Chapter 1: Limits
Section 1.7: Intermediate Value Theorem
A simple closed path is walked in rugged terrain. Suppose the path is parametrized by s, with 0≤s≤1. and suppose further that the heights along the path are given by hs. Prove that there are two points along the path where ha+1/2=ha.
The astute reader will see that Examples 1.7.3 and 1.7.2 are essentially the same. In fact, the curve in Figure 1.7.2(a) serves as a representation of the height function hs in Figure 1.7.3(a). The starting and ending heights are taken as zero.
The curve in Figure 1.7.3(a) represents the heights hs,s∈0,1.
The black line represents a span s=1/2. The endpoints sit at two equal heights.
The endpoints of the black line are solutions of the equation ha+1/2=ha , 0≤a≤1/2, or the alternative equation
Figure 1.7.3(a) Representative heights hs
If f0=0, then h1/2=h0, so the heights are equal for s=0 and s=1/2.
If f0≠0, then f1/2=h1−h1/2=h0−h1/2=−h1/2−h0=−f0
Assuming hs is continuous, so fa is continuous, and f0⋅f1/2<0, the Intermediate Value theorem can be applied to f and there is an a^≠0 for which fa^=0=ha^+1/2−ha^. Thus, the heights are the same at a^ and a^+1/2.
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