The approximation rule alike to the Trapezoid Rule is the Midpoint Rule. The rule gives the...

 

Transcribed Image Text:

The approximation rule alike to the Trapezoid Rule is the Midpoint Rule. The rule gives the approximation for the integral ſå f(t)dt by (b − a)ƒ([b+a]/2). Present a rigorous proof assuming the function f : [a, b] → R is continuous and that its restriction ƒ : (a, b) → R has a 2nd derivative, then for a point & € (a,b), + - (b − a)³ƒ" (p) 1 24 = (b − a)/2, n = (a+b)/2, & let G(t) = [fm+z ƒ] − 2xƒ(n) — ( 2 )³ fn+% ƒ − 25 (ƒ)n for −xo ≤ x ≤ xo, then use - Tip: Let x = [*ƒ(t)dt = (b − a)ƒ(ª+b) 2 Generalized Rolle's Theorem having m = 1 to the function G : [-o, To] → R The approximation rule alike to the Trapezoid Rule is the Midpoint Rule. The rule gives the approximation for the integral ſå f(t)dt by (b − a)ƒ([b+a]/2). Present a rigorous proof assuming the function f : [a, b] → R is continuous and that its restriction ƒ : (a, b) → R has a 2nd derivative, then for a point & € (a,b), + - (b − a)³ƒ" (p) 1 24 = (b − a)/2, n = (a+b)/2, & let G(t) = [fm+z ƒ] − 2xƒ(n) — ( 2 )³ fn+% ƒ − 25 (ƒ)n for −xo ≤ x ≤ xo, then use - Tip: Let x = [*ƒ(t)dt = (b − a)ƒ(ª+b) 2 Generalized Rolle's Theorem having m = 1 to the function G : [-o, To] → R

Answer rating: 100% (QA)

leef a b IR che a continuous function on cb has such that its restriction Besond ... View the full answer