Composite quadrature

1.

Use the loop on page 122 of the book to evaluate the Composite Midpoint rule for the same f(x), using 2, 4, and 8 intervals. Notice that unlike for the other Composite rules, the Midpoint rule uses n in a weird way that does not correspond to the number of subintervals. I suggest that you create a variable called intervals, and set n := 2 * (intervals-1);

2.

Write a similar loop to evaluate the Composite Trapezoid rule. Hints: The Maple interface for editing is not as wonderful as we would like, so it might be a good idea to write your loop on paper before you type it in. Also, note that the calculation of h and the x_i is different for the Trapezoid rule, so don't just look at what's in the blue box.

3.

Use your loop to evaluate the Trapezoid rule with 2, 4 and 8 intervals.

4.

Make a table that shows all eight estimates (two rules, 1, 2, 4, and 8 intervals), and for each estimate, calculate the absolute error and the number of times f(x) was evaluated to get the estimate (again, be careful about the fact that for the Midpoint rule n is not the number of subintervals. Overall, which method seems to be more efficient?

5.

Use either your loop or the book's loop to answer Question 11 on page 124. Notice that the function is symmetric around zero, if that gives you any ideas.