Cauchy Mean Value Theorem/Example

From ProofWiki
Jump to navigation Jump to search

Example of Use of Cauchy Mean Value Theorem

In the 2012 Olympics Usain Bolt won the 100 metres gold medal with a time of $9.63$ seconds.

By definition, his average speed was the total distance travelled divided by the total time it took:

\(\ds V_a\) \(=\) \(\ds \frac {\map d {t_2} - \map d {t_1} } {t_2 - t_1}\)
\(\ds \) \(=\) \(\ds \frac {100 \ \mathrm m} {9.63 \ \mathrm s}\)
\(\ds \) \(=\) \(\ds 10.384 \ \mathrm {m/s}\)
\(\ds \) \(=\) \(\ds 37.38 \ \mathrm {km/h}\)


The Mean Value Theorem gives:

$\map {f'} c = \dfrac {\map f b - \map f a} {b - a}$

Hence, at some point Bolt was actually running at the average speed of $37.38 \ \mathrm {km/h}$

Asafa Powell was participating in that same race.

He achieved a time of $11.99 \ \mathrm s = 1.245 \times 9.63 \ \mathrm s$.

So Bolt's average speed was $1.245$ times the average speed of Powell.


The Cauchy Mean Value Theorem gives:

\(\ds \frac {\map {f'} c} {\map {g'} c}\) \(=\) \(\ds \frac {\map f b - \map f a} {\map g b - \map g a}\)
\(\ds \) \(=\) \(\ds \frac {\dfrac {\map f b - \map f a} {b - a} } {\dfrac {\map g b - \map g a} {b - a} }\)

Hence, at some point, Bolt was actually running at a speed exactly $1.245$ times that of Powell's.


Sources