N Suppose for the integral T|[a,b] () = f f(x)dx, we approximate it with the midpoint...

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N Suppose for the integral T|[a,b] () = f f(x)dx, we approximate it with the midpoint method, which for mesh {x;} where xj = a + , = b-a N j=0 is given by N-1 1 A[a,b] (f) = 5x f (xj+1/2), where xj+1/2 = (x j + x j+1) = x; + 2 = Xj+1 2 j=0 a)Prove using Taylor-series based methods that the local error is O(dx). Hint, you'll need to introduce T (x; xj+1/2) where T(x; xj+1/2) = f(xj+1/2) + f'(xj+1/2)(x xj+1/2). And then of course you'll need the Taylor Remainder Theorem. b)Prove that the global error is O(5x) c)Choose a test function and provide numerical experiments which confirm that the theoretical global error is O(5x). N Suppose for the integral T|[a,b] () = f f(x)dx, we approximate it with the midpoint method, which for mesh {x;} where xj = a + , = b-a N j=0 is given by N-1 1 A[a,b] (f) = 5x f (xj+1/2), where xj+1/2 = (x j + x j+1) = x; + 2 = Xj+1 2 j=0 a)Prove using Taylor-series based methods that the local error is O(dx). Hint, you'll need to introduce T (x; xj+1/2) where T(x; xj+1/2) = f(xj+1/2) + f'(xj+1/2)(x xj+1/2). And then of course you'll need the Taylor Remainder Theorem. b)Prove that the global error is O(5x) c)Choose a test function and provide numerical experiments which confirm that the theoretical global error is O(5x).