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Newton-MVT - 0.1 Newton's method and the Mean Value Theorem...

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0.1 Newton’s method and the Mean Value Theorem Newton’s 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 ˆ x such that f x ) = 0. In Newton’s method, which is geometrical, we consider the curve y = f ( x ). Then the curve crosses the x -axis at the point (ˆ x,f x )). Let x 0 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 0 ,f ( x 0 )) on the curve. This tangent line satisfies the equation y f ( x 0 ) = f ( x 0 )( x x 0 ) . The tangent line crosses the x -axis at the point x 1 = x 0 f ( x 0 ) f ( x 0 ) , and we take x 1 as our improved estimate of the root ˆ 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 ˆ x will be a fixed point of T , ( T x ) = ˆ x ) if and only if f x ) = 0. To get the growth rate for the iteration
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