Chapter 3.2, Problem 119AYU

Bisection Method for Approximating Zeros of a Function $f$ We begin with two consecutive integers, $a$ and $a+1$, such that $f\left(a\right)$ and $f\left(a+1\right)$ are of opposite sign. Evaluate $f$ at the midpoint $m_1$ of $a$ and $a+1$. If $f\left(m_1\right)=0$. then $m_1$ is the zero of $f$, and we are finished. Otherwise, $f\left(m_1\right)$ is of opposite sign to either $f\left(a\right)$ or $f\left(a+1\right)$. Suppose that it is $f\left(a\right)$ and $f\left(m_1\right)$ that are of opposite sign. Now evaluate $f$ at the midpoint $m_2$ of $a$ and $m_1$. Repeat this process until the desired degree of accuracy is obtained. Note that each iteration places the zero in an interval whose length is half that of the previous interval. Use the bisection method to approximate the zero of $f\left(x\right)=8x^4-2x^2+5x-1$ in the interval $\left[0,1\right]$ correct to three decimal places. Verify your result using a graphing utility.

[Hint: The process ends when both endpoints agree to the desired number of decimal places.]