Today I was having a discussion with a colleague about the three-door problem. I'm sure you've all heard of it in some form or another. Most commonly it is given as a game show concept, where a contestant has the choice between three doors. One conceals some prize (a car, or a large sum of money) and the other two don't (they contain a goat or nothing at all). After choosing one of the doors, the game show host opens one of the remaining two doors and shows that the big prize is not behind it. Then...

So the idea of a linear approximation, is to describe a function f in the neighborhood of some point a by a linear function. In terms of the graph of f, we want to find a straight line, such that in the vicinity of a, we can approximate the function by the line. If you think about this for a moment, you will realize that we are talking about the tangent line here. Thus you quickly arrive at the equation
y = f(a) + f'(a) (x - a).
Staring at this, you may realize that f'(a)(x - a) is...

Generally, when you have some statement like "for all n, X is true", a proof by induction consists of two steps. First, you have to show that for some simple case (usually n = 0 or 1, depending on the question), X is true. Then you assume that X is true for all integers n up to some given value n₀, and you prove that under that assumption, X is also true for n₀ + 1.

The reasoning is then as follows: you have checked by hand that it is true for n = 1. You have proven that...

Using LaTeX is a bit like programming. You need a "compiler" (TeX distribution) to convert the documents you type to a final format, such as PDF. For Windows, MikTeX is the most common and easiest to install, in my experience. For *NIX (e.g. Ubuntu) texlive is straightforward to install using apt.
The documents themselves can be typed in whatever program you like, you can do it in Notepad for all anyone cares. However, there are several programs available which make editing TeX...

So after posting here for a while, I find that I have written some extensive posts on different subjects which often seem to help people out. I also find myself looking for them all the time. And I found out about the blog function. So even though, technically, this might not be the exact function of the blogs, I'll be collecting some of those posts here so I can easily find them when needed.

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From thread: Significance of Special Relativity...