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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="centermath"><?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="centermath"><?php echo(tex("\$f(x)=ax\$")); ?></p>
<p>
In this example, 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>
<p class="centermath"><?php echo(tex("\$f(0)=a\\cdot0=0\$")); ?></p>
<p class="centermath"><?php echo(tex("\$f(1)=a\\cdot1=a\$")); ?></p>
<p class="centermath"><?php echo(tex("\$f(2)=a\\cdot2=2a\$")); ?></p>
<p class="centermath"><?php echo(tex("\$f(3)=a\\cdot3=3a\$")); ?></p>
<p>
As x increases by increments of 1, y increases by increments of a. This gives rise to
the familiar slope-intercept form of a line,
<?php echo(tex("\$y=mx+b\$")); ?>. m, here, is the <i>slope</i> of the line,
the ratio between the growth of y and x, or the rate of change of the function mapping
the two. Since b is a constant term, it has no impact on the rate of change of the function.
It is this understanding that we will build upon. Consider two points in the form (x, y):
(0, 0) & (3, 9). As x increases by 3, y increases by 9. We refer to these as delta x
and delta y. We obtain the rate of change as follows:
</p>
<p class="centermath"><?php echo(tex("\$\\Delta x = 3,\ \\Delta y = 9\$")); ?></p>
<p class="centermath"><?php echo(tex("\$m = \\frac{\\Delta y}{\\Delta x} = \\frac{9}{3} = 3$")); ?></p>
<p>
The implication of this is that a line drawn between the two points would have a slope
of 3. We note two functions upon whose graphs these points fall:
<?php echo(tex("\$y=3x\$")); ?>, and <?php echo(tex("\$y=x^2\$")); ?>. If graphed,
the path of <?php echo(tex("\$y=3x\$")); ?> would appear identical to a straight line drawn
between the points, except
that its line would continue infinitely in either direction. Recall that it is a linear
function, and so every segment of its path has the same inclination as the whole. Compare
this to <?php echo(tex("\$y=x^2\$")); ?>, a parabolic curve. Though both points fall upon
its graph, the path it takes between them is not linear, but curved. It appears to follow a
much less direct route. However, as each function has a slope of 3 over the interval [0, 3],
we say that they share the same <i>average rate of change</i> over that interval. The linear
function appears to move towards its destination faster in the start, but as the parabola
steepens, it catches up, such that both functions reach (3, 9). If we continued observing
this competition, the parabola would continue accelerating more and more rapidly, its value
rapidly eclipsing the straight line.
</p>
<p>
Suppose we wish to know the slope of a function not over an interval but at a point.
This is easily accomplished with a linear function--it is observed that the slope of a
linear function is the same over any interval, and so it is reasonable to extend that
to a single point. It is more difficult, however, with non-linear functions.
Take <?php echo(tex("\$y=x^2\$")); ?> on [0, 1]:
</p>
<p class="centermath"><?php echo(tex("\$f(x) = x^2,\ x = 1,\ \\Delta x = 1\$")); ?></p>
<p class="centermath"><?php echo(tex("\$f(1) = 1,\ f(1+\\Delta x) = 4,\ \\Delta y = 4-1 = 3\$")); ?></p>
<p class="centermath"><?php echo(tex("\$m = \\frac{\\Delta y}{\\Delta x} = 3\$")); ?></p>
<p>Compare to the interval [0, 0.5]:</p>
<p class="centermath"><?php echo(tex("\$\\Delta x = 0.5\$")); ?></p>
<p class="centermath"><?php echo(tex("\$f(1+\\Delta x) = 2.25,\ \\Delta y = 1.25\$")); ?></p>
<p class="centermath"><?php echo(tex("\$m = \\frac{\\Delta y}{\\Delta x} = 2.5\$")); ?></p>
<p>Or to [0, 0.25]:</p>
<p class="centermath"><?php echo(tex("\$\\Delta x = 0.25\$")); ?></p>
<p class="centermath"><?php echo(tex("\$f(1+\\Delta x) = 1.5625,\ \\Delta y = 0.5625\$")); ?></p>
<p class="centermath"><?php echo(tex("\$m = 2.25\$")); ?></p>
<p>
As the interval becomes smaller, m decreases. One could use a computer to perform this
calculation with increasingly small values, and would find that m gets very close to 2.
As we can see, when moving from [0, 1] to [0, 0.5], m moved half of the way to 3.
On the next interval, it moved half of the distance that remains, again.
This would continue infinitely, with m never reaching 2. A new calculus is needed
to precisely describe the slope of non-linear functions at a point. This value,
the rate that the function will change over the next infinitesimally small interval,
is known as its <i>instant rate of change</i> at a point.
</p>
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