In Problem 3, show that Simpson's rule can be obtained by approximating the integrand with quadratic functions. A. Let a <b and let m be the midpoint of [a, b]. That is, let m = h = ba. Show that if f is a quadratic function, then (a + b)/2. Let h fo f (x)dx = 1½ (f(a) + 2f (m) + f(b)). 3 2 Hint: Show that this equation holds for fo (x) = 1, f₁(x) : = x, and f₂(x) = x², then use properties of the integral to show that the equation holds for any quadratic function f(x) = Ax² + Bx + C. Consider the following method for approximating for f (x)dx: divide the interval [a, b] into n equal subintervals. On each subinterval, approximate ƒ by a quadratic function that agrees with fat both endpoints and at the midpoint of the subinterval. B. Explain why the integral off on the subinterval [xi, xi+1] is approximately equal to h 3 2 1½ (F(x) + 2 ƒ (m²) + f (x + 1)), 2 C. rule: Where mĘ is the midpoint of the subinterval, m₁ = (xi + Xi+1)/2. Show that if we add up these approximations for each subinterval, we get Simpson's Så f(x)dx 2.mid(n)+trap(n) = 3