NEWTON'S METHOD

This slide show consists of graphs of various demonstrations of Newton's Method.

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A.  Method - Example 1                               2
     This shows how the method would work on f(x) = x , starting from x = -2,
and looking for x = 0.  First f(x) is drawn, then the vertical line x = -2 is
added, then the tangent line to f(x) at x = -2 is drawn, and so on.

B.  Method - Example 2
     This applies the method to f(x) = sin(x), looking for x = 0, and starting
with x = 11ã/32.

C.  Problem - Example 1
     This is the same as B, except the initial value is taken as ã/2 - .02.  A
solution will be found, (at x = -ã), but not the one that was sought (x = 0).

D.  Problem - Example 2
     This is the same as B, except the initial value is taken as ã/2.  A
solution is not found because the tangent line is horizontal, and never crosses
the x-axis. Consequently there is no second estimate for x.

E.  Problem - Example 3                 3
     This applies the method to f(x) = x  - x , looking for x = 0, and starting
with x = 1/û5.  No solution is found because the estimates oscillate.  However,
if a better starting value is used, the desired root can be found.

F.  Problem - Example 4
     This applies the method to f(x) = û³x³, looking for x = 0, and starting
with x = 1.  No solution is found because the estimates oscillate.  However,
even if a different starting value is used, the desired root can never be found.

G.  Problem - Example 5
     This applies the method to f(x) = x exp(-x), looking for x = 0, and
starting with x = 2.  No solution is found because the estimates diverge from x
= 0.  However, if a different starting value is used, the desired root can be
found.

H.  Problem - Example 6
     This deals with the quartic function
                       4      3      2
               f(x) = x  - 20x  - 25x  + 500x + 1 .
It is easily shown that   f(-4) = -863   and   f(4) = 577, so there is a root in
the interval (-4, 4).  The purpose of this example is to first show the
numerical values associated with the starting choices -4, 4, -3, 3, -2, and 2,
and then give a graphical interpretation of these choices.  This should indicate
why it is always sensible to sketch the function before using Newton's method.

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