Cauchy Mean Value Theorem/Example
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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
- Mathematics.StackExchange: Post 296194, revision 11