Newton's Method


Approximating Zeroes of Functions with Newton's Method

In this illustration of Newton's Method we start by entering a formula for the function in the input box. We also input an initial approximation of the zero (x-coordinate of the x-intercept) that we are looking for in the input box for . Check Step 1 to see how we use this initial estimate to produce a better estimate . We find the point on the curve and then draw the tangent line to the curve at that point. Since this is a differentiable function, the tangent line will follow the curve fairly closely. The equation of this tangent line is . We find the x-intercept of this line and call its x-coordinate , which is a better estimate of the zero than the initial guess. We see that we have . Solving for we see that . We then iteratively repeat this process using the estimate to get a better estimate . In general, we use the following recursive formula to get from one estimate to the next: . By doing this enough times we can get the zero to any desired degree of accuracy.