12.2 Determining an Inverse Function Accurately by Iterating the Linear Approximation

How?

Given a pair of numbers,
$({x}_{0},f({x}_{0}))$
the linear approximation,
$fL{x}_{0}$
to
$f$
defined at
${x}_{0}$
, allows us to compute
$fL{x}_{0}(x)$
as an approximation to
$f(x)$
.

If we know the inverse function to
$f$
, we can compute
${f}^{-1}(fL{x}_{0}(x))$
and that gives us a new pair of numbers,
$({f}^{-1}(fL{x}_{0}(x)),fL{x}_{0}(x))$
that we can call
$({x}_{1},f({x}_{1}))$
and repeat (or iterate) this operation to produce
${x}_{2}$
, then
${x}_{3},\dots ,$
until it converges.

This in the old days was so horribly boring a procedure that it could not be inflicted on students. Now it is duck soup for a spreadsheet, and can be set up and computed in a matter of minutes for all the inverse functions we encounter: which are the roots
$({x}^{1/j})$
, the (natural) logarithm, arcsin and arctan.

What has to be done?

The linear approximation
$fL{x}_{j}$
defined at
${x}_{j}$
evaluated at
$x$
is given by

12.1 Set up a general root finding spreadsheet so that you can input
$x$ and
$j$ and it will spit out the j-th root of
$x$ by using this method where the machine only computes integer powers. (Clues on how to do this are in the next section.)

12.2 Set up a spreadsheet to find
$\mathrm{ln}x$ using the ability of the machine to compute$\mathrm{exp}x$
.

12.3 Do the same for the inverse functions to the sine and tangent. These
are generally written as something like arcsine and arctangent or asin or atan
or something in between.