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.
<< Previous Example Section 1.7
Next Chapter >>
© Maplesoft, a division of Waterloo Maple Inc., 2021. All rights reserved. This product is protected by copyright and distributed under licenses restricting its use, copying, distribution, and decompilation.
For more information on Maplesoft products and services, visit www.maplesoft.com
Download Help Document
What kind of issue would you like to report? (Optional)