Edits in Chapters 1-4

ptx/sec_deriv_basic_rules.ptx

@@ -51,20 +51,18 @@
       <title>Derivatives of Common Functions</title>
       <statement>
         <p>
-          <dl>
-            <li>
+          <ol cols="2">
+            <li xml:id="constant-derivative-rule">
               <title>Constant Rule</title>
-
-                  <idx><h>derivative</h><h>Constant Rule</h></idx>
+	        <idx><h>derivative</h><h>Constant Rule</h></idx>
 
               <p>
                 <m>\lzoo{x}{c} = 0</m>, where <m>c</m> is a constant.
               </p>
             </li>
 
             <li xml:id="power-derivative-rule">
-              <title>Power Rule</title>
-
+		<title>Power Rule</title>
                   <idx><h>derivative</h><h>Power Rule</h></idx>
                   <idx><h>Power Rule</h><h>differentiation</h></idx>
 
@@ -73,26 +71,27 @@
                 where <m>n</m> is an integer, <m>n \gt 0</m>.
               </p>
             </li>
-
             <li>
-              <title>Other common functions</title>
               <p>
                 <m>\lzoo{x}{\sin(x)} = \cos(x)</m>
               </p>
-
+            </li>
+            <li>
               <p>
                 <m>\lzoo{x}{\cos(x)} = {-\sin(x)}</m>
               </p>
-
+            </li>
+            <li>
               <p>
                 <m>\lzoo{x}{e^x} = e^x</m>
               </p>
-
+            </li>
+            <li>
               <p>
                 <m>\lzoo{x}{\ln(x)} = \frac{1}{x}</m>, for <m>x \gt  0</m>.
               </p>
             </li>
-          </dl>
+          </ol>
 
           <idx><h>derivative</h><h>basic rules</h></idx>
 
@@ -224,8 +223,9 @@
           </ol>
         </p>
 
-        <figure xml:id="fig_xcubedwithderiv" vshift="-1">
-          <caption>A graph of <m>f(x) = x^3</m>, along with its derivative <m>\fp(x) = 3x^2</m> and its tangent line at <m>x=-1</m></caption>          <!-- START figures/fig_xcubedwithderiv.tex -->
+        <figure xml:id="fig_xcubedwithderiv" vshift="3">
+          <caption>A graph of <m>f(x) = x^3</m>, along with its derivative <m>\fp(x) = 3x^2</m> and its tangent line at <m>x=-1</m></caption>          
+          <!-- START figures/fig_xcubedwithderiv.tex -->
           <image width="47%">
             <shortdescription>
               Graph of function x^3, its derivative and a tangent line drawn at point (-1,-1).
@@ -286,22 +286,21 @@
       The following theorem helps with the first two of these examples
       (the third is answered in the next section).
     </p>
-
+		
     <theorem xml:id="thm_deriv_prop">
       <title>Properties of the Derivative</title>
       <statement>
         <p>
-          Let <m>f</m> and <m>g</m> be differentiable on an open interval <m>I</m> and let <m>c</m> be a real number.
+          Let <m>f</m> and <m>g</m> be differentiable on an open interval 
+					<m>I</m> and let <m>c</m> be a real number.
           Then:
-          <dl>
+          <ol cols="1">
             <li xml:id="sum-difference-derivative-rule">
               <title>Sum/Difference Rule</title>
-              <p>
-                <md>
-                  <mrow>\lzoo{x}{f(x) \pm g(x)} \amp= \lzoo{x}{f(x)} \pm \lzoo{x}{g(x)}</mrow>
-                  <mrow>\amp= \fp(x)\pm g'(x)</mrow>
-                </md>
-
+ 						 <p>
+                <me>\lzoo{x}{f(x) \pm g(x)} = \lzoo{x}{f(x)} \pm \lzoo{x}{g(x)} 
+								=\fp(x)\pm g'(x) 
+								</me>
                   <idx><h>derivative</h><h>Sum/Difference Rule</h></idx>
                   <idx><h>Sum/Difference Rule</h><h>of derivatives</h></idx>
 
@@ -311,20 +310,18 @@
             <li xml:id="constant-multiple-derivative-rule">
               <title>Constant Multiple Rule</title>
               <p>
-                <md>
-                  <mrow>\lzoo{x}{c\cdot f(x)} \amp= c\cdot\lzoo{x}{f(x)}</mrow>
-                  <mrow>\amp = c\cdot\fp(x)</mrow>
-                </md>.
-
+               <me>
+							   \lzoo{x}{c\cdot f(x)} =c\cdot\lzoo{x}{f(x)}=c\cdot\fp(x)
+							 </me>
                   <idx><h>derivative</h><h>Constant Multiple Rule</h></idx>
                   <idx><h>Constant Multiple Rule</h><h>of derivatives</h></idx>
 
               </p>
             </li>
-          </dl>
+          </ol>
         </p>
       </statement>
-    </theorem>
+    </theorem>		
 
     <figure xml:id="vid_deriv_basic_rules_const_sum_rule" component="video" vshift="2">
       <caption>Video presentation of <xref ref="thm_deriv_prop"/></caption>
@@ -343,6 +340,7 @@
       <video youtube="nVVpyilxZTw" label="vid_deriv_basic_rules_proofs"/>
     </figure>
 
+		<pagebreak-latex/>
 
     <p>
       <xref ref="thm_deriv_prop"/>
@@ -521,20 +519,6 @@
 
   <subsection>
     <title>Higher Order Derivatives</title>
-    <aside xml:id="aside-derivative-second-order-notation" vshift="2">
-      <p>
-        <em>Note:</em> The second derivative notation could be written as
-        <me>
-          \frac{d^2y}{dx^2}=\frac{d^2y}{(dx)^2}=\frac{d^2}{(dx)^2}\big(y\big)
-        </me>.
-      </p>
-
-      <p>
-        That is, we take the derivative of <m>y</m> twice
-        (hence <m>d^2</m>),
-        both times with respect to <m>x</m> (hence <m>(dx)^2=dx^2</m>).
-      </p>
-    </aside>
 
     <p>
       The derivative of a function <m>f</m> is itself a function,
@@ -582,14 +566,23 @@
       </statement>
     </definition>
 
-    <aside xml:id="aside-derivative-second-order-caveat" vshift="6">
-      <title>Higher Order Derivative Caveat</title>
+    <aside xml:id="aside-derivative-second-order-caveat" vshift="4">
+      <title>Higher Order Derivative Notes <!-- Caveat --></title>
       <p>
         <xref ref="def_Higher_Deriv"/>
         comes with the caveat <q>Where the corresponding limits exist.</q>
         With <m>f</m> differentiable on <m>I</m>,
         it is possible that <m>\fp</m> is <em>not</em>
         differentiable on all of <m>I</m>, and so on.
+      </p>
+			<p>
+			Also, the second derivative notation could be written as
+        <me>
+          \frac{d^2y}{dx^2}=\frac{d^2y}{(dx)^2}=\frac{d^2}{(dx)^2}\big(y\big)
+        </me>.
+        That is, we take the derivative of <m>y</m> twice
+        (hence <m>d^2</m>),
+        both times with respect to <m>x</m> (hence <m>(dx)^2=dx^2</m>).
       </p>
     </aside>
 
@@ -618,8 +611,11 @@
             <li><m>f(x) = \sin(x)</m></li>
             <li><m>f(x) = 5e^x</m></li>
           </ol>
-        </p>
+				</p>
       </statement>
+
+      <pagebreak-latex/>
+
       <solution>
         <p>
           <ol>
@@ -669,9 +665,7 @@
       What do higher order derivatives <em>mean</em>?
       What is the practical interpretation?
 
-      <!-- TODO: is ! the right markup here? -->
-
-          <idx><h>derivative</h><h>higher order!interpretation</h></idx>
+          <idx><h>derivative</h><h>higher order</h><h>interpretation</h></idx>
     </p>
 
     <p>

ptx/sec_deriv_chainrule.ptx

@@ -40,7 +40,7 @@
     So it can't be correct to say that <m>y'=-\sin(2x)</m>.
   </p>
 
-  <figure xml:id="fig_chain" vshift="-4">
+  <figure xml:id="fig_chain" vshift="0">
     <caption>A graph of <m>y=\cos(x^2)</m> and a tangent line at <m>\pi/2</m></caption>
     <image width="47%">
       <shortdescription>
@@ -152,7 +152,7 @@
     </solution>
   </example>
 
-  <aside vshift="0">
+  <aside vshift="-3">
     <p>
       When composing functions,
       we need to make sure that the new function is actually defined.
@@ -166,7 +166,7 @@
 
     <p>
       The statement of <xref ref="thm_chain_rule"/> takes care to ensure
-      this problem does not arise, but our focus is more on the derivative result than
+      this problem does not arise. We will focus more on the derivative result than
       on the domain/range conditions.
     </p>
   </aside>
@@ -440,7 +440,7 @@
         The tangent line is sketched along with <m>f</m> in <xref ref="fig_chain7"/>.
       </p>
 
-      <figure xml:id="fig_chain7" vshift="0">
+      <figure xml:id="fig_chain7" vshift="2.5">
         <caption><m>f(x) = \cos(x^2)</m> sketched along with its tangent line at <m>x=1</m></caption>
         <!-- START figures/fig_chain7.tex -->
         <image width="47%">
@@ -832,32 +832,7 @@
       That is, the rate at which the <m>u</m> gear makes a revolution is twice as fast as the rate at which the <m>x</m> gear makes a revolution.
     </p>
 
-    <sidebyside widths="47% 47%" margins="0%">
-
-      <stack><!-- Old paragraphs title: -->
-        <p>
-          Using the terminology of calculus,
-          the rate of <m>u</m>-change, with respect to <m>x</m>,
-          is <m>\lz{u}{x} = 2</m>.
-        </p>
-
-        <p>
-          Likewise, every revolution of <m>u</m> causes <m>3</m> revolutions of <m>y</m>:
-          <m>\lz{y}{u} = 3</m>.
-          How does <m>y</m> change with respect to <m>x</m>?
-          For each revolution of <m>x</m>,
-          <m>y</m> revolves <m>6</m> times; that is,
-          <me>
-            \frac{dy}{dx} = \frac{dy}{du}\cdot \frac{du}{dx} = 2\cdot 3 = 6
-          </me>.
-        </p>
-
-        <p>
-          We can then extend the <xref ref="thm_chain_rule" text="title"/> with more variables by adding more gears to the picture.
-        </p>
-      </stack>
-
-      <figure xml:id="fig_chainrulegears">
+      <figure xml:id="fig_chainrulegears"  vshift="0">
         <caption>A series of gears to demonstrate the Chain Rule. Note how <m>\lz{y}{x} = \lz{y}{u}\cdot\lz{u}{x}</m></caption>
         <!-- START figures/fig_chainrule_gears.tex -->
         <image>
@@ -915,7 +890,27 @@
         </image>
         <!-- figures/fig_chainrule_gears.tex END -->
       </figure>
-    </sidebyside>
+
+        <p>
+          Using the terminology of calculus,
+          the rate of <m>u</m>-change, with respect to <m>x</m>,
+          is <m>\lz{u}{x} = 2</m>.
+        </p>
+
+        <p>
+          Likewise, every revolution of <m>u</m> causes <m>3</m> revolutions of <m>y</m>:
+          <m>\lz{y}{u} = 3</m>.
+          How does <m>y</m> change with respect to <m>x</m>?
+          For each revolution of <m>x</m>,
+          <m>y</m> revolves <m>6</m> times; that is,
+          <me>
+            \frac{dy}{dx} = \frac{dy}{du}\cdot \frac{du}{dx} = 2\cdot 3 = 6
+          </me>.
+        </p>
+
+        <p>
+          We can then extend the <xref ref="thm_chain_rule" text="title"/> with more variables by adding more gears to the picture.
+        </p>
 
     <p>
       It is difficult to overstate the importance of the <xref ref="thm_chain_rule" text="title"/>.