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<?php
$title = "The Calculus of Infinitesimals";
require($_SERVER["DOCUMENT_ROOT"] . "/head.php");
require($_SERVER["DOCUMENT_ROOT"] . "/header.php");
include($_SERVER["DOCUMENT_ROOT"] . "/phplatex.php");
require($_SERVER["DOCUMENT_ROOT"] . "/vars.php");
function tex($latex) {
global $c_fg, $c_bg;
return texify($latex, 130, $c_fg, $c_bg, "", FALSE);
}
?>
<p>
<i>Calculus</i> is often used as the complete name of a branch of mathematics dealing with
rates of change and very small numbers. In fact, a calculus is a specific discipline
or method for the analysis of some set of problems. The fact that the <i>calculus of infinitesimals</i>
has come to be known in this way is a testament to its importance.
</p>
<p>
To understand calculus, it is critical to understand the derivative. Let us begin with
the definition of a function.
</p>
<p class="center"><?php echo(tex("\$y=f(x)\$")); ?></p>
<p>
This expression defines a relation between two quantities <i>x</i> and <i>y</i>. It can be said that
y is given <i>in terms of</i> x. Specifically, we have defined a function <i>f</i> that provides
a mapping for values of x. The actual mapping of a given function varies; what follows is
one example.
</p>
<p class="center"><?php echo(tex("\$f(x)=ax\$")); ?></p>
<p>
Here, the value of the function is said to be equal to the product of its argument, and
a constant <i>a</i>. Take note of the fact that this is a constant multiple, as it means that
the function is said to be linear--an important property. Evidently,
<?php echo(tex("\$y=f(x)=ax\\Rightarrow y=ax\$")); ?> by the transitive property.
Here, the relation between y and x becomes clear; y is greater than x by a multiple a.
Observe the pattern that forms when evaluating f at various x.
</p>
<?php require($_SERVER["DOCUMENT_ROOT"] . "/footer.php"); ?>
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