This program graphs the equation X^3/3 - 2*X + 5. When you click on the graph, it uses Newton's method to find a root of the equation, starting from the X value that you clicked.

' The function.
Private Function F(ByVal x As Double) As Single
    Return x * x * x / 3 - 2 * x * x + 5
End Function

' Draw the background graph.
Private Sub DrawGraph(ByVal gr As Graphics)
    gr.Clear(Me.BackColor)

    ' Scale for convenience.
    gr.ScaleTransform(SCALE_FACTOR, -SCALE_FACTOR)
    gr.TranslateTransform( _
        Me.ClientRectangle.Width / 2, _
        Me.ClientRectangle.Height / 2, _
        Drawing2D.MatrixOrder.Append)

    ' Draw the axes.
    Dim blue_pen As New Pen(Color.Blue, 0)
    For x As Integer = -10 To 10
        gr.DrawLine(blue_pen, x, -0.5!, x, 0.5!)
    Next x
    For y As Integer = -10 To 10
        gr.DrawLine(blue_pen, -0.5!, y, 0.5!, y)
    Next y
    gr.DrawLine(blue_pen, -10, 0, 10, 0)
    gr.DrawLine(blue_pen, 0, -10, 0, 10)
    blue_pen.Dispose()

    ' Draw the curve.
    Dim red_pen As New Pen(Color.Red, 0)
    Dim y1 As Single
    Dim y2 As Single = F(-10)
    Const STEP_SIZE As Single = 0.1
    For x As Single = -10 + STEP_SIZE To 10 Step STEP_SIZE
        y1 = y2
        y2 = F(x)
        gr.DrawLine(red_pen, x - STEP_SIZE, y1, x, y2)
    Next x
    red_pen.Dispose()
End Sub

    epsilon = -F(x) / dFdx(x)

The method then sets its next guess for x to be the current value plus epsilon (X(n+1) = X(n) + epsilon). It repeats this process until epsilon is smaller than some cutoff value.

' The function's derivative.
Private Function dFdx(ByVal x As Double) As Single
    Return x * x - 4 * x
End Function

' Find the roots by using Newton's method.
Private Sub Form1_MouseDown(ByVal sender As Object, ByVal e _
     As
System.Windows.Forms.MouseEventArgs) Handles _
    MyBase.MouseDown
    ' Clear previous results.
    Dim gr As Graphics = Me.CreateGraphics()
    DrawGraph(gr)

    ' Convert the X coordinate into world coordinates.
    Dim m As Matrix = gr.Transform()
    m.Invert()
    Dim pts() As PointF = {New PointF(e.X, e.Y)}
    m.TransformPoints(pts)
    Dim x0 As Single = pts(0).X

    Const CUTOFF As Single = 0.0000001
    Dim epsilon As Single
    Dim the_pen As New Pen(Color.Black, 0)
    Dim iterations As Integer = 0
    Do
        iterations += 1
        gr.DrawLine(the_pen, x0, -1.5!, x0, 1.5!)
        epsilon = -F(x0) / dFdx(x0)
        x0 = x0 + epsilon
    Loop While Abs(epsilon) > CUTOFF
    gr.DrawLine(the_pen, x0, -3.0!, x0, 3.0!)
    the_pen.Dispose()

    Me.Text = x0 & " +/-" & epsilon & " in " & iterations & _
        " iterations"
End Sub

For more information on Newton's method, see Eric W. Weisstein's article Newton's Method from MathWorld--A Wolfram Web Resource.