/sqt/ Stupid Questions Thread: Oxford Capacity Analysis Edition

sin(x) + cos(x) = root2 sin(pi/4 + x)

that absolute value in the denominator is absolute bullshit

when I get take the function f(x)=1/x and rotate it about the x axis from 1 to infinity I get a solid of revolution with finite volume. My question is can I take the function f(x)=1/x^2 and also rotate it about the x axis on a interval from 1 to infinity to get a solid of revolution with finite volume and why?

>My question is can I take the function f(x)=1/x^2 and also rotate it about the x axis on a interval from 1 to infinity to get a solid of revolution with finite volume and why?
Have you tried applying the same reasoning you used to conclude that you get a solid with finite volume by rotating 1/x?

Well something tells me the volume is going to be finite too, but truth be told I am not capable of checking myself, so I figured I'd ask here because I am very curious

should be [math]\frac{\pi}{3}[/math]

A more complete answer. What we're looking for is the value of [eqn]\pi \cdot \int_0^{\infty} x^{-2n}dx[/eqn] for [math]n>0[/math]. Skipping a bunch of limit bullshit this is just [math]\frac{\pi}{2n-1}[/math].

should be [eqn]\pi \cdot \int_1^{\infty} x^{-2n}dx[/eqn] sorry about that.

stupid fucking question
why is this 2+4/5 and not 2(4/5)?

because you're in pre-algebra where people still write mixed fractions instead of improper fractions WHICH TOTALLY IRONICALLY are the preferred form to write rational numbers.

how do i express [math]\mathbb{R}\setminus\mathbb{Z}[/math] as a union of several open intervals in the same way that [math]\mathbb{R}\setminus\mathbb{N}=\bigcup_{n=1}^{\infty} (n,n+1)[/math]

take the union over all integers instead of positive ones

How to compute the coefficient [math][A_1^{p_1}..A_q^{p_q}][/math] of [math](A_1 + ... + A_q)^2 (A_i^2 + ... + A_q^2)^{14}[/math] ?

Example: for [math]q = 2[/math] and coefficient [math][A_1^2A^{28}][/math] the answer would is [math]15[/math]

thanks I think I figured it out

cos(x)=e^ix+e^-ix / 2
sin(x)=e^ix-e^-ix / 2i

cos(x)+sin(x) = e^ix(1-i) + e^-ix (1+i) / 2
= e^ix(√2 e^-iπ/4) + e^-ix (√2 e^+iπ/4) / 2
= √2[ e^(i(x-π/4)) + e^(-i(x-π/4)) ]/ 2
= √2cos(x-π/4)

how?

So does derivative have a special "number" counterpart too?

*equivalence class of functions

Just do it™

You have a unit vector at angle x and another offset by -90° .
Algebraically, the vector addition is .
Visually, you know that the unit vectors are at right angles so the resulting one is √2 long and splits the angle in half so it is √2

cos(x)+sin(x) = √2 cos(x-45°)

antiderivative is self-explanatory. integral is a number, the signed area under a graph of function. the relation between antiderivative and integral is that the integral can be computed by taking the antiderivative and plugging in the endpoints. this is the fundamental theorem of calculus. for this reason, the antiderivative is usually also called "integral", but it's kind of misleading.

No because the derivative f'(x) only depends on the local nature of f at x.
The integral depends on a domain, the antiderivative is what you get when you allow that domain to vary.

What is the point of fourier series if they're used to describe functions you already know? We're doing it in my calculus class and I don't see the application

Fourier series arise naturally by solving certain PDEs.

Easier to take derivatives

but in the creation of the series you have to integrate the original function to get An and Bn terms correct? So I don't see how it fixes anything

Can someone explain the difference between countably/uncountably infinite to me? Not sure I understand

countably as the name implies refers to a count (of objects or subjects for examples) that goes potentially to infinity.
uncountable infinity is just the domain of r

[math] \bigcup\limits_{n \in \mathbb{Z}} (n,n+1) [/math]

If you can list all elements of an infinite set in this way: [math] \{a_1 , a_2 , a_3 , \ldots \} [/math] , then the set is countably infinite; if you can't then, it's uncountably infinite.

In an infinite series, can grouping successive terms affect the convergence? I know that if I group terms by changing the order in some way that can affect convergence. But what about just putting parenthesis all over the infinite series?

countable same size as N
uncountable literally everything else that's not finite or countable

What do you exactly mean by "grouping successive terms"?
Can you give a concrete example?

Like grouping by pairs or n-tuples?
No, they're equal at infinite many points so you can rerun the same convergence argument.

>The set [math]\{1,2,3\}[/math] is not countable because it's not the same size as [math]\mathbb{N}[/math]

This is one of the things that distinguishes conditionally convergent series from unconditionally convergent series, whether grouping or reordering the terms affects the limit. Conditionally convergent series can be manipulated to sum to any number at all.

>No, they're equal at infinite many points so you can rerun the same convergence argument.
[math] \sum (-1)^n [/math] doesn't converge.
Grouping by pairs:
[math] \sum ((-1)^n + (-1)^{n+1}) [/math] converges to 0.

Okay, I am sorry guys. I had a severe brain fart. It is true that grouping terms can change convergence and I knew this but I started freaking out because I thought this would fuck up an argument I used.

But nevermind, my brain has calmed down and I have proven that the specific grouping I used is perfectly Kosher. Sorry guys.

How do I calculate the net electric field at point p in unit vector notation?
I solved each vector for it’s magnitude and have the unit vector notation for each of them.
Do I just add them up?
For example, q1 q2 and q3 is(1.2 x 10^3)i , (1.4x10^2)j , (7.2x10^2)i respectively.
Add i together and leave j alone?

Can someone please help with this?

>Do I just add them up?
yes
>Add i together and leave j alone?
bingo

>uncountable infinity is just the domain of r
Wrong.

wtf are bloch equations

Does anyone have any good study guides for doing well on the GRE math subject test?

How the fuck do people isomorphism fast? It took me like a whole fucking hour to do this one, but I definitely need to be able to do these in a few minutes max.

>Prove that the product of two orthogonal matrices is orthogonal.

Is this correct? I'm not sure if I can assume that A inverse * B inverse = (AB) inverse.

>Is this correct?
Have you tried to prove it?

AHHHH I'M A FUCKING BRAINLET

SOMEONE FUCKING HELP ME PLEASEEE

I think i have to use P=IV, V=IR, and inductance for henry but I'm a fucking brainlet so idk

>Have you tried to prove it?
Yes, that's what's written in the picture. I'm just not sure if my proof is valid.

>Yes, that's what's written in the picture. I'm just not sure if my proof is valid.
I meant prove that "A inverse * B inverse = (AB) inverse."

It doesn't seem like it would be true (at least for most matrices), but since A^T * B^T = (AB)^T for symmetrical matrices, I feel like the same could maybe be true for inverses of original matrices.

(AB)^T = B^T A^T
(AB)^-1 = B^-1 A^-1

P=IV, V=IR
P=I^2R
400^2 amps * 1 Ohm = 160,000Watts

how long have you been waiting to use that photo user?

do the rest of the problem you fucking faggot brainlet

b) be all damn hot damn fast yo nigga
dat 1 Ohm better be like all over that wire a long way or it be all meltin and shieet
c) 3T^2 * 1m^3 / 2/1.26*10^6 H/m = 3.57 Mega Joules!
d) 3.57MJ / 2.5kJ/L = 1.43kL of liquid He -> 1.43kL liquid *700 gas/liquid = 1,000,000 liters of gas!
dat shieet is gonna go boom nigga, run!

>and inductance for henry but I'm a fucking brainlet so idk
>en.wikipedia.org/wiki/Henry_(unit)
H=J/A^2
>en.wikipedia.org/wiki/Tesla_(unit)
T=J/m^2/A
T^2 m^3 /(H/m) = (J/m^2/A)^2 m^3 /(J/A^2) * m = J

How new are you?

Someone pls

where can I compute initial value difference equations online?

The examples I want to compute have subscripted elements, like such:

[math]x_n = x_{n-1} + 2n + 1 [/math]

[math] x_0 = 7 [/math]

I've tried to use wolfram alpha:

wolframalpha.com/input/?i=x(0)=1, x(n)=(5x(n-1))-4n + 1

but when I input my problem, I don't know how to input the subscriped elements. I've tried inputing them as functions like:

x(0)=1, x(n)=(5x(n-1))-4n + 1

but im afraid its not recognizing the subscriped elements correctly,

But user the solution is on the page you linked

i was just a little confused when it displays my input in the link i posted, it displays my difference equation as:

x(0) = 7
x(n) = 1 x(n - 1) + 2 n + 1

is my notation with subscripted (n-1) the same as treating it as a function of x?

I'm not sure what exactly your question is

It's treating it as a function of n, which is why the solution x(n) is in terms of n and not x.

[eqn]x_n = x_0 + \sum_{i=1}^{n} 2i + 1[/eqn]
[math] x_n = x_0 + n(n+1) + n = x_0 + n^2 + 2n [/math]

protip: if [math]x_n = a x_{n-1} + f(n)[/math]
[eqn]x_n = a^n x_0 + \sum_{i=1}^{n} a^{n-i}f(i)[/eqn]

>with subscripted (n-1) the same as treating it as a function of x?

It's a function of n and yes, "sequence x_n" is a fancy pants way of saying the function x:N->R.

I am stuck. How do I prove this?:

If
[math] a \in \mathbb{N} [/math] and [math] \lambda \in \mathbb{C}: \lvert \lambda \rvert

math.stackexchange.com/questions/55468/how-to-prove-that-exponential-grows-faster-than-polynomial

Thank you!

How is anything that isn't in anyone's mind real?

Ahoy Veeky Forums anons. Could anyone please explain to me why do we use Newton Cotes formulas?

I think I understand that why we use quadratic interpolation. We can solve integrals easier using approximation => integral of the Lagrange polynomial can be calculated fast. We just need an n number of fix points. This should deal with straight and curved functions.

So Newton Cotes formulas use fixpoints which are all in equal distances from each other. Why?

>So Newton Cotes formulas use fixpoints which are all in equal distances from each other. Why?

Because symmetry makes things better. That's way mid point rectangle rule can do linear functions perfectly and Simpson's rule (2nd degrre) can do cubics perfectly or Boole's rule (4th degree) can do quintics perfectly.

thats nice, but the problem asks me to find x, not a. but it is a step forward
(i am aware of the possibility of x being any degree possible)

can't you just say that the slope of the function at a certain point is the "number counterpart" of the derivative

are you a brainlet?
x = arccos(a/√2) + π/4

How would an dfa that accepts the L = {0^k | k

Can someone explain to me the difference between a scalar and a vector field? Both mathematically and physically.

So, pic related. I have to find the eigenvalues and eigenfunctions of the differential equations with the given boundary conditions. I got to the point where I have the null points of the equations, which are -5 +- sqrt(25-λ). Now I think I have to look at three cases, namely:
1)sqrt(25-λ)>0
2)sqrt(25-λ)=0
3)sqrt(25-λ)

I guess, but not in the same sense as the integral.

I'm guessing you meant >=

Just have five states that represent the number mod 5 - you increment when you hit a 0, and fail on any other character.

Mathematically a scalar field assigns a number to each point in space, a vector field assigns a vector.
An example would be e.g. mass distribution for a scalar field and gravitational force for a vector field.

Why are some units wrote with a subtraction? Such as m^3 being m^-3.
I've done maths up to calc 2 and still don't know this.

>Such as m^3 being m^-3.
As in cubic meters, not an algebraic unknown.

1 m^-3 is the same as 1/m^3 and means "1 per meter cubed"

It could mean, say, you have 1 atom of uranium in every cubic meter of space.

Okay so, 2.5 x 10^25 molecules m^-3 is equivalent to saying 2.5 x 10^25 per m^3.

yes

REEEEE SOMEONE PLS

>REEEEE SOMEONE PLS
What have you tried?

I'm a CS student, I've done some linear algebra, vector calculus, discrete math and probability at uni. I'm interested in more math disciplines, outside of CS focused of course, where should I look in terms of material/textbooks?

What is the common ratio for 1/n^2?
I can't understand how you can get a common ratio from when the n is squared.

1+1/2^2+1/3^2 etc
because it changes at all times.

>Oxford Capacity Analysis
I'm a Scientologist since 1994. Ask me.