Aproximarea functionalelor liniare

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Integrând avem
xk+1
unde
iar
Deci
xk
¸si xk+1
f(x)dx = h
2 (fk + fk+1) + R1(f),
R1(f) =
xk+1
xk
K1(t)f ′′ (t)dt,
K1(t)= (xk+1 − t) 2
−
2
h
2 [(xk − t)+ + (xk+1 − t)+]
= (xk+1 − t) 2
−
2
h(xk+1 − t)
2
= 1
2 (xk+1 − t)(xk − t) ≤ 0.
xk
R1(f) = − h3
12 f ′′ (ξk), ξk ∈ (xk, xk+1)
f(x)dx = h
2 (fk + fk+1) − 1
12 h3 f ′′ (ξk). (20)