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Racetrack principle

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In calculus, the racetrack principle describes the movement and growth of two functions in terms of their derivatives.

This principle is derived from the fact that if a horse named Frank Fleetfeet always runs faster than a horse named Greg Gooseleg, then if Frank and Greg start a race from the same place and the same time, then Frank will win. More briefly, the horse that starts fast and stays fast wins.

In symbols:

if for all , and if , then for all .

or, substituting ≥ for > produces the theorem

if for all , and if , then for all .

which can be proved in a similar way

Proof

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This principle can be proven by considering the function . If we were to take the derivative we would notice that for ,

Also notice that . Combining these observations, we can use the mean value theorem on the interval and get

By assumption, , so multiplying both sides by gives . This implies .

Generalizations

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The statement of the racetrack principle can slightly generalized as follows;

if for all , and if , then for all .

as above, substituting ≥ for > produces the theorem

if for all , and if , then for all .

Proof

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This generalization can be proved from the racetrack principle as follows:

Consider functions and . Given that for all , and ,

for all , and , which by the proof of the racetrack principle above means for all so for all .

Application

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The racetrack principle can be used to prove a lemma necessary to show that the exponential function grows faster than any power function. The lemma required is that

for all real . This is obvious for but the racetrack principle can be used for . To see how it is used we consider the functions

and

Notice that and that

because the exponential function is always increasing (monotonic) so . Thus by the racetrack principle . Thus,

for all .

References

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  • Deborah Hughes-Hallet, et al., Calculus.