Chapter 1.4, Problem 80A

a.

To determine

To explain: The reason for the root to exist by the Intermediate Value Theorem.

b.

To determine

To show: The value of the function $f\left(x\right)=x^2-2$ is negative at $x=0$ and positive at $x=2$ . Also the relevance of the Intermediate Value Theorem for the above.

c.

To determine

To explain: The Intermediate Value Theorem for the function $f\left(x\right)=x^2-2$ on the interval $\left(0,1\right)$ and $\left(1,2\right)$ .

d.

To determine

To calculate: The interval of length $\frac{1}{2}$ such that the function $f\left(x\right)=x^2-2$ has a root in that interval.

e.

To determine

To describe: The relevance of Bisection Method to approximate the root of a function. Also, the approximation to the root of the function $f\left(x\right)=x^2-2$ as $\sqrt{2}$ .