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In [[calculus]],
the '''racetrack principle''' describes the movement and growth of two functions in terms of their [[derivative]]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. 
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In symbols:
:if <math>f'(x)>g'(x)</math> for all <math>x>0</math>, and if <math>f(0)=g(0)</math>, then <math>f(x)>g(x)</math> for all <math>x>0</math>.  
or, substituting ≥ for > produces the theorem
:if <math>f'(x) \ge g'(x)</math> for all <math>x>0</math>, and if <math>f(0)=g(0)</math>, then <math>f(x) \ge g(x)</math> for all <math>x>0</math>.
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
 
:<math> h'= f'-g'>0.</math>
 
Also notice that h(0) = 0. Combining these observations, we can use the [[mean value theorem]] on the interval [0, x] and get
 
:<math> h'(x_0)= \frac{h(x)-h(0)}{x-0}= \frac{f(x)-g(x)}{x}>0.</math>
 
Since x > 0 for the mean value theorem to work then we may conclude that f(x) - g(x) > 0. This implies f(x) > g(x).
 
==Generalizations==
 
The statement of the racetrack principle can slightly generalized as follows;
:if <math>f'(x)>g'(x)</math> for all <math>x>a</math><!--where a≥0 **don't need this restriction**-->, and if <math>f(a)=g(a)</math>, then <math>f(x)>g(x)</math> for all <math>x>a</math>.
 
as above, substituting ≥ for > produces the theorem
:if <math>f'(x) \ge g'(x)</math> for all <math>x>a</math><!--where a≥0 **don't need this restriction**-->, and if <math>f(a)=g(a)</math>, then <math>f(x) \ge g(x)</math> for all <math>x>a</math>.
 
===Proof===
This generalization can be proved from the racetrack principle as follows:
 
Given <math>f'(x)>g'(x)</math> for all <math>x>a</math> where a≥0, and <math>f(a)=g(a)</math>,
 
Consider functions <math>f_2(x)=f(x-a)</math> and <math>g_2(x)=g(x-a)</math>
 
<math>f_2'(x)>g_2'(x)</math> for all <math>x>0</math>, and <math>f_2(0)=g_2(0)</math>, which by the proof of the racetrack principle above means <math>f_2(x)>g_2(x)</math> for all <math>x>0</math> so <math>f(x)>g(x)</math> for all <math>x>a</math>.
 
==Application==
The racetrack principle can be used to prove a [[lemma (mathematics)|lemma]] necessary to show that the [[exponential function]] grows faster than any power function. The lemma required is that
:<math> e^{x}>x </math>
for all real x. This is obvious for x<0 but the racetrack principle is required for x>0. To see how it is used we consider the functions
:<math> f(x)=e^{x}</math>
and
:<math> g(x)=x+1.</math>
Notice that f(0) = g(0) and that
:<math> e^{x}>1</math>
because the exponential function is always increasing ([[monotonic]]) so <math>f'(x)>g'(x)</math>. Thus by the racetrack principle f(x)>g(x). Thus,
:<math> e^{x}>x+1>x</math>
for all x>0.
 
==External links==
*[http://www.matheverywhere.com/mei/candm/race.php3 Usage of Racetrack Principle] (Math Everywhere)
 
[[Category:Calculus]]
[[Category:Pedagogy]]
[[Category:Mathematical principles]]

Latest revision as of 18:40, 28 February 2014

The main advantage of using the blog is that anyone can use the Word - Press blog and customize the elements in the theme regardless to limited knowledge about internet and website development. You can either install Word - Press yourself or use free services offered on the web today. This is a service where people write articles using a specific keyword you have given them. s and intelligently including a substantial amount of key words in the title tags, image links, etc. If you are happy with your new look then click "Activate 'New Theme'" in the top right corner.

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