Midpoint panel error kernel and second-derivative bound

Given: Reals $a<b$ and $f\in C^2([a,b])$.

Higher derivatives and the classes $C^k$ and $C^{\mathrm{\infty }}$: $C^2([a,b])$ means that $f$, $f^{\mathrm{\prime }}$, and $f^{\mathrm{\prime }\mathrm{\prime }}$ are continuous on the closed interval, with endpoint derivatives interpreted from within the interval.

If $u,v$ are differentiable on $[a,b]$ with $u^{\mathrm{\prime }},v^{\mathrm{\prime }}$ integrable, then ${\int }_a^{b}u{v}^{\mathrm{\prime }}=u(b)v(b)-u(a)v(a)-{\int }_a^b{u}^{\mathrm{\prime }}v$: for differentiable $u,v$ with integrable derivatives, $\int u{v}^{\mathrm{\prime }}=[uv]-\int u^{\mathrm{\prime }}v$ on a closed interval.

For $a<c<b$: $f$ is integrable on $[a,b]$ if and only if it is integrable on $[a,c]$ and on $[c,b]$, and then ${\int }_a^{b}f={\int }_a^{c}f+{\int }_c^{b}f$; with the oriented form for arbitrary $a,b,c$: an integrable function's integral over two adjacent intervals is the sum of the two integrals.

Every continuous function on a closed nondegenerate rectangle in ${\mathbb{R}}^m$ is Riemann integrable: every continuous real function on a closed nondegenerate rectangle in ${\mathbb{R}}^q$, $q\ge 1$, is Riemann integrable; here $q=1$ and the rectangle is $[a,b]$.

Heine-Borel by bisection: every closed bounded interval $[a,b]$ is compact: the closed interval $[a,b]$ is compact.

Extreme value theorem: a continuous real function on a nonempty compact subset of $\mathbb{R}$ attains a greatest and a least value: a continuous real function on a nonempty compact subset of $\mathbb{R}$ attains its maximum and minimum.

If $f\le g$ on $[a,b]$ and both are integrable then ${\int }_a^{b}f\le {\int }_a^{b}g$; and $m(b-a)\le {\int }_a^{b}f\le M(b-a)$: pointwise order between integrable functions on an increasing interval gives the same order between their integrals.

Integrable functions on $[a,b]$ form a set closed under sums and scalar multiples, and ${\int }_a^b(\lambda f+\mu g)=\lambda {\int }_a^{b}f+\mu {\int }_a^{b}g$: the integral of a scalar multiple of an integrable function is the same scalar multiple of its integral.

The second fundamental theorem: if $G$ is differentiable on $[a,b]$ with $G^{\mathrm{\prime }}=f$ and $f$ is integrable, then ${\int }_a^{b}f=G(b)-G(a)$: if $G$ is differentiable on $[a,b]$, $G^{\mathrm{\prime }}=g$, and $g$ is integrable, then ${\int }_a^{b}g=G(b)-G(a)$.

Sums, scalar multiples, products, absolute values, maxima, minima and quotients with nonvanishing denominator of continuous functions are continuous, as are constants, the identity and every polynomial function: constants and the identity are continuous; sums, scalar multiples and products preserve continuity, so every polynomial function and its product with $f^{\mathrm{\prime }\mathrm{\prime }}$ are continuous on each interval where the polynomial formula for $K$ applies.

The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g(c)$, then $f\circ g$ is differentiable at $c$ with $(f\circ g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(g(c))g^{\mathrm{\prime }}(c)$: the derivative of a differentiable composition is the outer derivative at the image multiplied by the inner derivative.

For a natural $n\ge 1$ the function $x\mapsto x^n$ is differentiable everywhere with derivative $\iota (n)x^{n-1}$; for $n=0$ it is the constant $1$, with derivative $0$; for a natural $n\ge 1$ the function $x\mapsto x^{-n}$ is differentiable at every $x\mathrm{\ne }0$ with derivative $-\iota (n)x^{-n-1}$; consequently every polynomial function is differentiable at every real, with the derivative computed term by term: $(x^k{)}^{\mathrm{\prime }}=k{x}^{k-1}$ for each integer $k\ge 1$.

Sums, scalar multiples, products and quotients: $(f+g{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)+g^{\mathrm{\prime }}(c)$, $(\alpha f{)}^{\mathrm{\prime }}(c)=\alpha f^{\mathrm{\prime }}(c)$, $(fg{)}^{\mathrm{\prime }}(c)=f^{\mathrm{\prime }}(c)g(c)+f(c)g^{\mathrm{\prime }}(c)$, and $(f\mathrm{/}g{)}^{\mathrm{\prime }}(c)=(f^{\mathrm{\prime }}(c)g(c)-f(c)g^{\mathrm{\prime }}(c))\mathrm{/}g(c{)}^2$ when $g(c)\mathrm{\ne }0$: derivatives of sums and scalar multiples are the corresponding sums and scalar multiples of the derivatives.