collection-valued attribute?

I can provide some context if this does not make sense on its own.

Why is it that people in my classes can be frequently disruptive (talking in class while the lecture is going) or just not show up to class frequently, and talk about having not prepared for exams or left assignments to the last minute, then pass the course without issue while I make a reasonable effort to show up to every class and do my work but I don't do very well? Is there some trick I'm not getting

Not really depends on your field. If you want to make games you need algebra, calculus, physics, and geometry

Are there any situations that arise in the universe where matter travels at a significant fraction of the speed of light?

I'm thinking something around a fraction of say 1/10 or maybe 1/5

en.wikipedia.org/wiki/Oh_my_god_particle

99.99999999999999999999951% of the speed of light.

>friends
>form group
>group studies together
>collection of minds with varying knowledge
>each fill each others gaps of misunderstanding
>all whilst enjoying witty banter
>many minds> one mind ???

My book defines the limit of a function at a point p as: Let f : D -> R, D a subset of R, and the point p in D an accumulation point of D. Then f has a limit as x approaches p iff there exists L such that for every epsilon>0 there exists delta>0, such that x in D and 0 < |x-p| < delta implies |f(x) - L| < epsilon.

I understand that p doesn't have to be in the domain for the limit to exist, and the motivation for defining it this way is clear to me. My confusion arises from the fact we don't require a deleted epsilon neighborhood around p (on at least one side) as a subset of the domain. This way we still wouldn't care what goes on with f at p, but could be ensured f is defined at all points leading up to p. Requiring only that p is an accumulation point in the domain allows for holes or gaps in the domain approaching p. any sort of insight or help would be most appreciated

My CS course had a fairly substantial chunk of discrete math: propositional logic, predicate logic, temporal logic, formal grammars, Turing machines, lambda calculus.

That's aside from the math-related courses (graphics, computational geometry, optimisation and linear programming).

not him but do we need actual geometry (like studying The Elements) or would the analytic geometry encountered in calculus plus some linear algebra suffice? Euclid was of course very axiomatic, which is awesome, but doesn't appear to me to be all that important if all you are wanting to do is make games.

Also, formally speaking, is analytic geometry a subset of euclidean geometry? or maybe rather an extension of it?

What is the point of making students prove trig identities during a test when we can just assume they were proven centuries ago?

Okay. cos(x)/(1+tan(x))=cos(x)-sin(x)

Having to prove this is tedious:
cos(x)/(1+(sinx/cosx))-sinx/(1+(cosx/sinx))

Okay, I'm already to bored to type the rest out. Do most professors make you do this shit or is she just a bitch? What is the point in the big picture of modern mathematics now?

> Requiring only that p is an accumulation point in the domain allows for holes or gaps in the domain approaching p.
Like how you can define an irrational number as the limit of a sequence of rationals?

Your Uni miles ahead of mine then (but sadly that isn't saying much). All we require of our CS undergrads math-wise is calculus 1 and 2, and their choice between discrete math or some watered down "equivalent" the CS department offers called data structures, or something like that. No linear algebra, no number theory or logic, and certainly no lambda calculus or optimisation.

Well, a lot of that was for optional courses (I tended to choose theoretical courses over the more vocational stuff).

Basic discrete math (mostly logic) was mandatory, calculus wasn't. But subjects which had some underlying mathematical theory would actually teach it (e.g. databases used relational algebra a lot, concurrent programming used temporal logic for expressing invariants, etc).

AFAICT, places where CS is offered by the math faculty tend to do more theory than where CS (or Software Engineering) is part of the engineering faculty.

>Requiring only that p is an accumulation point in the domain allows for holes or gaps in the domain approaching p.
This is exactly the point. Limiting behaviour does not have anything to do with whether you contain a complete interval of the metric space or not.

All you care about is whether the function itself is getting close to something. All that depends on is the values of the function. All _those_ depend on are the domain. Why would numbers outside the domain make a difference?

You ought to take some time to reread (or get some professor's help with) the earlier sections of your book on limit points of sets and limits of sequences (presumably your book does sequences before limits of functions). The fact that you got through those and think a limit point has anything to do with "gaplessness" indicates you didn't really understand those sections very well.

I'm in a babby's first analysis course and the instructor completely glossed over the construction of the reals, could you provide another hint or direct me to some requisite material?

This is precisely what I needed, thank you. Yes I sort of put this course on the back burner and ended up doing very poorly on the first exam, so i am trying to go back from the beginning and get caught up. thanks again

>the instructor completely glossed over the construction of the reals
really makes you think

Retard here. Just want to make sure I understand something.

Mythbusters did pic related a long time ago. They fired a ball out the back of a car with the same speed as the car was moving, cancelling the momentum. For clarification's sake, let's say the car was moving at 20mph and the ball was launched at 20mph to make this happen.

From the frame of reference of an observer not in the car, the ball's speed is 0, since the momentum was cancelled.

But from the frame of reference of someone in the car, the ball was launched at 20mph. From their point of view, the ball's behavior was no different at all than if it had been launched from a stationary car. But this observation is wrong because we know that the ball's speed is 0mph.

Now, the only reason we have these two frames of reference is because we have the background and context to figure that out. But let's say there was no way of figuring that out; let's say the car was in a featureless void and the only observers were located in the car. These observers have no clue how fast the car is moving, or if the car is moving at all.

If the ball were launched from this void-car at 20mph, how would the observers know the true speed of the ball? It would be moving 20mph away from the car, but is it the car that's moving, the ball, or both? To what degree? How could something like that be measured without an outside point of reference?

And to extrapolate from that, doesn't the same concept apply to measuring the speed of any given celestial body in the real world, since the only reference are other celestial bodies that we also don't know the speed of? Is there no such thing as true speed, but only relative speed instead?

And given all of this, how do we know what the actual speed of light is?

>But this observation is wrong because we know that the ball's speed is 0mph.
No. There is no privileged reference frame.

>If the ball were launched from this void-car at 20mph, how would the observers know the true speed of the ball?
There is no such thing as "true speed".

>And given all of this, how do we know what the actual speed of light is?
The speed of light is constant in all reference frames and is an exception to the rule.

Are animals aware they have organs?

So far I have 2....4,5....7,8....10,11....13,14....16,17

sort of a pattern maybe.

How the hell do I do this?

>It would be moving 20mph away from the car, but is it the car that's moving, the ball, or both?
depends on how you define your reference frame.

>To what degree? How could something like that be measured without an outside point of reference?
it can't, speed is only a meaningful concept with some reference frame. in a total void there's no physical difference between it being the car moving or the ball moving

>And to extrapolate from that, doesn't the same concept apply to measuring the speed of any given celestial body in the real world, since the only reference are other celestial bodies that we also don't know the speed of?
yes
>Is there no such thing as true speed, but only relative speed instead?
yes
>And given all of this, how do we know what the actual speed of light is?
You can calculate it experimentally, it also falls out of the equations of E&M as the speed of electromagnetic radiation in a vacuum.
I'm sure it falls out of the relativity equations as well, but I've only had a few brief lectures on that and cbf to self study it.

If I have a block of wood that is partialy sumerged in both water and oil (the oil being on top lf the water) of length 10, what would be the pressure at the bottom of the block?

Find the derivative of the curve at an arbitrary point construct the general equation of the tangent line at point p and equate it to to thag point. Solve the system. Considering that the point must be on the curve.

...

What about this guy? I got the first part right but I can't get the cos(x) correct. I tried doing a*b=|a||b|cos(x) where a and b are the values at the derivative and it doesn't work

What other value of t you got?

None

Oh, I got 3 and -7

Can you maintain a geosynchronous orbit at points other than the equator and what's the difference exactly between geosynchronous and geostationary?

Depending on your course, yes. If the course aims to teach proof techniques, then it's obvious that you would be expected to use those techniques on the test.

The point is developing your skills.

Geosynchronous means the time of orbit is 1 day. This is possible for an infinite amount of orbits, it does not have to be "parallel" to the equator it just has to be at the right distance away.

Geostationary means the satellite remains above the same spot on earth. You're still moving with some orbital speed, but that orbital speed matches the speed of the earths rotation. This is only possible if you're in geosynchronous orbit above the equator moving in the direction of earths orbit.

It's having an input that gives an undefined output. Like 1÷x and x=infinity. The function is indeterminant when x is zero, but we can see it approaches zero. y = 0 is the limit. But I'm just now taking calc I. I'm no math major.

For the tangent vectors, I get [-20,0,-10] and [20,0,110] with magnitudes 10*sqrt(5) and 50*sqrt(5), and dot-product -1500. So cos(a)=-1500/2500 = -3/5.