What a continuous function can do...
In preparation for my next test I came across a very fascinating mathematical problem.
"Prove that there always two subtended points on the Earth's equator where you are able to measure the exact same temperature." (Adapted from the Book: Repetiorium der Analysis Teil 1 (en. Revision Course for Analysis Part 1) from Timmann).
I have tried to visualize the proposition. The two red dots represent two subtended points on the equator. Remember: We want to show that there are always two subtended points (the red ones) on the Earth's equator where you are able to measure the exact same temperature.
If you believe it or not: That can be proven by combining simple mathematical concepts.
Let the points of the equator be the interval
, 
a point on the equator and )
a function which returns the temperature at the given point. is a continuous function, obviously. \; :=\;T\left( p \right)\; -\; T\left( p+\pi \right))
returns the difference between two subtended points on the equator. Like the temperature function this function is continuous too. Because of \; =\; T\left( 2\pi \right)\; is\; f\left( \pi \right)\; =\; -f\left( 0 \right))
. If \; =\; -f\left( \pi \right)\; =\; 0)
then the two subtended points on the equator must have the same temperature. If \; =\; -f\left( \pi \right)\; \neq \; 0)
then \; and\; f\left( \pi \right))
have different signs and therefore because of the "intermediate value theorem" there exists a \; =\; 0)
and this means that \; =\; T\left( p\; +\; \pi \right))
which was we wanted to show.
When I first came across this problem I couldn't believe what one can prove with math. I hope you enjoyed my first post. More will come.
"Prove that there always two subtended points on the Earth's equator where you are able to measure the exact same temperature." (Adapted from the Book: Repetiorium der Analysis Teil 1 (en. Revision Course for Analysis Part 1) from Timmann).
I have tried to visualize the proposition. The two red dots represent two subtended points on the equator. Remember: We want to show that there are always two subtended points (the red ones) on the Earth's equator where you are able to measure the exact same temperature.
If you believe it or not: That can be proven by combining simple mathematical concepts.
Let the points of the equator be the interval
When I first came across this problem I couldn't believe what one can prove with math. I hope you enjoyed my first post. More will come.