Math_137_Winter_2010_Week_10_Notes

Math_137_Winter_2010_Week_10_Notes - Math 137 Week 10 Notes...

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Math 137 Week 10 Notes – Dr. Paula Smith 4.8 Newton’s Method Given a function f(x), the values of x that make f(x) = 0 are called the roots of function f. If f is a quadratic function, the quadratic formula finds its roots. There are formulae for finding the roots of cubic (polynomials of degree 3) and quartic (polynomials of degree 4) functions as well, but in general finding the roots of f is not simple. Newton’s (or Newton-Raphson) method for finding the approximate value of a function’s roots is recursive, that is, we repeat the method over and over until our successive answers agree to as many decimal places as we like. It uses an arbitrary value of x to start and the equation of the tangent line to f at the point x. Say we choose value x 1 ; the equation of the tangent line to f at that point is y = f(x 1 ) + f (x 1 ) (x – x 1 ). We let the tangent line stretch till it intersects the x-axis so that y = 0, and we then solve for x (we are finding the x-intercept of the tangent line). Thus 0 = f(x 1 ) + f (x 1 ) (x – x 1 ) x = x 1 – [f(x 1 ) / f (x 1 )]. The new value x is (generally) a better approximation of a root for f than x 1 was. But to improve the value still more, we let x 2 = x and find the tangent line to the function f at the point x 2 , and then solve for the new x. More mechanically, we use a value to start that close to a suspected root, and the formula x n+1 = x n – [f(x n ) / f (x n )] over and over again until successive answers agree. Warning:
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Math_137_Winter_2010_Week_10_Notes - Math 137 Week 10 Notes...

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