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Bisection - Bisection Method James Keesling 1 The...

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Bisection Method James Keesling 1 The Intermediate Value Theorem The Bisection Method is a means of numerically approximating a solution to an equation. f ( x ) = 0 The fundamental mathematical principle underlying the Bisection Method is the In- termediate Value Theorem . Theorem 1.1. Let f : [ a, b ] [ a, b ] be a continuous function. Suppose that d is any value between f ( a ) and f ( b ) . Then there is a c , a < c < b , such that f ( c ) = d . In particular, the Intermediate Value Theorem implies that if f ( a ) · f ( b ) < 0, then there is a point c , a < c < b such that f ( c ) = 0. Thus if we have a continuous function f on an interval [ a, b ] such that f ( a ) · f ( b ) < 0, then f ( x ) = 0 has a solution in that interval. The Intermediate Value Theorem not only guarantees a solution to the equation, but it also provides a means of numerically approximating a solution to arbitrary accuracy. Proceed as follows. 1. Let > 0 be the upper bound for the error required of the answer.
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