# Assignment 9

You will write a *function* script called `chebexpansion`
which computes a Chebyshev polynomial approximation to any
function we specify. Specifically your function will satisfy the
following criteria.

- Your function will require three arguments: a handle to the
function to be approximated; the degree of the polynomial approximate;
and a vector of points where the polynomial is to be evaluated. Thus,
to run your function we will be able to type something like:
`chebexpansion(@sin,9,[-1,0,1])`. - Your function will have one output: a vector of values of
the polynomial at the points given in the input
`x`. Thus, in the previous example, the output would be a vector`[p(-1),p(0),p(1)]`, where`p`is the Chebyshev polynomial approximation you calculated. - Your function can call a function
provided here to calculate the
coefficients of the expansion in Chebyshev polynomials.
In other words, your polynomial will be
\[\sum_{k=0}^K a_kT_k(x),\]
where \(K\) is the number you gave as a second argument to
your function, \(x\) is essentially the vector you gave as
the third argument, and \(a_k\) is calculated by the
little function we provide. Note that the Chebyshev polynomials
of the first kind \(T_k\) are computed in Matlab
using the built-in function
`chebyshevT(d,x)`where`d`is the degree of the Chebyshev basis polynomial, and`x`is the vector of x coordinates (same as above).

Once your function is working, you will use it to plot Chebyshev polynomial approximations of degree 12 to the three functions you created in Assignment 8. The plots will look something like that in Figure 1. Observe the following things about the plot.

Figure 1: Chebyshev approximation to the Cooper function.

- It has a title. The title tells the degree of the polynomial
- The axes are labeled.
- The curves are different colors. It does not matter what colors you choose.
- The plot has a legend.

You will turn in two things for this assignment: the function script
file called `chebexpansion.m`; and a script in
which you call that function in the course of plotting
those three degree-twelve approximations.
The assignment is turned in when the two .m files containing the
scripts are received as attachments to an email message in
the instructor's inbox.
The assignment is worth 40 points, and
is due at 9:00 on Tuesday, 7 November.

A solution for the
final is available.