This problem is our first introduction to Newton's Method for
the solution of nonlinear equations.
The positive solution of is obviously How might one calculate a numerical value of ?
The idea of Newton's method is to pick a guess , compute the
tangent to the graph of at , and then
replace the guess with which is the intercept of
the tangent. (You should draw a picture of this idea.)
Suppose . The tangent to the graph of at the point
can be written in slope intercept form as
.
The tangent intercepts the -axis at
.
Now let's repeat the process.
The tangent to the graph of at the point
can be written in slope intercept form as
.
The tangent intercepts the -axis at
.
Note the accuracy of this approximation:
.
Note: To obtain the same result in your last answer as ww you need to
compute all intermediate answers to the full accuracy of your
calculator. To accomplish this store all intermediate results in your
calculator. Do not copy them on paper and then reenter them by hand
later. Doing so introduces rounding errors and compromises the
accuracy of your calculations and solutions. In fact, you should make
a habit of this every time you use your calculator: store intermediate
results and avoid having to reenter them. Most calculators keep more
digits internally than they can display, so losing accuracy by copying
and reentering is inevitable.