An applet to look at the Newton-Raphson method for finding the root of
a
function.
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:
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.
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.
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.
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.
Describe what happens in the second example if we start with
X0=0. Find 2 more starting points that produce an infinite loop.
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.