Calculus is comprised of two related kinds of procedures that cannot be performed by algebra alone. These are

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.

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

DEFINITION - A

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

DEFINITION - The

DEFINITION - The

DEFINITION - The

DEFINITION - The

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) = x

DEFINITION - A set of numbers, S, is

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.

DEFINITION - D(x) = y is the

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

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

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

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) = x

(f(x + h) - f(x))/h = ((x + h)

= (x^{2} + 2hx + h^{2} - x^{2})/h

= (2hx + h^{2})/h

= 2x + h

= (2hx + h

= 2x + h

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

Eqe = 2x +h

Tre = 2x.

Therefore, D(x) = 2x.

DEFINITION - A(x) is an

DEFINITION - I(x) is an

DEFINITION - U(x) is the

Distinct rational numbers

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

U(x) = A(b) - A(a)

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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) = x

D(x) = 2x

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f(x) = x

D(x) = 3x

The exponent could be any constant, n.

f(x) = x

D(x) = nx

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f(x) = sin x = x - (x

D(x) = 1 - (x

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f(x) = e

D(x) = 0 + 1 + x + (x

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