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How?
Given a pair of numbers, (x0, f(x0))
the linear approximation allows us to compute a better guess
than f(x0) for f(x); call that linear approximation
at x0 by the name f(x1). If we know
how to compute the inverse function to f, that means given
f(x1) we know how to compute x1, and
thus find a new and better pair (x1, f(x1)).
We can then “iterate” this procedure, producing x2,
x3, …, until we reach x, and are done.
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 seconds for all the inverse functions we encounter: which
are roots (x1/j), the (natural) logarithm, arcsin
and arctan.
Exercises:
12.1 Set up a general root finding spreadsheet so that
you can input x and j and it will spit out the jth root of
x by using this method where the machine only computes integer
powers.
12.2 Set up a spreadsheet to find ln x using the ability
of the machine to compute 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.
12.4 Can this method fail? If so how?
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