[2] [Derivation of Simpon's Rule] In this exercise will derive Simpson's rule using Taylor's theorem. Suppose ƒ € C¹¹ ([a, b]) and consider three equispaced points {xo, x₁, x2} with xo = a, x₁ = a + h and £₂ b, where h = = (b-a)/2. (a) Use Taylor's theorem to write f(x) as a third order polynomial plus a remainder term using the point of expansion ₁. (b) Insert the expression for f(x) from (a) into the expression ² f(x)dx and explicitly compute the integral of each term in the Taylor polynomial to derive Cx2 x2 [*"* ƒ (x) dx = 2hƒ(x1) + ²/²_ƒ″(x1) + 12/17 [** ƒ(¹) ({(x)) (x − x1) ª da 3 24 XO (hint: two of the integrals will vanish identically because of parity, i.e. be- cause they are antisymmetric functions) (c) Justify that the Weighted Mean Value Theorem can be used for the remainder term. Then, apply the theorem to obtain: x2 1 27/4 * * ƒ (¹) (§ (x)) (x − x₁) ªdx - хо f(4) (c) h5 60