SOLVED:An approximation rule similar to the Trapezoid Rule is the Midpoint Rule. This rule approximates the integral ∫c^d f(x) d x by (d-c) f([d+c] / 2) . Prove that if the function f:[c, d] →ℝ is continuous and its restriction f:(c, d) →ℝ has a second derivative, then for some point ζin (c, d) ∫c^d f(x) d x=(d-c) f((c+d)/(2))+(1)/(24)(d-c)^3 f^''(ζ) [Hint: Let m=(c+d) / 2, let t0=(d-c) / 2 and define H(t)=[∫m-t^m+t f]-2 t f(m)-((t)/(t0))^3 ∫m-t0^m+t0 f-2 (t^3)/(t0^2)(f) m for -t0 ≤t ≤t0 Appl
SOLVED:An approximation rule similar to the Trapezoid Rule is th…
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