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The Race Track Principle

The Race Track Principle says that if horse A always runs faster than horse B, and if they start a race at the same place and time, horse A is bound to win. That's not too difficult is it? In terms of functions and their derivatives, it says if f'[x] > g'[x] for all x > 0, and if f[0] = g[0], then f[x] > g[x] for all x > 0. For you math types, it's the mean value theorem in its active voice. (There's no there exists in it anywhere.) Students in Calculus&Mathematica courses use it to get inequalities like x^1/2 - y^1/2 < (x - y)^1/2 for x > y. They also use it to get a good idea about why Euler's method for approximate integration of functions and differential equations gives reasonably accurate results. All of that from a simple little principle that looks like this: