lec13 - .01 Fall 2006 Lecture 13 Newton’s Method and...

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Unformatted text preview: Lecture 13 18.01 Fall 2006 Lecture 13: Newton’s Method and Other Applications Newton’s Method Newton’s method is a powerful tool for solving equations of the form f ( x ) = 0. Example 1. f ( x ) = x 2- 3. In other words, solve x 2- 3 = 0. We already know that the solution to this is x = √ 3. Newton’s method, gives a good numerical approximation to the answer. The method uses tangent lines (see Fig. 1). x =1 x 1 (1,-2) y = x 2-3 Figure 1: Illustration of Newton’s Method, Example 1. The goal is to find where the graph crosses the x-axis. We start with a guess of x = 1. Plugging that back into the equation for y , we get y = 1 2- 3 =- 2, which isn’t very close to 0. Our next guess is x 1 , where the tangent line to the function at x crosses the x-axis. The equation for the tangent line is: y- y = m ( x- x ) When the tangent line intercepts the x-axis, y = 0, so- y = m ( x 1- x ) y- m = x 1- x y x 1 = x- m Remember: m is the slope of the tangent line to y = f ( x ) at the point ( x , y ). 1 In terms of f : y = f ( x ) m = f ( x ) Therefore, f ( x ) x 1 = x- f ( x ) x 1 x x 2 Figure 2: Illustration of Newton’s Method, Example 1. In our example, f ( x ) = x 2- 3 , f ( x ) = 2 x . Thus, ( x 2- 3) 1 3 x 1 = x- 2 x = x- 2 x + 2 x 1 3 x 1 = x + 2 2 x The main idea is to repeat (iterate) this process: 1 3 x 2 = x 1 + 2 2 x 1 1 3 x 3 = x 2 + 2 2 x 2 and so on. The procedure approximates √ 3 extremely well....
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lec13 - .01 Fall 2006 Lecture 13 Newton’s Method and...

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