1. Explain why the trapezoidal rule can integrate a straight line exactly and the Simpsoifs rule can integrate a cubic exactly [213 pts}. 1 2. An example ofa peak-y function is the Lorentz prolem + x' I a" _ For largea, it is well behaved but for smalls! , it is strongly pea-zed near the origin. For n = {3.1 and l= {a} plot the function om D to l (5 pts) and integrate this mction om IE! to 1 using {b} NewtonCotes methods of various orders (H=2, 3, and ti} with xed number of intervals {number of intervals = 32) (ID pts), and (c) Trapezoidal rule {H = l} with variable number of intervals (number ofintervafs = 5, I2, and 24'} (I!) pts). (d) Plot the error versus the order n for {b} and versus the number of intervals for (c) {11:} pts}