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# 14.2 Applications

14.6   Example. We know that is continuous on , and that .(Cf equation (5.78).) It follows that there is a number in such that

14.7   Example. Two points on a sphere are called antipodal points if and are opposite ends of the same diameter of the sphere. We will consider the surface of the earth to be a sphere of radius . At any fixed time, let denote the temperature of the earth at the point on the surface of the earth. (More precisely, let be the number such that the temperature at is ). We will show that there are two antipodal points on the surface of the earth such that In fact, we will show that there are two antipodal points on the equator with the same temperature. We first introduce a coordinate system so that the center of the earth is at the origin, and the plane of the equator is the - plane, and the point on the equator passing through the Greenwich meridian is the point . Then the points on the equator are the points

Define a function by

Thus

We suppose that is a continuous function on . If then , so and are a pair of antipodal points with the same temperature. Now

so if then and have opposite signs. Hence by the intermediate value property, there is a number such that , i.e.

Then and are a pair of antipodal points with the same temperature.

14.8   Example. Let

where and are real numbers, and . Then there exists some number such that .

Proof:     I will suppose that for all and derive a contradiction. Let

Since for all , is continuous on R. We know that

Hence for some . Then and have opposite signs, so by the intermediate value property there is a number such that . This contradicts our assumption that for all

14.9   Exercise. A Let Show that there are at least three different numbers such that

14.10   Exercise. A Three wires are joined at a common point .
Let be the Y-shaped figure formed by the three wires. Prove that at any time there are two points in with the same temperature.

14.11   Exercise. A Six wires are joined to form the figure shown in the diagram.
Show that at any time there are three points in that have the same temperature. To simplify the problem, you may assume that the temperatures at ,, , and are all distinct.

Next: 14.3 Inverse Functions Up: 14. The Inverse Function Previous: 14.1 The Intermediate Value   Index
Ray Mayer 2007-09-07