Newtons_Method_2020.pdf - CALC 1501B Newtonu2019s Method...

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CALC 1501B Newton’s Method 2020 Newton’s Method Newton’s Method or the Newton–Raphson Method is a method for finding a root of the equation f ( x ) = 0. We start with a guess x 0 for the root r . Draw a tangent line L to the graph y = f ( x ) at the point x 0 . Let x 1 be the point at which the line L intersects the x -axis. If the initial guess x 0 is “close” to an actual root r , then x 1 is a better approximation to r . The process can be repeated to produce a sequence of numbers x 2 , x 3 , . . . converging to the root r . The number x n +1 is the x -intercept of the tangent line to y = f ( x ) at ( x n , f ( x n )). Newton’s method for finding roots. We can find the equation for x n +1 by noticing that the line L passes through the point ( x n +1 , 0) and ( x n , f ( x n )) and hence has the slope 0 - f ( x n ) x n +1 - x n . But it is also tangent to y = f ( x ) at x = x n and hence slope equals f 0 ( x n ), giving us f 0 ( x n ) = 0 - f ( x n ) x n +1 - x n = x n +1 = x n - f ( x n ) f 0 ( x n ) (1) Remark 1. A formula like (1) that gives x n +1 in terms of the previous values is called a recursive formula for the sequence x 0 , x 1 , x 2 , . . . . 1
CALC 1501B Newton’s Method 2020 Example 2.Approximate5 using Newton’s method. Example 3.Find an approximate root off(x) =x3+x-1 using Newton’s method. 2
CALC 1501B Newton’s Method 2020 Thus the number x 5 = 0 . 682327803828019 is an approximate root of x 3 + x - 1.

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