Plotting Newton Method

Recall that Newton's Method, finds the place where a function crosses the axis, by drawing a tangent line to the original function at a specified (possibly random) StartingPoint and then repeating by replacing the StartingPoint by the x coordinate of intersection of the tangent line and the X axis. We first define a preliminary function

• show_newton(Function,StartingPoint,NumberOfIterations) which
  takes three arguments:
  • an Function is an expression x, ,
  • a point to start at StartingPoint,
  • a number of iterations of drawing the tangent line, and looking for its intercept with the x axis.
  It calculates and plots the curve as well as the straight lines. See code .

• show_newton(x^2-2,7,4); This shows 4 tangent lines, using different colors lines are listed in order on the left. To do more lines replace the 4 by a higher number. The graph you see has a tangent line at the start point x=7, and then another one where that tangent line meets the x axis, and so on. The last line strikes the x axis very close to sqrt(2) where the function crosses. When you get the graph you may want to go into the config, and add more colors at the end, and click on 'replot'. Result gives
• show_newton(x^3+sin(x)+1,3,4); evaluates to Result gives
• show_newton(x^3+3*sin(x)-1,-.2,4); evaluates to Result gives
• show_newton(x^3+3*sin(x)-2,.3275,4); produces Result gives
• show_newton(x^3-4,.3275,6); evaluates to Result

Go here to simply compute some approximations.