Here's a good explanation of this.......http://tutorial.math.lamar.edu/Classes/CalcI/NewtonsMethod.aspx
Basically, we're picking [ guessing] some point on a curve, drawing a tangent line to the curve at that point, and then using the coordinate of the x intercept of this tangent line as our next "guess." Then, we repeat the process to get the next x-intercept point, etc. As long as the derivative exists in each case [ and is non-zero ], we should arrive at a pretty good approximation of the solution to our function after several iterations.
