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**Unformatted text preview: **0.1 Newtons method and the Mean Value Theorem Newtons method for computing the zeros of functions is a good example of the practical application of the Mean Value Theorem. Let f ( x ) be a real- valued function on the real line that has two continuous derivatives. We are looking for a root of f , i.e., a point x such that f ( x ) = 0. In Newtons method, which is geometrical, we consider the curve y = f ( x ). Then the curve crosses the x-axis at the point ( x, f ( x )). Let x be an initial guess for the root. To improve on the guess we construct the tangent line to the curve y = f ( x ) that passes through the point ( x , f ( x )) on the curve. This tangent line satisfies the equation y f ( x ) = f ( x )( x x ) . The tangent line crosses the x-axis at the point x 1 = x f ( x ) f ( x ) , and we take x 1 as our improved estimate of the root x . Now we repeat this procedure with x 1 to get an improved estimate x 2 , and so on. Thus we have a sequence { x n } such that x n +1 = x n f ( x n ) f ( x n ) , n = 0 , 1 , . We need to give conditions that will guarantee that the sequence will converge to a root of f ( x ), and will provide information about the rate of convergence. To analyze this procedure we define an updating function T ( x ) by T ( x ) = x f ( x ) f ( x ) . We will not yet fix the domain D of this function, but it is clear that we must require f ( x ) negationslash = 0 for all x D . Then x will be a fixed point of T , ( T ( x ) = x ) if and only if f ( x ) = 0. To get the growth rate for the iteration) = 0....

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