### Section 7.2

Problems 1c, 5bc, 6bc, 7, 10, 12, 15-17

Written problem 1a: Prove the following lemma:

Lemma: Assume p is the quadratic polynomial that interpolates the points (x0,f(x0)), (x1,f(x1)), and (x2,f(x2)). Assume f has three continous derivatives. Then for each x in [x0,x2] there is a c_x in [x0,x2] such that
f(x)-p(x) = (x-x0)(x-x1)(x-x2) f'''(c_x)/6

Written problem 1b: Recall Simpson's rule:
\int_x0^x2 f(x) - S_1(f) = \int_x0^x2 f(x) - \int_x0^x2 p(x)
= \int_x0^x2 (x-x0)(x-x1)(x-x2) f'''(c_x)/6

Assume the point c_x is approximately fixed, so you can pull f'''(c_x) out of the integral. What do you end up with for the error? How does this differ from the error for the trapezoidal rule?