The bisection method will converge on the root of almost any function, even very

Question

The bisection method will converge on the root of almost any function, even very complicated functions with unusual shapes.  Describe a type of function for which the bisection method will not work, and show the graph of such a function, and state why it won The bisection method will converge on the root of almost any function, even very complicated functions with unusual shapes.  Describe a type of function for which the bisection method will not work, and show the graph of such a function, and state why it won

Explanation / Answer

This method can be used with either a spreadsheet or a graphing calculator with spreadsheet capabilities. Basically, the bisection method squeezes in on the root from both sides by making use of the Intermediate Value Theorem. In essence, this theorem says that if f is a continous function on [a, b] and the sign of f(a) is different from the sign of f(b), then there must be some point, c, in the interval such that f(c) = 0, and is thus a root of the function. The Bisection method uses this to squeeze in on the root in the following way. First, we can select an interval that contains the root we are looking for by looking at the graph. Second, we find the midpoint of the given interval, in this case (b - a)/2 = m, and then look at the function f evaluated at a, b, and m. Third we compare the signs of f(a), f(m) and f(b). f(m) will be one of three things. It will either have the sign of f(a), the sign of f(b), or it will be 0. If it is 0 (it usually isn't), then we stop because we have found the root we are looking for. Now, suppose that the sign of f(m) is equal to the sign of f(a). This implies that the root will be between m and b, and so we choose [m, b] as our new interval and repeat the process. Similarly, we replace b with m if the sign of f(m) equals the sign of f(b). Which ever the case we know that the root is in our new interval, and the endpoints of that interval have different signs. To "squeeze" in on the root we continue the process, until the desired accuracy is achieved.

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