In single-variable
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
, the difference quotient is usually the name for the expression
:
which when taken to the
limit
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
as ''h'' approaches 0 gives the
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
of the
function ''f''.
The name of the expression stems from the fact that it is the
quotient
In arithmetic, a quotient (from lat, quotiens 'how many times', pronounced ) is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics, and is commonly referred to as the integer part of a ...
of the
difference of values of the function by the difference of the corresponding values of its argument (the latter is (''x'' + ''h'') - ''x'' = ''h'' in this case).
The difference quotient is a measure of the
average
In ordinary language, an average is a single number taken as representative of a list of numbers, usually the sum of the numbers divided by how many numbers are in the list (the arithmetic mean). For example, the average of the numbers 2, 3, 4, 7 ...
rate of change of the function over an
interval (in this case, an interval of length ''h'').
The limit of the difference quotient (i.e., the derivative) is thus the
instantaneous rate of change.
By a slight change in notation (and viewpoint), for an interval
'a'', ''b'' the difference quotient
:
is called
the mean (or average) value of the derivative of ''f'' over the interval
'a'', ''b'' This name is justified by the
mean value theorem
In mathematics, the mean value theorem (or Lagrange theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It i ...
, which states that for a
differentiable function
In mathematics, a differentiable function of one real variable is a function whose derivative exists at each point in its domain. In other words, the graph of a differentiable function has a non- vertical tangent line at each interior point in ...
''f'', its derivative ''f′'' reaches its
mean value at some point in the interval.
Geometrically, this difference quotient measures the
slope
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
of the
secant line
Secant is a term in mathematics derived from the Latin ''secare'' ("to cut"). It may refer to:
* a secant line, in geometry
* the secant variety, in algebraic geometry
* secant (trigonometry) (Latin: secans), the multiplicative inverse (or recipr ...
passing through the points with coordinates (''a'', ''f''(''a'')) and (''b'', ''f''(''b'')).
Difference quotients are used as approximations in
numerical differentiation,
but they have also been subject of criticism in this application.
Difference quotients may also find relevance in applications involving
Time discretization, where the width of the time step is used for the value of h.
The difference quotient is sometimes also called the Newton quotient
(after
Isaac Newton
Sir Isaac Newton (25 December 1642 – 20 March 1726/27) was an English mathematician, physicist, astronomer, alchemist, Theology, theologian, and author (described in his time as a "natural philosophy, natural philosopher"), widely ...
) or Fermat's difference quotient (after
Pierre de Fermat
Pierre de Fermat (; between 31 October and 6 December 1607 – 12 January 1665) was a French mathematician who is given credit for early developments that led to infinitesimal calculus, including his technique of adequality. In particular, he ...
).
[Donald C. Benson, ''A Smoother Pebble: Mathematical Explorations'', Oxford University Press, 2003, p. 176.]
Overview
The typical notion of the difference quotient discussed above is a particular case of a more general concept. The primary vehicle of
calculus
Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape, and algebra is the study of generalizati ...
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 (Δ''P''), as is the difference in their function result, the particular notation being determined by the direction of formation:
*Forward difference: Δ''F''(''P'') = ''F''(''P'' + Δ''P'') − ''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
A finite difference is a mathematical expression of the form . If a finite difference is divided by , one gets a difference quotient. The approximation of derivatives by finite differences plays a central role in finite difference methods for t ...
, with specific denotations of DP and DF(P);
*If , ΔP, is ''
infinitesimal
In mathematics, an infinitesimal number is a quantity that is closer to zero than any standard real number, but that is not zero. The word ''infinitesimal'' comes from a 17th-century Modern Latin coinage ''infinitesimus'', which originally re ...
'' (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 "difference quotient":
:
If ΔP is infinitesimal, then the difference quotient is a ''
derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
'', otherwise it is a ''
divided difference
In mathematics, divided differences is an algorithm, historically used for computing tables of logarithms and trigonometric functions. Charles Babbage's difference engine, an early mechanical calculator, was designed to use this algorithm in ...
'':
:
:
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 ± (0.5) 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 (''P''
''i''), where LB = ''P''
0 and UB = ''P''
''ń'', the ''n''th point, equaling the degree/order:
LB = P
0 = P
0 + 0Δ
1P = P
ń − (Ń-0)Δ
1P;
P
1 = P
0 + 1Δ
1P = P
ń − (Ń-1)Δ
1P;
P
2 = P
0 + 2Δ
1P = P
ń − (Ń-2)Δ
1P;
P
3 = P
0 + 3Δ
1P = P
ń − (Ń-3)Δ
1P;
↓ ↓ ↓ ↓
P
ń-3 = P
0 + (Ń-3)Δ
1P = P
ń − 3Δ
1P;
P
ń-2 = P
0 + (Ń-2)Δ
1P = P
ń − 2Δ
1P;
P
ń-1 = P
0 + (Ń-1)Δ
1P = P
ń − 1Δ
1P;
UB = P
ń-0 = P
0 + (Ń-0)Δ
1P = P
ń − 0Δ
1P = P
ń;
ΔP = Δ
1P = P
1 − P
0 = P
2 − P
1 = P
3 − P
2 = ... = P
ń − P
ń-1;
ΔB = UB − LB = P
ń − P
0 = Δ
ńP = ŃΔ
1P.
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 P
0 essentially equals P
1 = P
2 = ... = P
ń (as the differences are infinitesimal), the
Leibniz notation and derivative expressions do not distinguish P to P
0 or P
ń:
:::
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, P
ã 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, P
ã is found in the
mean value theorem
In mathematics, the mean value theorem (or Lagrange theorem) states, roughly, that for a given planar arc between two endpoints, there is at least one point at which the tangent to the arc is parallel to the secant through its endpoints. It i ...
of calculus, which says:
::''For any function that is continuous on
B,UBand differentiable on (LB,UB) there exists some P
ã in the interval (LB,UB) such that the secant joining the endpoints of the interval
B,UBis parallel to the tangent at P
ã.''
:Essentially, P
ã 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/P
0 and UB/P
ń, the Leibniz and derivative expressions ''do'' require
divarication of the function argument.
Higher-order difference quotients
Second order
:
:
:
Third order
:
:
:
''N''th order
:
:
:
:
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
ASCII ( ), abbreviated from American Standard Code for Information Interchange, is a character encoding standard for electronic communication. ASCII codes represent text in computers, telecommunications equipment, and other devices. Because ...
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 integral''s (ΔA = AU − AL, ΔB = BU − BL, ΔC = CU − CL):
:
Hence,
:
and
:
See also
*
Divided differences
*
Fermat theory
*
Newton polynomial
*
Rectangle method
In mathematics, a Riemann sum is a certain kind of approximation of an integral by a finite sum. It is named after nineteenth century German mathematician Bernhard Riemann. One very common application is approximating the area of functions or lin ...
*
Quotient rule
In calculus, the quotient rule is a method of finding the derivative of a function that is the ratio of two differentiable functions. Let h(x)=f(x)/g(x), where both and are differentiable and g(x)\neq 0. The quotient rule states that the deriva ...
*
Symmetric difference quotient
References
External links
Saint Vincent College: Br. David Carlson, O.S.B.—''MA109 The Difference Quotient''*Mathworld:
*University of Wisconsin:
Thomas W. Reps and Louis B. Rall �
''Computational Divided Differencing and Divided-Difference Arithmetics''
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Differential calculus
Numerical analysis