Solution 1.

1 (a).  Using the nodes  .

Now plot the function    and polynomial  .

1 (b).  Using the nodes  .  

Now plot the function    and polynomial  .

1 (c).  Using the nodes  .

Now plot the function    and polynomial  .

Notice that the three polynomials of degree n = 1 were different.

Compare the various graphs.

Notice that the three polynomials of degree n = 1 are different.  The error in approximating  f[x]  will also be different.

Which polynomial has the smallest overall error on the entire interval  .  Later we will discover that    was based on "Chebyshev's nodes."

(c) John H. Mathews 2004