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4.6 Newton’s method y = f(x) x2 x1 x1 x1 x1 is initial guess at root (or zero) The equation of tangent line to graph of f(x) at (x1, f(x1)) is Let (x2, 0) denote the place where the tangent line crosses the x-axis. So (x2, 0) is on the tangent line and thus we have Solving for x2, we get Repeat and get x3 In general xn 1 xn f (xn ) f (xn ) EXAMPLE Use Newton’s method with initial guess of x1 = 1 to approximate a zero of f (x) x 3 x 2 1 . EXAMPLE Use Newton’s method to find EXAMPLE Find, correct to six decimal places, the root of the equation cos x = x. 6 2 correct to eight decimal places.