calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizations of arithm ...

, the racetrack principle describes the movement and growth of two functions in terms of their

derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...

s. 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

f'(x)>g'(x)

for all

x>0

, and if

f(0)=g(0)

, then

f(x)>g(x)

for all

x>0

. or, substituting ≥ for > produces the theorem :if

f'(x) \ge g'(x)

for all

x>0

, and if

f(0)=g(0)

, then

f(x) \ge g(x)

for all

x \ge 0

. which can be proved in a similar way

Proof

This principle can be proven by considering the function

h(x) = f(x) - g(x)

. If we were to take the derivative we would notice that for

x>0

, :

h'= f'-g'>0.

Also notice that

h(0) = 0

. Combining these observations, we can use the

mean value theorem In mathematics, the mean value theorem (or Lagrange theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It i ...

on the interval

, x The comma is a punctuation mark that appears in several variants in different languages. It has the same shape as an apostrophe or single closing quotation mark () in many typefaces, but it differs from them in being placed on the baseline o ...

/math> and get :

0 < h'(x_0)= \frac= \frac.

By assumption,

x>0

, so multiplying both sides by

x

gives

f(x) - g(x) > 0

. This implies

f(x) > g(x)

Generalizations

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

f'(x)>g'(x)

for all

x>a

, and if

f(a)=g(a)

, then

f(x)>g(x)

for all

x>a

. as above, substituting ≥ for > produces the theorem :if

f'(x) \ge g'(x)

for all

x>a

, and if

f(a)=g(a)

, then

f(x) \ge g(x)

for all

x>a

Proof

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

f_2(x)=f(x+a)

and

g_2(x)=g(x+a)

. Given that

f'(x)>g'(x)

for all

x>a

, and

f(a)=g(a)

f_2'(x)>g_2'(x)

for all

x>0

, and

f_2(0)=g_2(0)

, which by the proof of the racetrack principle above means

f_2(x)>g_2(x)

for all

x>0

f(x)>g(x)

for all

x>a

Application

The racetrack principle can be used to prove a

lemma Lemma may refer to: Language and linguistics * Lemma (morphology), the canonical, dictionary or citation form of a word * Lemma (psycholinguistics), a mental abstraction of a word about to be uttered Science and mathematics * Lemma (botany), a ...

necessary to show that the

exponential function The exponential function is a mathematical function denoted by f(x)=\exp(x) or e^x (where the argument is written as an exponent). Unless otherwise specified, the term generally refers to the positive-valued function of a real variable, a ...

grows faster than any power function. The lemma required is that :

e^>x

for all real

x

. This is obvious for

x<0

but the racetrack principle can be used for

x>0

. To see how it is used we consider the functions :

f(x)=e^

and :

g(x)=x+1.

Notice that

f(0) = g(0)

and that :

e^>1

because the exponential function is always increasing (

monotonic In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order ...

) so

f'(x)>g'(x)

. Thus by the racetrack principle

f(x)>g(x)

. Thus, :

e^{x}>x+1>x

for all

x>0

References

* Deborah Hughes-Hallet, et al., ''Calculus''. Differential calculus Mathematical principles