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 [math]x_0[/math]. [br][br]Check Step 1 to see how we use this initial estimate to produce a better estimate [math]x_1[/math]. We find the point [math]\left(x_0,f\left(x_0\right)\right)[/math] 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 [br][math]y=f'\left(x_0\right)\left(x-x_0\right)+f\left(x_0\right)[/math]. We find the x-intercept of this line and call its x-coordinate [math]x_1[/math], which is a better estimate of the zero than the initial guess. We see that we have [math]0=f'\left(x_0\right)\left(x_1-x_0\right)+f\left(x_0\right)[/math]. Solving for [math]x_1[/math] we see that [math]x_1=x_0-\frac{f\left(x_0\right)}{f'\left(x_0\right)}[/math].[br][br]We then iteratively repeat this process using the estimate [math]x_1[/math] to get a better estimate [math]x_2[/math]. In general, we use the following recursive formula to get from one estimate to the next:[br][math]x_n=x_{n-1}-\frac{f\left(x_{n-1}\right)}{f'\left(x_{n-1}\right)}[/math].[br][br]By doing this enough times we can get the zero to any desired degree of accuracy.