From 3dcc07703a9c73df4ba89770849787965c4aef31 Mon Sep 17 00:00:00 2001
From: Kai Stevenson <kai@kaistevenson.com>
Date: Wed, 2 Aug 2023 07:08:47 -0700
Subject: wrote more calculus, fixed latex positioning problem

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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, 260, $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>
+<h2>Functions</h2>
+<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 wished 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>instantantaneous rate of change</i> at a point.
+</p>
+<h2>Limits</h2>
+<p>
+	The </i>limit</i> of a function is the value L it approaches as its argument approaches
+	c.
+</p>
+<p class="centermath"><?php echo(tex("\$\\lim_{x\\to c} f(x) = L\$")); ?></p>
+<p>
+	Limits enable us to make statements about the behaviour of functions near a point,
+	even when the behaviour differs <i>at</i> that point. For example, 1/0 is undefined,
+	but <?php echo(tex("\$\\lim_{x\\to 0} \\frac{1}{x} = \\infty$")); ?>. This is fairly
+	self-evident--division by an infinitely small number will produce an infinitely large one.
+	The limit is formally defined as follows:
+</p>
+<p class="centermath"><?php echo(tex("\$\\lim_{x\\to c}f(x)=L\\ \\textup{if}\\ \\forall\\epsilon\\in\\mathbb{R},\\ \\epsilon>0\\ \\exists\\delta\\in\\mathbb{R},\\newline|f(x)-L|<\\epsilon\\ \\textup{whenever}\\ 0 < |x-c| < \\delta\$")); ?></p>
+</p>
+<?php require($_SERVER["DOCUMENT_ROOT"] . "/footer.php"); ?>
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