 Three-point asymmetric second-derivative formula

Matching the monomials $1,x,x^2$ gives $f^{\prime \prime }(2)\approx (2/3)f(0)-f(1)+(1/3)f(3)$. With error defined as the true derivative minus this approximation, its Peano kernel is $K(\theta )=21_{\{\theta <2\}}+(1-\theta {)}_{+}^2-(3-\theta {)}^2/3$ on $[0,3]$. The derivative evaluation causes a jump at two. Applying the Peano kernel theorem gives $\lambda (f)=\frac{1}{2}{\int }_0^{3}K{f}^{(3)}$.