- Difference quotient
The primary vehicle of calculus and other higher mathematics is the function. Its "input value" is its argument, usually a point ("P") expressible on a graph. The difference between two points, themselves, is known as their Delta (ΔP), as is the difference in their function result, the particular notation being determined by the direction of formation:
- Forward difference: ΔF(P) = F(P + ΔP) - F(P);
- Central difference: δF(P) = F(P + ½ΔP) - F(P - ½ΔP);
- Backward difference: ∇F(P) = F(P) - F(P - ΔP).
The general preference is the forward orientation, as F(P) is the base, to which differences (i.e., "ΔP"s) are added to it. Furthermore,
- If |ΔP| is finite (meaning measurable), then ΔF(P) is known as a finite difference, with specific denotations of DP and DF(P);
- If |ΔP| is infinitesimal (an infinitely small amount—ι—usually expressed in standard analysis as a limit: ), then ΔF(P) is known as an infinitesimal difference, with specific denotations of dP and dF(P) (in calculus graphing, the point is almost exclusively identified as "x" and F(x) as "y").
Defining the point range
Regardless if ΔP is infinitesimal or finite, there is (at least—in the case of the derivative—theoretically) a point range, where the boundaries are P ± (.5)ΔP (depending on the orientation—ΔF(P), δF(P) or ∇F(P)):
- LB = Lower Boundary; UB = Upper Boundary;
Derivatives can be regarded as functions themselves, harboring their own derivatives. Thus each function is home to sequential degrees ("higher orders") of derivation, or differentiation. This property can be generalized to all difference quotients.
As this sequencing requires a corresponding boundary splintering, it is practical to break up the point range into smaller, equi-sized sections, with each section being marked by an intermediary point ("Pi"), where LB = P0 and UB = Pń, the nth point, equaling the degree/order:
LB = P0 = P0 + 0Δ1P = Pń - (Ń-0)Δ1P; P1 = P0 + 1Δ1P = Pń - (Ń-1)Δ1P; P2 = P0 + 2Δ1P = Pń - (Ń-2)Δ1P; P3 = P0 + 3Δ1P = Pń - (Ń-3)Δ1P; ↓ ↓ ↓ ↓ Pń-3 = P0 + (Ń-3)Δ1P = Pń - 3Δ1P; Pń-2 = P0 + (Ń-2)Δ1P = Pń - 2Δ1P; Pń-1 = P0 + (Ń-1)Δ1P = Pń - 1Δ1P; UB = Pń-0 = P0 + (Ń-0)Δ1P = Pń - 0Δ1P = Pń;
ΔP = Δ1P = P1 - P0 = P2 - P1 = P3 - P2 = ... = Pń - Pń-1;
ΔB = UB - LB = Pń - P0 = ΔńP = ŃΔ1P.
The primary difference quotient (Ń = 1)
As a derivative
- The difference quotient as a derivative needs no explanation, other than to point out that, since P0 essentially equals P1 = P2... = Pń (as the differences are infinitesimal), the Leibniz notation and derivative expressions do not distinguish P to P0 or Pń:
There are other derivative notations, but these are the most recognized, standard designations.
As a divided difference
- A divided difference, however, does require further elucidation, as it equals the average derivative between and including LB and UB:
- In this interpretation, Pã represents a function extracted, average value of P (midrange, but usually not exactly midpoint), the particular valuation depending on the function averaging it is extracted from. More formally, Pã is found in the mean value theorem of calculus, which says:
- For any function that is continuous on [LB,UB] and differentiable on (LB,UB) there exists some Pã in the interval (LB,UB) such that the secant joining the endpoints of the interval [LB,UB] is parallel to the tangent at Pã.
- Essentially, Pã denotes some value of P between LB and UB—hence,
- which links the mean value result with the divided difference:
- As there is, by its very definition, a tangible difference between LB/P0 and UB/Pń, the Leibniz and derivative expressions do require divarication of the function argument.
Higher order difference quotients
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):
- Saint Vincent College: Br. David Carlson, O.S.B.—MA109 The Difference Quotient
- University of Birmingham: Dirk Hermans—Divided Differences
Wikimedia Foundation. 2010.
Look at other dictionaries:
Quotient rule — In calculus, the quotient rule is a method of finding the derivative of a function that is the quotient of two other functions for which derivatives exist. If the function one wishes to differentiate, f(x), can be written as :f(x) =… … Wikipedia
Quotient Intellectuel — « QI » redirige ici. Pour les autres significations, voir QI (homonymie) … Wikipédia en Français
QUOTIENT INTELLECTUEL — (Q.I.) Indice exprimant les capacités intellectuelles d’un individu. Il existe deux procédures de détermination de cet indice, lesquelles sont d’inspiration très différente l’une de l’autre. L’une, introduite par Stern puis par Terman, fait… … Encyclopédie Universelle
Quotient conjugal — Foyer fiscal La notion de foyer fiscal est utilisée dans certains pays comme la France pour le calcul de l impôt sur le revenu des personnes physiques. Il consiste à imposer non pas les individus isolément mais les ménages qu ils composent.… … Wikipédia en Français
Quotient familial — Foyer fiscal La notion de foyer fiscal est utilisée dans certains pays comme la France pour le calcul de l impôt sur le revenu des personnes physiques. Il consiste à imposer non pas les individus isolément mais les ménages qu ils composent.… … Wikipédia en Français
Quotient intellectuel — « QI » redirige ici. Pour les autres significations, voir QI (homonymie). Le quotient intellectuel ou QI, est le résultat d un test psychométrique qui, lorsqu il est corrélé avec les autres éléments d un … Wikipédia en Français
Quotient space (linear algebra) — In linear algebra, the quotient of a vector space V by a subspace N is a vector space obtained by collapsing N to zero. The space obtained is called a quotient space and is denoted V / N (read V mod N ). Definition Formally, the construction is… … Wikipedia
Difference operator — In mathematics, a difference operator maps a function, f ( x ), to another function, f ( x + a ) − f ( x + b ).The forward difference operator :Delta f(x)=f(x+1) f(x),occurs frequently in the calculus of finite differences, where it plays a role… … Wikipedia
Quotient — The result of mathematical division. The I.Q. (Intelligence Quotient) is arrived at by dividing the person s mental age (as determined on the Binet test) by the person s chronologic age and multiplying by 100. So if a child scores at the 8 year… … Medical dictionary
difference ring — Math. See quotient ring. * * * … Universalium