From 1ebf6a6467167ae9ec0db0c069cfe6170dbcc18f Mon Sep 17 00:00:00 2001
From: Kai Stevenson <kai@kaistevenson.com>
Date: Tue, 1 Aug 2023 20:36:04 -0700
Subject: started calculus, fixed error in colorspace for phplatex

---
 src/writing/derivative.php | 25 +++++++++++++++++++++++--
 1 file changed, 23 insertions(+), 2 deletions(-)

(limited to 'src/writing/derivative.php')

diff --git a/src/writing/derivative.php b/src/writing/derivative.php
index a7a369a..17f2e7f 100644
--- a/src/writing/derivative.php
+++ b/src/writing/derivative.php
@@ -3,7 +3,11 @@ $title = "The Calculus of Infinitesimals";
 require($_SERVER["DOCUMENT_ROOT"] . "/head.php");
 require($_SERVER["DOCUMENT_ROOT"] . "/header.php");
 include($_SERVER["DOCUMENT_ROOT"] . "/phplatex.php");
-include($_SERVER["DOCUMENT_ROOT"] . "/vars.php");
+require($_SERVER["DOCUMENT_ROOT"] . "/vars.php");
+function tex($latex) {
+	global $c_fg, $c_bg;
+	return texify($latex, 130, $c_fg, $c_bg, "", FALSE);
+}
 ?>
 <p>
 	<i>Calculus</i> is often used as the complete name of a branch of mathematics dealing with
@@ -12,6 +16,23 @@ include($_SERVER["DOCUMENT_ROOT"] . "/vars.php");
 	has come to be known in this way is a testament to its importance.
 </p>
 <p>
-	Cube root of two: <?php echo(texify("$\\sqrt[3]{2}\$", 160, $c_fg, $c_bg, "", TRUE)); ?>
+	To understand calculus, it is critical to understand the derivative. Let us begin with
+	the definition of a function.
+</p>
+<p class="center"><?php echo(tex("\$y=f(x)\$")); ?></p>
+<p>
+	This expression defines a relation between two quantities <i>x</i> and <i>y</i>. It can be said that
+	y is given <i>in terms of</i> x. Specifically, we have defined a function <i>f</i> that provides
+	a mapping for values of x. The actual mapping of a given function varies; what follows is
+	one example.
+</p>
+<p class="center"><?php echo(tex("\$f(x)=ax\$")); ?></p>
+<p>
+	Here, the value of the function is said to be equal to the product of its argument, and
+	a constant <i>a</i>. Take note of the fact that this is a constant multiple, as it means that
+	the function is said to be linear--an important property. Evidently, 
+	<?php echo(tex("\$y=f(x)=ax\\Rightarrow y=ax\$")); ?> by the transitive property.
+	Here, the relation between y and x becomes clear; y is greater than x by a multiple a.
+	Observe the pattern that forms when evaluating f at various x.
 </p>
 <?php require($_SERVER["DOCUMENT_ROOT"] . "/footer.php"); ?>
-- 
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