Editing Difference quotient (section)

==Applying the divided difference==
The quintessential application of the divided difference is in the presentation of the definite integral, which is nothing more than a finite difference:

: <math>
\begin{align}
\int_{LB}^{UB} G(p) \, dp & = \int_{LB}^{UB} F'(p) \, dp=F(UB)-F(LB), \\[10pt]
& =F[LB,UB]\Delta B, \\[10pt]
& =F'(LB < P < UB)\Delta B, \\[10pt]
& =\ G(LB < P < UB)\Delta B.
\end{align}
</math>

Given that the mean value, derivative expression form provides all of the same information as the classical integral notation, the mean value form may be the preferable expression, such as in writing venues that only support/accept standard [[ASCII]] text, or in cases that only require the average derivative (such as when finding the average radius in an elliptic integral).
This is especially true for definite integrals that technically have (e.g.) 0 and either <math>\pi\,\!</math> or <math>2\pi\,\!</math> as boundaries, with the same divided difference found as that with boundaries of 0 and <math>\begin{matrix}\frac{\pi}{2}\end{matrix}</math> (thus requiring less averaging effort):

: <math>
\begin{align}
\int_0^{2\pi} F'(p) \, dp & =4\int_0^{\frac{\pi}{2}} F'(p)\, dp=F(2\pi)-F(0)=4(F(\begin{matrix}\frac{\pi}{2}\end{matrix})-F(0)), \\[10pt]
& =2\pi F[0,2\pi]=2\pi F'(0 < P < 2\pi), \\[10pt]
& =2\pi F[0,\begin{matrix}\frac{\pi}{2}\end{matrix}] =2\pi F'(0 < P < \begin{matrix}\frac{\pi}{2}\end{matrix}).
\end{align}
</math>

This also becomes particularly useful when dealing with ''iterated'' and [[multiple integral|''multiple integral''s]] (ΔA = AU − AL, ΔB = BU − BL, ΔC = CU − CL):

: <math>
\begin{align}
& {} \qquad \int_{CL}^{CU}\int_{BL}^{BU} \int_{AL}^{AU} F'(r,q,p)\,dp\,dq\,dr \\[10pt]
& =\sum_{T\!C=1}^{U\!C=\infty}\left(\sum_{T\!B=1}^{U\!B=\infty}
\left(\sum_{T\!A=1}^{U\!A=\infty}F^{'}(R_{(tc)}:Q_{(tb)}:P_{(ta)})\frac{\Delta A}{U\!A}\right)\frac{\Delta B}{U\!B}\right)\frac{\Delta C}{U\!C}, \\[10pt]
& = F'(C\!L < R < CU:BL < Q < BU:AL < P <\!AU)
\Delta A\,\Delta B\,\Delta C.
\end{align}
</math>

Hence,

: <math>F'(R,Q:AL < P < AU)=\sum_{T\!A=1}^{U\!A=\infty}
\frac{F'(R,Q:P_{(ta)})}{U\!A};\,\!</math>

and
:<math>F'(R:BL < Q < BU:AL < P < AU)=\sum_{T\!B=1}^{U\!B=\infty}\left(\sum_{T\!A=1}^{U\!A=\infty}\frac{F'(R:Q_{(tb)}:P_{(ta)})}{U\!A}\right)\frac{1}{U\!B}.\,\!</math>