Chapter 4.5, Problem 1CFU

To determine

To write: A convincing argument that the Location Principle works. Include a labeled graph.

Answer to Problem 1CFU

The graph

Explanation of Solution

Given information: Location Principle

Calculation:

  $f\left(x\right)=x^2-2$

  

The “Location Principal” is really a logic statement. If you choose 1 value, say x = 1and plug it in $\left(f\left(1\right)\right)$ and you get a negative answer, and when you try another number next to 1, say x = 2, and plug it in $\left(f\left(2\right)\right)$ and you get a positive answer, then it’s logical to say that at some point between x = 1 and x = 2, the graph must have crossed the x-axis.

Therefore the zero must be somewhere between them.

This is of course, assuming the function is continuous and defined at all points where you’re testing.

  $\begin{array}{l}f\left(1\right)=-1\\ f\left(2\right)=2\end{array}$

Therefore, the location principle correctly shows that somewhere between x = 1 and x = 2 the graph = 0.

Zero occurs at x = 1.414