How do we know that the inverse tangent function arctan x
is not a polynomial? State at least one property of this function
that could not be a property of any polynomial.
We have seen that the geometric polynomials
approximate the function 1/(1 - x)
for |x| < 1. Substitute x = - t2
into these polynomials to find approximating polynomials for the function
1/(1 + t2). What is the interval
of convergence for these new polynomials? Explain how you know.
Integrate the typical polynomial of degree 2n (from 0 to x)
to find a polynomial of degree 2n + 1 that should approximate
arctan x. Why should it approximate arctan x?
Use your computer algebra system to confirm that the new polynomials are in
fact Taylor polynomials for arctan x.
For each of the Taylor polynomials p1(x), p3(x),
and p5(x),
- plot pn(x) and arctan x
together;
- plot the error function, arctan x - pn(x);
- describe the extent to which pn(x) does
and does not approximate arctan x.
You may need to experiment with the ranges of the error plots to get the most
information out of each one.
Repeat step 4 for n = 10, 20,
and 40, using your computer algebra system to generate the polynomials.
What do you think is the interval of convergence of the Taylor polynomials
for arctan x ? Explain.
