Newton-Raphson Method of Finding Roots

An applet to look at the Newton-Raphson method for finding the root of a function.

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Min X = Max X = Min Y = Max Y =
  f(x) = X0 =
Newton's method finds approximations of a root of a function by starting with an initial guess for the value of the root of the function.  You then find the next approximation by finding where the tangent line intersects the x-Axis.  By increasing the number of steps, this applet lets you visualize the first 8 approximations.  Note that X0 is draggable.

Questions to explore:
  1. The roots of the loaded example, 2*x*cos(x)-0.84 are nice in the sense that if X0 starts close to a root, this method will find the nearest root.  Find the region around each root visible on the screen where the method converges to the obvious root.
  2. If X0 starts close to a critical point, the Newton-Raphson method gets sent for a trip.  What is the biggest root you can find with this method with a starting point in the viewing window.
  3. In first example from the example list x^4-2*x^3-x^2-2*x+2, the first guess needs to be close enough for the answer to converge by the maximimum number of steps.  Find the region where the in the window where the precess does not converge in 10 steps.  Do you think it would converge with more steps.  Justify your answer.
  4. In the second example x^4-3*x^3+x^2+2*x-2 we have a region where Newton's method fails.  Describe the feature of the graph that causes the failure.
  5. Describe what happens in the second example if we start with X0=0.  Find 2 more starting points that produce an infinite loop.
  6. In the last example, cbrt(x), Newton's method fails to find a root.  Describe the behavior in this case.  Describe the feature of the graph that causes this failure.  Test your description on cbrt(x)+1.

Created by Mike May, S.J., with GeoGebra
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Last updated By Mike May, S.J., December 4, 2007.