please answer

By-hand problem Approximate the above integral using the composite Simp- son rule with 2 intervals of equal length (take n = 4). How many evaluations of the function f [2:] are required? Why is composite Simpson O(h)? First analyze basic Simpson's rule by Taylor Series: (around a) flat h) = f thf'+zhif"+zh + In3( " + 4 ! + Insf (5 ) + ... f (a + 2h) = f + 2hf' + 2h-f"+ = 4 15 f ( 5 ) + ... 3" 15 This gives 3 If ( a ) + 4f ( a + h ) + f (b ) ] = 2hf + 2h?f' + = hof"+z 5 15 f ( 4) 3" 18 J. B. Schroder (UNM) Math/CS 375 27/31 Why is composite Simpson O(h+)? Consider the antiderivative of f (x): F ( x ) = [f ( x ) dx Taylor series of F: F(a + 2h) = F(a) + 2hF' (a) + 212F" (a) + 2h3F"" (a) + zhaF(4) + # 15F (5) +... Noting that F(a + 2h) = =a+2h It2" f(x) dx, F(a) = 0, F' = f, F" = f' and so on, b=a+2h f(x) dx = 2hf + 2h f' + 213f"+ =haf" + # hof(4)+... Comparing this equation with the one on previous slide, basic Simpson's Rule gives an error of 90 2 "f(I ( E ) J. B. Schroder (UNM) Math/CS 375 28/31\f