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# Difference between revisions of "Discrete calculus article"

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It has two major branches, [[differential calculus]] and [[integral calculus]]. Differential calculus concerns incremental rates of change and the slopes of discrete curves. Integral calculus concerns accumulation of quantities and the areas under and between such curves. These two branches are related to each other by the [[fundamental theorem of discrete calculus]]. | It has two major branches, [[differential calculus]] and [[integral calculus]]. Differential calculus concerns incremental rates of change and the slopes of discrete curves. Integral calculus concerns accumulation of quantities and the areas under and between such curves. These two branches are related to each other by the [[fundamental theorem of discrete calculus]]. | ||

− | + | These concepts of change start in their discrete form. Then by making the increment smaller and smaller, we find their contonuous counterparts via this ''limit'': | |

$$\newcommand{\ra}[1]{\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!} | $$\newcommand{\ra}[1]{\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!} | ||

\begin{array}{ccccc} | \begin{array}{ccccc} | ||

\begin{array}{|cc|}\hline\text{ discrete }\\ \text{ calculus }\\ \hline\end{array}& \ra{\quad\Delta x\to 0\quad} &\begin{array}{|cc|}\hline\text{ infinitesimal }\\ \text{ calculus }\\ \hline\end{array} | \begin{array}{|cc|}\hline\text{ discrete }\\ \text{ calculus }\\ \hline\end{array}& \ra{\quad\Delta x\to 0\quad} &\begin{array}{|cc|}\hline\text{ infinitesimal }\\ \text{ calculus }\\ \hline\end{array} | ||

\end{array}$$ | \end{array}$$ | ||

+ | |||

+ | == History == | ||

+ | |||

+ | See also [[History of calculus]]. | ||

+ | |||

+ | ==Calculus of sequences== | ||

+ | |||

+ | |||

+ | |||

+ | ==Principles== | ||

+ | |||

+ | Differential calculus is the study of the definition, properties, and applications of the [[difference quotient]] of a function. The process of finding the difference quotient is called ''differentiation''. Given a function and a point in the domain, the difference quotient at that point is a way of encoding the small-scale (i.e., from the point to the next) behavior of the function. By finding the difference quotient of a function at every point in its domain, it is possible to produce a new function, called the ''difference quotient function'' or just the ''difference quotient'' of the original function. In formal terms, the difference quotient is a [[linear operator]] which takes a function as its input and produces a second function as its output. This is more abstract than many of the processes studied in elementary algebra, where functions usually input a number and output another number. For example, if the doubling function is given the input three, then it outputs six, and if the squaring function is given the input three, then it outputs nine. The derivative, however, can take the squaring function as an input. This means that the derivative takes all the information of the squaring function—such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on—and uses this information to produce another function. The function produced by deriving the squaring function turns out to be the doubling function. | ||

+ | |||

+ | In more explicit terms the "doubling function" may be denoted by {{math|''g''(''x'') {{=}} 2''x''}} and the "squaring function" by {{math|''f''(''x'') {{=}} ''x''<sup>2</sup>}}. The "difference quotient" now takes the function {{math|''f''(''x'')}}, defined by the expression "{{math|''x''<sup>2</sup>}}", as an input, that is all the information—such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on—and uses this information to output another function, the function {{math|''g''(''x'') {{=}} 2''x''}}, as will turn out. | ||

+ | |||

+ | The most common symbol for a difference quotient is: | ||

+ | $$\frac{\Delta f}{\Delta x}.$$ | ||

+ | This notation is known as [[Leibniz's notation]]. | ||

+ | |||

+ | If the input of the function represents time, then the difference quotient represents change with respect to time. For example, if {{math|''f''}} is a function that takes a time as input and gives the position of a ball at that time as output, then the derivative of {{math|''f''}} is how the position is changing in time, that is, it is the [[velocity]] of the ball. | ||

+ | |||

+ | If a function is [[linear function|linear]] (that is, if the [[Graph of a function|graph]] of the function is a straight line), then the function can be written as {{math|''y'' {{=}} ''mx'' + ''b''}}, where {{math|''x''}} is the independent variable, {{math|''y''}} is the dependent variable, {{math|''b''}} is the ''y''-intercept, and: | ||

+ | |||

+ | :<math>m= \frac{\text{rise}}{\text{run}}= \frac{\text{change in } y}{\text{change in } x} = \frac{\Delta y}{\Delta x}.</math> | ||

+ | |||

+ | This gives an exact value for the slope of a straight line. If the graph of the function is not a straight line, however, then the change in {{math|''y''}} divided by the change in {{math|''x''}} varies. The difference quotient give an exact meaning to the notion of change in output with respect to change in input. To be concrete, let {{math|''f''}} be a function, and fix a point {{math|''a''}} in the domain of {{math|''f''}}. {{math|(''a'', ''f''(''a''))}} is a point on the graph of the function. If {{math|''h''}} is the increment of $x$, then {{math|''a'' + ''h''}} is the value of $x$ after (or before) {{math|''a''}}. Therefore, {{math|(''a'' + ''h'', ''f''(''a'' + ''h''))}} is the increment of {{math|(''a'', ''f''(''a''))}}. The slope between these two points is | ||

+ | |||

+ | :<math>m = \frac{f(a+h) - f(a)}{(a+h) - a} = \frac{f(a+h) - f(a)}{h}.</math> | ||

+ | |||

+ | So {{math|''m''}} is the slope of the line between {{math|(''a'', ''f''(''a''))}} and {{math|(''a'' + ''h'', ''f''(''a'' + ''h''))}}. | ||

+ | |||

+ | Here is a particular example, the difference quotient of the squaring function at the input 3. Let {{math|''f''(''x'') {{=}} ''x''<sup>2</sup>}} be the squaring function. | ||

+ | |||

+ | [[File:Sec2tan.gif|thumb|300px|The difference quotient {{math|''f′''(''x'')}} of a curve at a point is the slope of the line tangent to that curve at that point. We considering the slopes of line. Here the function involved (drawn in red) is {{math|''f''(''x'') {{=}} ''x''<sup>3</sup> − ''x''}}. The tangent line (in green) which passes through the point {{nowrap|(−3/2, −15/8)}} has a slope of 23/4. Note that the vertical and horizontal scales in this image are different.]] | ||

+ | |||

+ | :<math>\begin{align}f'(3) &={(3+h)^2 - 3^2\over{h}} \\ | ||

+ | &={9 + 6h + h^2 - 9\over{h}} \\ | ||

+ | &={6h + h^2\over{h}} \\ | ||

+ | &= (6 + h) | ||

+ | \end{align} | ||

+ | </math> |

## Revision as of 16:25, 28 August 2019

**Discrete calculus** or "the calculus of discrete functions", is the mathematical study of *incremental* change, in the same way that geometry is the study of shape and algebra is the study of generalizations of arithmetic operations. The word *calculus* is a Latin word, meaning originally "small pebble"; as such pebbles were used for calculation, the meaning of the word has evolved and today usually means a method of computation. Meanwhile, **calculus**, originally called **infinitesimal calculus** or "the calculus of infinitesimals", is the study of *continuous* change.

It has two major branches, differential calculus and integral calculus. Differential calculus concerns incremental rates of change and the slopes of discrete curves. Integral calculus concerns accumulation of quantities and the areas under and between such curves. These two branches are related to each other by the fundamental theorem of discrete calculus.

These concepts of change start in their discrete form. Then by making the increment smaller and smaller, we find their contonuous counterparts via this *limit*:
$$\newcommand{\ra}[1]{\!\!\!\!\!\xrightarrow{\quad#1\quad}\!\!\!\!\!}
\begin{array}{ccccc}
\begin{array}{|cc|}\hline\text{ discrete }\\ \text{ calculus }\\ \hline\end{array}& \ra{\quad\Delta x\to 0\quad} &\begin{array}{|cc|}\hline\text{ infinitesimal }\\ \text{ calculus }\\ \hline\end{array}
\end{array}$$

## History

See also History of calculus.

## Calculus of sequences

## Principles

Differential calculus is the study of the definition, properties, and applications of the difference quotient of a function. The process of finding the difference quotient is called *differentiation*. Given a function and a point in the domain, the difference quotient at that point is a way of encoding the small-scale (i.e., from the point to the next) behavior of the function. By finding the difference quotient of a function at every point in its domain, it is possible to produce a new function, called the *difference quotient function* or just the *difference quotient* of the original function. In formal terms, the difference quotient is a linear operator which takes a function as its input and produces a second function as its output. This is more abstract than many of the processes studied in elementary algebra, where functions usually input a number and output another number. For example, if the doubling function is given the input three, then it outputs six, and if the squaring function is given the input three, then it outputs nine. The derivative, however, can take the squaring function as an input. This means that the derivative takes all the information of the squaring function—such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on—and uses this information to produce another function. The function produced by deriving the squaring function turns out to be the doubling function.

In more explicit terms the "doubling function" may be denoted by Template:Math and the "squaring function" by Template:Math. The "difference quotient" now takes the function Template:Math, defined by the expression "Template:Math", as an input, that is all the information—such as that two is sent to four, three is sent to nine, four is sent to sixteen, and so on—and uses this information to output another function, the function Template:Math, as will turn out.

The most common symbol for a difference quotient is: $$\frac{\Delta f}{\Delta x}.$$ This notation is known as Leibniz's notation.

If the input of the function represents time, then the difference quotient represents change with respect to time. For example, if Template:Math is a function that takes a time as input and gives the position of a ball at that time as output, then the derivative of Template:Math is how the position is changing in time, that is, it is the velocity of the ball.

If a function is linear (that is, if the graph of the function is a straight line), then the function can be written as Template:Math, where Template:Math is the independent variable, Template:Math is the dependent variable, Template:Math is the *y*-intercept, and:

- [math]m= \frac{\text{rise}}{\text{run}}= \frac{\text{change in } y}{\text{change in } x} = \frac{\Delta y}{\Delta x}.[/math]

This gives an exact value for the slope of a straight line. If the graph of the function is not a straight line, however, then the change in Template:Math divided by the change in Template:Math varies. The difference quotient give an exact meaning to the notion of change in output with respect to change in input. To be concrete, let Template:Math be a function, and fix a point Template:Math in the domain of Template:Math. Template:Math is a point on the graph of the function. If Template:Math is the increment of $x$, then Template:Math is the value of $x$ after (or before) Template:Math. Therefore, Template:Math is the increment of Template:Math. The slope between these two points is

- [math]m = \frac{f(a+h) - f(a)}{(a+h) - a} = \frac{f(a+h) - f(a)}{h}.[/math]

So Template:Math is the slope of the line between Template:Math and Template:Math.

Here is a particular example, the difference quotient of the squaring function at the input 3. Let Template:Math be the squaring function.

- [math]\begin{align}f'(3) &={(3+h)^2 - 3^2\over{h}} \\ &={9 + 6h + h^2 - 9\over{h}} \\ &={6h + h^2\over{h}} \\ &= (6 + h) \end{align} [/math]