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Rational Calculus

by James Adrian


Introduction

      Calculus is comprised of two related kinds of procedures that cannot be performed by algebra alone. These are Differential Calculus and Integral Calculus. They are simple. This article will define what they are and explain how to use them.

      Despite the theoretical foundations most preferred by mathematicians who teach mathematics and write mathematical papers, and following the traditions of scientist and engineers who successfully use calculus while approximating irrational and transcendental numbers with rational numbers, this presentation will include and use only rational numbers.


Background

      You may recall that integers include positive whole numbers, their negatives, and zero. The numbers -2, -1, 0, 1 and 2 are integers. For a formal development of the integers, see Mathematics From The Beginning.

DEFINITION - A Rational Number is the ratio of two integers, p divided by q (notated p/q) where q is not zero. In the ratio p/q, p is called the numerator and q is called the denominator.

      Nowhere in mathematics is division by zero given a definition. Wherever division by zero might be written, it is undefined and without meaning.

DEFINITION - A Function is a rule that associates each permissible input number with one and only one output number.

DEFINITION - The Domain of a Function is the set of the function's permissible input numbers.

DEFINITION - The Range of a Function is the set of the function's output numbers.

DEFINITION - The Independent Variable of a Function is a variable whose value is selected from the domain of the function.

DEFINITION - The Dependent Variable of a Function is a variable whose value is selected from the range of the function.

      If a function, f, has an independent variable, x, then f is said to be a function of x and this is notated f(x). If y is the dependent variable of f, then we say that f(x) = y. As an example, f(x) could associate each x in its domain with a y value by means of a polynomial such as y = f(x) = x2 + x - 1. The range is the set of y values calculated by f(x) where x is in the domain of f(x).

DEFINITION - A set of numbers, S, is Dense if and only if for each pair, a and b, of distinct numbers in S, there is another number x in S such that a < x < b.

      In order for quantities to be discerned through calculus, the domains of the functions involved must be dense. Any set of rational numbers that includes all rational numbers that are both greater than or equal to some rational number j, and less than or equal to some rational number k, is dense. This property is not true of the integers.

      In the olden days, using time or the order of events as a necessary element in a mathematical definition was taboo. Computer programs seem to have made that prohibition a thing of the past. In any event, this article disregards any such restriction. Specifying procedures or the order of events is regarded as reasonable within definitions.


Differential Calculus

DEFINITION - D(x) = y is the Derivative or Rate of Change of function f(x) = y if and only if the following statements are true:

Each number x in the domain of f(x) is a rational number such that x is greater than or equal to rational number a and x is less than or equal to rational number b.

The function f(x) is such that an expression, Eqe, equal to (f(x + h) - f(x))/h, is found such that Eqe is defined for h = 0. This expression, Eqe, may be called the Equivalent Expression.

Tre is an expression identical to Eqe except that h = 0 in expression Tre. This expression, Tre, may be called the Transformed Expression

D(x) is then defined as a function such that each x in the domain of D(x) is also in the domain of f(x), and each x in the domain of f(x) is also in the domain of D(x), and y = D(x) = Tre.
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      Here is an example:

      If y = f(x) = x2, then

      (f(x + h) - f(x))/h = ((x + h)2 - x2)/h
= (x2 + 2hx + h2 - x2)/h
= (2hx + h2)/h
= 2x + h

      The expression 2x + h satisfies the requirements of equivalent expression, E.

      Eqe = 2x +h
      Tre = 2x.

      Therefore, D(x) = 2x.



Integral Calculus

DEFINITION - A(x) is an Antiderivative of function f(x) if and only if f(x) is the derivative of A(x).

DEFINITION - I(x) is an Indefinite Integral of function f(x) if and only if f(x) is the derivative of I(x).

DEFINITION - U(x) is the Definite Integral of function f(x) if and only if the following statements are true:

Distinct rational numbers a and b are in the domain of f(x).

A(x) is an antiderivative of f(x).

U(x) = A(b) - A(a)
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Derivatives and Integrals

      To calculate a definite integral of f(x) over the interval between a and b, you must know the indefinite in integral of f(x). To know that the derivative of g(x) is D(x) is to know that g(x) is the antiderivative of D(x). Knowing a great number of derivatives can help you identify antiderivatives; however, knowing some key derivatives can make the process a lot less tedious.

      Here are some key derivatives, D(x) for f(x), that you can verify with the definition of the derivative:

f(x) = 5
D(x) = 0
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The number 5 could instead be any constant, b, that you choose.

f(x) = b
D(x) = 0
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f(x) = x
D(x) = 1
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The multiplier, 1, could be any constant, m.

f(x) = mx
D(x) = m
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f(x) = mx + b
D(x) = m
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f(x) = x2 + b
D(x) = 2x
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f(x) = x3 + b
D(x) = 3x2


The exponent could be any constant, n.

f(x) = xn + b
D(x) = nxn - 1
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f(x) = sin x = x - (x3)/3! + (x5)/5! - (x7)/7! + . . .
D(x) = 1 - (x2)/2! + (x4)/4! - (x6)/6! + . . . = cos x
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f(x) = ex = 1 + x + (x2)/2! + (x3)/3! + (x4)/2! + . . .
D(x) = 0 + 1 + x + (x2)/2! + (x3)/3! + (x4)/2! + . . . = ex
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Contact

      Please feel free to write to me directly for more information or to make suggestions or comments. My email address is jim@futurebeacon.com. You can also go to my contact page to get my full contact information. Suggestions, questions, additional information and critiques are very welcome.