Science:Math Exam Resources/Courses/MATH100/December 2010/Question 08
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Full-Solution Problems. In questions 2-8, justify your answers and show all your work. If a box is provided, write your final answer there. Simplification of answers is not required unless explicitly requested.
Two points on the surface of the Earth are called antipodal if they are at exactly opposite points (for example, the North Pole and South Pole are antipodal points). Prove that, at any given moment, there are two antipodal points on the equator with exactly the same temperature.
Hint: Let be the temperature, at any given moment, at the point on the equator with longitudinal angle measured in radians, (that is, in one complete trip around the equator, goes from 0 to ) and consider
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There are two key facts to this problem.
1) The function T is periodic (like ).
2) We are looking for a root of f.
What theorem helps us guarantee the existence of roots? Use it!
The intermediate value theorem states that if a function is continuous on and or , then there exists a point such that .
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Notice that the function is periodic since the equator is shaped like a circle. Mathematically this in particular means
We exploit this fact by looking at
At this point we want to distinguish two cases:
Case 1. f(0) = 0. Then
and hence , which is what we wanted to show.
Case 2. f(0) ≠ 0. Then we have that one of and is positive and one is negative. As T is continuous, we have that f is continuous and hence we may invoke the intermediate value theorem to see that there exists a point such that . Thus, we have and this completes the proof.