Recent content by LumenPlacidum

So, is this appropriate? The general solution for a non-homogeneous system of differential equations is analogous to the +C of integration. Because of the superposition principle, any function of the form of the general solution COULD be a part of the solution for the ODE since it would have...

If the purpose of the general form of the solution to a differential equation is to represent a formula with parameters for the solutions to that differential equation, why is it that we typically want to add some particular solution to the general one? Solution = General Solution + Particular...

I've been looking around for information on Saturn's moon, Titan, to present to my class. One thing that wikipedia seems to say about the moon is that it has distinctly fewer impact craters than other moons of its size and position in the solar system. Supposedly, this is attributed to some...

Those gas samples are at a low pressure? I suppose that makes sense. Is it just that you're heating up the gas, but it's not generating a continuous spectrum because it's not dense enough to do so?

Update: while I find the grease spot photometer to be awesome, I just did some scrounging in the depths of the science closet at my school. Just turned up some basic photometers and some nice spectroscopes as well as a couple chemical spectrophotometers (though I'm not sure I can use these)...

I'm an astronomy teacher on a very tight budget. Before I really get into the meat of astronomical information, I want to make sure that my students have some understanding of WHY it is that we know what we do about astronomical bodies, and that means getting them to work with some of the tools...

All functions have domains and ranges. Functions are items to which you give them a value and they give you a value (usually these are numbers). For example, all of the following are functions, and as such they have domains and ranges (which are also given, based on the typical real number...

Great, thanks so much!

The alternating series test contains two conditions for convergence. The first condition is that the nth term (extracting the power of -1) is always positive and monotonically decreasing. The second is that the limit of that nth term goes to 0 as n goes to infinity. I've seen a proof...

You sure? It seems to be working for me, unless I'm making an arithmetic error. You don't need four integrals, by the way, but it should still work (you can go straight from one function to another without having the vertical x=pi/2 and x=3pi/2 in between). What are you checking it against?

You broke up the function into working sections. The only thing is, in the original, the part of the shaded region that falls below the y-axis is treated as negative. The integrals you have will give you a positive value for the integral of that region.

Well done!

Well, what is the form for the slope of a secant line to the graph of a function? How does that relate to the slope of the tangent line at some point?

Ah, then CRGreathouse was right; the quotient space on that relation is constructing the real numbers modulo 1. When it's flat out saying that that's an equivalence relation, then it's saying that it's reflexive, transitive and symmetric. Essentially, this is saying that the part of any real...

If your 'a' is simply any element of the rational numbers, then your r_n_1 is going to be the supremum of the rationals, which is not in the rationals. Also, after you show that something's strictly increasing, you have to show that it's also unbounded, meaning that for any M, you can pick...