Difference Quotient - Applying The Divided Difference

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:

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 or as boundaries, with the same divided difference found as that with boundaries of 0 and (thus requiring less averaging effort):

This also becomes particularly useful when dealing with iterated and multiple integrals (ΔA = AU - AL, ΔB = BU - BL, ΔC = CU - CL):

=\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},\,\!
= F'(C\!L\!<\!R\!<\!CU:BL\!<\!Q\!<\!BU:AL\!<\!P\!<\!AU) \Delta A\,\Delta B\,\Delta C .\,\!

Hence,

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

and

Read more about this topic:  Difference Quotient

Famous quotes containing the words applying the, applying, divided and/or difference:

    “There are always those who are willing to surrender local self-government and turn over their affairs to some national authority in exchange for a payment of money out of the Federal Treasury. Whenever they find some abuse needs correction in their neighborhood, instead of applying the remedy themselves they seek to have a tribunal sent on from Washington to discharge their duties for them, regardless of the fact that in accepting such supervision they are bartering away their freedom.”
    —Calvin Coolidge (1872–1933)

    “The people shall not be restrained from peacefully assembling and consulting for their common good, nor from applying to the legislature by petitions, or remonstrances for redress of their grievances.”
    —James Madison (1751–1836)

    “The old parties are husks, with no real soul within either, divided on artificial lines, boss-ridden and privilege-controlled, each a jumble of incongruous elements, and neither daring to speak out wisely and fearlessly on what should be said on the vital issues of the day.”
    —Theodore Roosevelt (1858–1919)

    “Consider the difference between looking and staring. A look is voluntary; it is also mobile, rising and falling in intensity as its foci of interest are taken up and then exhausted. A stare has, essentially, the character of a compulsion; it is steady, unmodulated, “fixed.””
    —Susan Sontag (b. 1933)