Consider the following piece-wise polynomial interpolation. The domain [a; b] is divided into ` subintervals [ai1; ai] with ai = a + i(b a)=` for i = 0; 1; : : : ; `. On each subinterval [ai1; ai], the function f : [a; b] ! R in interpolated by a polynomial pi at points ai1, ci = (ai1 + ai)=2, and ai, each of multiplicity 1. Let p be the corresponding piece-wise polynomial.

PROBLEM 1 (5 pts): Consider the following piece-wise polynomial interpolation. The domain (a, b) is divided into l subintervals (ai1, ai] with a; = a + i(b a)/l for i = 0,1,..., l. On each subinterval (ai1, ai], the function f : [a, b] R in interpolated by a polynomial pi at points Qi-1, C; = (ai1 +a;)/2, and ai, each of multiplicity 1. Let p be the corresponding piece-wise polynomial. (i): Derive a formula for the upper bound on the error max |f(x) p(x). a