Difference quotient
Encyclopedia
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:
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,
The function difference divided by the point difference is known as the difference quotient (attributed to Isaac Newton
, it is also known as Newton's quotient):
If ΔP is infinitesimal, then the difference quotient is a derivative
, otherwise it is a divided difference
:
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.
There are other derivative notations, but these are the most recognized, standard designations.
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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):
Calculus
Calculus is a branch of mathematics focused on limits, functions, derivatives, integrals, and infinite series. This subject constitutes a major part of modern mathematics education. It has two major branches, differential calculus and integral calculus, which are related by the fundamental theorem...
and other higher mathematics is the function
Function (mathematics)
In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can...
. 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
Delta (letter)
Delta is the fourth letter of the Greek alphabet. In the system of Greek numerals it has a value of 4. It was derived from the Phoenician letter Dalet...
(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 infinitesimalInfinitesimalInfinitesimals have been used to express the idea of objects so small that there is no way to see them or to measure them. The word infinitesimal comes from a 17th century Modern Latin coinage infinitesimus, which originally referred to the "infinite-th" item in a series.In common speech, an...
(an infinitely small amount—
—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").
The function difference divided by the point difference is known as the difference quotient (attributed to Isaac Newton
Isaac Newton
Sir Isaac Newton PRS was an English physicist, mathematician, astronomer, natural philosopher, alchemist, and theologian, who has been "considered by many to be the greatest and most influential scientist who ever lived."...
, it is also known as Newton's quotient):
If ΔP is infinitesimal, then the difference quotient is a derivative
Derivative
In calculus, a branch of mathematics, the derivative is a measure of how a function changes as its input changes. Loosely speaking, a derivative can be thought of as how much one quantity is changing in response to changes in some other quantity; for example, the derivative of the position of a...
, otherwise it is a divided difference
Divided differences
In mathematics divided differences is a recursive division process.The method can be used to calculate the coefficients in the interpolation polynomial in the Newton form.-Definition:Given n data points,\ldots,...
:
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 notationLeibniz notationIn calculus, Leibniz's notation, named in honor of the 17th-century German philosopher and mathematician Gottfried Wilhelm Leibniz, uses the symbols dx and dy to represent "infinitely small" increments of x and y, just as Δx and Δy represent finite increments of x and y...
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 theoremMean value theoremIn calculus, the mean value theorem states, roughly, that given an arc of a differentiable curve, there is at least one point on that arc at which the derivative of the curve is equal to the "average" derivative of the arc. Briefly, a suitable infinitesimal element of the arc is parallel to the...
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:
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- 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.
Second order
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Third order
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Ńth Order
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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: |
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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
ASCII
The American Standard Code for Information Interchange is a character-encoding scheme based on the ordering of the English alphabet. ASCII codes represent text in computers, communications equipment, and other devices that use text...
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):
 |
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This also becomes particularly useful when dealing with iterated and multiple integrals
Multiple integral
The multiple integral is a type of definite integral extended to functions of more than one real variable, for example, ƒ or ƒ...
(ΔA = AU - AL, ΔB = BU - BL, ΔC = CU - CL):
