Can any of you help me understand how the inequality comes into existence here?
[t]https://dl.dropboxusercontent.com/u/30829668/Screenshots/Screenshot%202014-02-18%2011.57.55.png[/t]
It's kind of annoying to be given some solutions without justification like this, given that I'd be slated in an exam for exactly this.
[editline]18th February 2014[/editline]
Wait nevermind I worked out how the n ≤ n^(3/4) ≤ n^(1/2) comes into play now
Yay learnin'
I think this is the first time I've actually laughed at a textbook joke.
[img_thumb]http://i.imgur.com/33aWVPS.jpg[/img_thumb]
My Calculus teacher that I also TA for has got me making assignments for some 7th grade math course so he doesn't have to
So far I made point-plotting practice that makes a 3D snake, and like 40 fraction addition/subtraction problems with different denominators that had to have only 20 different answers for some game
[QUOTE=ZenX2;44081049]My Calculus teacher that I also TA for has got me making assignments for some 7th grade math course so he doesn't have to
So far I made point-plotting practice that makes a 3D snake, and like 40 fraction addition/subtraction problems with different denominators that had to have only 20 different answers for some game[/QUOTE]
You're giving 7th graders points in R3?
Only in R^3? Plebs. Make it C^3
[QUOTE=Falubii;44087735]You're giving 7th graders points in R3?[/QUOTE]
No, it's just R2 and they connect into a 2D representation of a 3D snake
[QUOTE=ThisIsTheOne;44087865]Only in R^3? Plebs. Make it C^3[/QUOTE]
lol complex analysis is for scrub-tier mathematicians. The theorems are so overpowered
[QUOTE=JohnnyMo1;43712274]Yes it is! I just found out that my real analysis professor's advisor's advisor's advisor's advisors were Weierstrass and Kronecker the other day.
[editline]29th January 2014[/editline]
Damn, and my topology professor is math-descended from Lagrange, Laplace, more than one Bernoulli, and Euler.[/QUOTE]
One of my lecturers is Math descended from Rutherford, Routh, Stokes and [I]Newton[/I] himself.
So I just had my first taste of Analysis...holy shit. It feels like all of the mathematics I've studied until now has been monkey work.
First week of semester over; looks like some smooth sailing ahead.
[QUOTE]Calculus of Science and Engineering
[url]http://www.newcastle.edu.au/course/MATH2310[/url]
Linearity and Continuity (doing this for the shits & giggles):
[url]http://www.newcastle.edu.au/course/MATH2340[/url]
Operations Research 1
[url]http://www.newcastle.edu.au/course/MATH2730[/url][/QUOTE]
I'm so dead at Analytic Geometry.
Does anyone know how I'd solve something like this?
[img]http://latex.codecogs.com/gif.download?%5Cddot%7Bx%7D%20+%20R%5Cdot%7Bx%7D%5E2%20%3D%200[/img]
I've looked through the resources in the OP but I haven't seen anything like this just yet. It's hard to google too.
Not homework, I was wondering about the muzzle energy of a BB gun so I did some science with a curtain, a tape measure and the sound recorder on my phone to find the time taken to travel a given distance.
The problem is that if you assume that the velocity is constant then you underestimate the muzzle energy (by about a factor of two as it turned out) because of the effects of air resistance.
I was modelling drag due to dynamic pressure and ended up with a differential in that form (I made a bunch of assumptions about shape and cross section, and I let [img]http://latex.codecogs.com/gif.latex?R%20%3D%20%5Cfrac%7B%5Crho%20A%7D%7B2m%7D[/img]).
So I made a spreadsheet (Euler integration, or difference equations) to find x against time and then used Excel's goalseek to find the muzzle velocity that would travel the given distance in the measured time:
[img]http://i.imgur.com/ZcDOcyJ.png[/img]
Which is absolutely awesome! But I would really like to know how to calculate this analytically, if it's at all possible (I can find the Taylor but was wondering if there was a closed form solution).
Does anyone know?
Stick it in WolframAlpha, it says there's a closed-form solution. I admit I don't know how it solve it. It's been years since I took differential equations and nonlinear second-order differential equations are a huge bitch.
Using a little bit of [URL="http://www.sosmath.com/diffeq/second/nonlineareq/nonlineareq.html"]help[/URL] found by a quick google search I've managed to solve it with the result
x = ln((t + C)/(DR))/R
where C and D are some unknown constants. When I differentiate it and plug it back into the initial equation it checks out, so it seems to be correct.
[QUOTE=JohnnyMo1;44175720]Stick it in WolframAlpha, it says there's a closed-form solution. I admit I don't know how it solve it. It's been years since I took differential equations and nonlinear second-order differential equations are a huge bitch.[/QUOTE]
[QUOTE=pebkac;44178168]Using a little bit of [URL="http://www.sosmath.com/diffeq/second/nonlineareq/nonlineareq.html"]help[/URL] found by a quick google search I've managed to solve it with the result
x = ln((t + C)/DR)/R
where C and D are some unknown constants. When I differentiate it and plug it back into the initial equation it checks out, so it seems to be correct.[/QUOTE]
Thank you! I tried that this morning, using v as my variable, and found I could just use separation of variable :) (thanks pebkac!)
Don't quite seem to be getting the right constants out of my conditions but it does look right. The answer I got from the numeric stuff looks sensible (the average speed was measured at 35.8m/s +/- about 2.5) so I'll know when the analytic stuff has worked.
[video=youtube;5iUh_CSjaSw]http://www.youtube.com/watch?v=5iUh_CSjaSw[/video]
Also, I found this gem:
[t]http://i.imgur.com/WYQtVSN.png[/t]
[QUOTE=Bradyns;44231971][video=youtube;5iUh_CSjaSw]http://www.youtube.com/watch?v=5iUh_CSjaSw[/video][/QUOTE]
How many complaints concerning semantics can you fit into one video? Normally I like ViHart but that video was kind of stupid.
[QUOTE=Bradyns;44231971]
Also, I found this gem:
[t]http://i.imgur.com/WYQtVSN.png[/t][/QUOTE]
Pfft, the phi part is easy, I bet the rest is too :v:
I normally dislike Vihart and I thought that video was decent. How were they all semantic complaints? Have you ever encountered all these people who think pi is incredible and amazing because "woah man any number you can think of is in it?" (unproven, and a property held by most real numbers)
I'm mostly just glad she didn't whine about tau the whole time because [I]that[/I] is stupid
[editline]14th March 2014[/editline]
Also:
[url]http://www.playbuzz.com/adamf11/which-fundamental-theorem-are-you[/url]
I'm the fundamental theorem on homomorphisms. Pretty accurate.
[QUOTE=JohnnyMo1;44233382]I normally dislike Vihart and I thought that video was decent. How were they all semantic complaints? Have you ever encountered all these people who think pi is incredible and amazing because "woah man any number you can think of is in it?" (unproven, and a property held by most real numbers)
I'm mostly just glad she didn't whine about tau the whole time because [I]that[/I] is stupid
[editline]14th March 2014[/editline]
Also:
[URL]http://www.playbuzz.com/adamf11/which-fundamental-theorem-are-you[/URL]
I'm the fundamental theorem on homomorphisms. Pretty accurate.[/QUOTE]
One of the first things she said was pi is not infinite. Obviously nobody thinks that pi is infinite and are only referring to its irrationality. And yes there are an infinite number of irrational numbers, so I suppose that's a fair compliant and pi is certainly not unique in that respect. I've always thought the reason people find pi interesting was because of how ubiquitous it is. It pops up in all sorts of places that you wouldn't expect to find anything related to circles. What I think was stupid is that she was ripping on pi in the video, instead of ripping on the stupid reasons people think pi is cool.
She didn't mention the "Pi contains every story ever/Library of Babel" thing, which is interesting at least. (well, more interesting than yapping about reals imho)
I think I'm just a little grumpy about pi day because no one cared when I memorized the continued fraction and series expansions of e.
[editline]14th March 2014[/editline]
Oh, and k*pi contains every finite story at least k times (with k a positive integer and pi normal).
[QUOTE=Krinkels;44235446]and pi normal[/QUOTE]
prove it u scrub
I have never ever seen a proof of any number's normality. I don't even know where to start.
[QUOTE=Krinkels;44235697]I have never ever seen a proof of any number's normality. I don't even know where to start.[/QUOTE]
No one has proved it, so there would probably be some fame in fortune in it for you if you could.
[QUOTE=JohnnyMo1;44233382]I normally dislike Vihart and I thought that video was decent. How were they all semantic complaints? Have you ever encountered all these people who think pi is incredible and amazing because "woah man any number you can think of is in it?" (unproven, and a property held by most real numbers)
I'm mostly just glad she didn't whine about tau the whole time because [I]that[/I] is stupid
[editline]14th March 2014[/editline]
Also:
[url]http://www.playbuzz.com/adamf11/which-fundamental-theorem-are-you[/url]
I'm the fundamental theorem on homomorphisms. Pretty accurate.[/QUOTE]
Fundamental theorem of Algebra for me.
[editline]15th March 2014[/editline]
On a related note, I prefer e to pi. Pi is probably only more well known because it's easier to explain what it is to the general public, but e is still more elegant in it's meaning.
Anyone know a good way to evaluate sum from 1 to infinity of n/(2^n)? It's obvious it does converge (it's always less than n/n^3 after finitely many terms and that obviously converges). Wolfram gives the sum as 2 and gives a partial sum formula which obviously evaluates to 2 in the limit, but it's not obvious where it came from.
[QUOTE=JohnnyMo1;44244614]Anyone know a good way to evaluate sum from 1 to infinity of n/(2^n)? It's obvious it does converge (it's always less than n/n^3 after finitely many terms and that obviously converges). Wolfram gives the sum as 2 and gives a partial sum formula which obviously evaluates to 2 in the limit, but it's not obvious where it came from.[/QUOTE]
Oh, that's a real bugger isn't it. n/(2^n) would obviously converge because 2^n is a really fast growing function... but. Hmm, let me try to work it out.
[QUOTE=JohnnyMo1;44244614]Anyone know a good way to evaluate sum from 1 to infinity of n/(2^n)? It's obvious it does converge (it's always less than n/n^3 after finitely many terms and that obviously converges). Wolfram gives the sum as 2 and gives a partial sum formula which obviously evaluates to 2 in the limit, but it's not obvious where it came from.[/QUOTE]
We know it converges, so it's relatively safe* to mess around with the terms as much as we like.
We want to evaluate:
[; \frac{1}{2} + \frac{2}{4} + \frac{3}{8} + \frac{4}{16} + ...;]
Now we can split this up into a sum of sums:
[; \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + ...;]
[; + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + ...;]
[; + \frac{1}{8} + \frac{1}{16} + ...;]
[;+ \frac{1}{16} + ...;]
[; + ...;]
Then each of these is easily summable to leave
[; 1 + \frac{1}{2} + \frac{1}{4} + \frac{1}{8} + \frac{1}{16} + ...;]
Which you can then sum again to get your final answer.
*never say this
[QUOTE=JohnnyMo1;44244614]Anyone know a good way to evaluate sum from 1 to infinity of n/(2^n)? It's obvious it does converge (it's always less than n/n^3 after finitely many terms and that obviously converges). Wolfram gives the sum as 2 and gives a partial sum formula which obviously evaluates to 2 in the limit, but it's not obvious where it came from.[/QUOTE]
1/2 + 2/4 + 3/8 + 4/16...
1/2 + (1/4 + 1/4) + (1/4 + 1/8) + (1/8 + 1/8) + (1/8 + 1/32)...
Tada, at the point it's pretty much solved.
[editline]15th March 2014[/editline]
Ha, beaten to it. I love infinite summations, they are just fun to calculate, to see what converges and what doesn't. Infinite summations are up there with fast growing functions as the most fun things to play around with, for me.
[editline]15th March 2014[/editline]
Eternal respect to anyone who correctly finds what is next in the sequence 2,2,4,4,8,7,5,4,6,6... and gives an equation for it. If anyone needs more terms I'll supply it.
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