Taylor's theorem for functions $f \in C^{k+1}[a, b]$ is given in the form

$$f(x)=f(a)+(x-a) f^{\prime}(a)+\cdots+\frac{(x-a)^{k}}{k !} f^{(k)}(a)+R(x) .$$

Use integration by parts to show that

$$R(x)=\frac{1}{k !} \int_{a}^{x}(x-\theta)^{k} f^{(k+1)}(\theta) d \theta$$

Let $\lambda_{k}$ be a linear functional on $C^{k+1}[a, b]$ such that $\lambda_{k}[p]=0$ for $p \in \mathbb{P}_{k}$. Show that

$$\lambda_{k}[f]=\frac{1}{k !} \int_{a}^{b} K(\theta) f^{(k+1)}(\theta) d \theta$$

where the Peano kernel function $K(\theta)=\lambda_{k}\left[(x-\theta)_{+}^{k}\right] . \quad[$ You may assume that the functional commutes with integration over a fixed interval.]

The error in the mid-point rule for numerical quadrature on $[0,1]$ is given by

$$e[f]=\int_{0}^{1} f(x) d x-f\left(\frac{1}{2}\right)$$

Show that $e[p]=0$ if $p$ is a linear polynomial. Find the Peano kernel function corresponding to $e$ explicitly and verify the formula ( $\dagger$ ) in the case $f(x)=x^{2}$.