This applet demonstrates the Newton method,  which is a rapidly converging root finding routine.

The text fields on the bottom of the window are used to enter the function f(x),  a range for x : x min ... x max,  # of  points at which the function is to be displayed and an initial guess for a root : x initial.  The first derivative required by the Newton method is computed symbolically and need not be entered. 

The Plot f   button graphs the function on the specified domain.  After viewing the graph, the user can type in the initial guess. Alternatively, she or he can use the fact that clicking the right mouse button anywhere on the graph area inserts the x coordinate of the point under the mouse pointer in the x initial field. The Start button initiates the iterative process of solving the equation f(x)=0 by computing the value of the function at the x initial point and displaying the result in the output area on the right.  Clicking on the Next button produces an iteration sequence of root approximations. 

Start and Next will also draw the tangent from the current point on the graph to the x-axis,  which is displayed as a thick blue line.  Due to the large convergence rate of the method, the tangents may become invisible after the first few iterations.

Finally,  recall that the convergence of the Newton method is guaranteed only when the initial guess is "sufficiently" close to the actual root.