Problem 6: Use the definition of precision directly to determine the coefficients and test overall degree of precision.
Problem 14: Note this should be essentially the same as problem 6, so here you do the theoretical work. To derive the error term, break it into three parts. The integral of the error term from x0 to x2 is exactly the same as Simpson's rule. You can use the weighted mean value theorem directly on the outer two integrals. However, remember that getting the third degree of precision from an n=2 method relies on symmetry. So, you'll have to combine these two integrals together. This should let you combine the two fourth order divided differences into a single fifth order divided difference.
Problem 7: Verify convergence without using the exact value of the integral.