Newton&acirc;€™s Method

# Newton&acirc;€™s Method - x 2 = x 1 – f(x 1 f...

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Newton’s Method Newton’s Method Section 3.8

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Newton’s Method is a technique for approximating the real zeros of a function. It uses tangent lines to approximate the graph of the function near its x-intercepts.
Consider a function f that is continuous on [a,b] and differentiable on (a,b). If f(a) and f(b) differ in sign, by the Intermediate Value Thm., f must have at least one zero in (a,b). Estimate where you think the zero should occur: x = x 1 x x 1 2 (x , f(x ) 1 1

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The tangent line passes thru the point (x 1 , f(x 1 ) ) with a slope of f ’(x 1 ). y – f(x 1 ) = f ’(x 1 )(x – x 1 ) y = f ’(x 1 )(x – x 1 ) + f(x 1 ) x-intercept: Let y = 0 and solve for x. 0 = f ’(x 1 )x – f ’(x 1 )x + f(x 1 ) f ’(x )x = f ’(x 1 )x - f(x 1 ) x = x - f(x 1 ) f ’(x )
From the initial estimate x 1 , you obtain a new estimate:

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Unformatted text preview: x 2 = x 1 – f(x 1 ) f ’(x 1 ) Repeated application of this process is called Newton’s Method. Each successive application of this procedure is called an iteration . Newton’s Method for Approximating Newton’s Method for Approximating the Zeros of a Function the Zeros of a Function 1. Make an initial estimate x 1 that is “close to” c. (A graph is helpful.) 2. Determine a new approximation x n+1 = x n – f(x n ) f ’(x n ) 3. If x n-x n+1 is within the desired accuracy, let x n+1 serve as the final approximation. Otherwise, return to step 2 and calculate a new approximation. Joke Time Joke Time Who was the dancing Vice-President in 2000? Al-gore rhythum What does a lumberjack do to trees? axi-oms...
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