diff --git a/ptx/sec_deriv_basic_rules.ptx b/ptx/sec_deriv_basic_rules.ptx
index d4d0f8081..b02d65f06 100644
--- a/ptx/sec_deriv_basic_rules.ptx
+++ b/ptx/sec_deriv_basic_rules.ptx
@@ -51,11 +51,10 @@
       <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.
@@ -63,8 +62,7 @@
             </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>
diff --git a/ptx/sec_deriv_chainrule.ptx b/ptx/sec_deriv_chainrule.ptx
index 021980b46..48abdef63 100644
--- a/ptx/sec_deriv_chainrule.ptx
+++ b/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"/>.
diff --git a/ptx/sec_deriv_implicit.ptx b/ptx/sec_deriv_implicit.ptx
index eabb5b2a7..7c21f4439 100644
--- a/ptx/sec_deriv_implicit.ptx
+++ b/ptx/sec_deriv_implicit.ptx
@@ -47,7 +47,7 @@
 
     </p>
 
-    <figure xml:id="fig_implicit1" vshift="-3">
+    <figure xml:id="fig_implicit1" vshift="2.5">
       <caption>A graph of the implicit relationship <m>\sin(y)+y^3=6-x^3</m></caption>
       <!-- START figures/fig_implicit1.tex -->
       <image width="47%">
@@ -240,7 +240,7 @@
           The curve and this tangent line are shown in <xref ref="fig_implicit2"/>.
         </p>
 
-        <figure xml:id="fig_implicit2" vshift="0">
+        <figure xml:id="fig_implicit2" vshift="2.5">
           <caption>The function <m>\sin(y) +y^3 = 6-x^3</m> and its tangent line at the point <m>(\sqrt[3]{6},0)</m></caption>
           <!-- START figures/fig_implicit2.tex -->
           <image width="47%">
@@ -504,9 +504,9 @@
             <latex-image label="img_implicit5">
 
               \begin{tikzpicture}
-              \begin{axis}[xmin=-1.95,xmax=1.99,
+              \begin{axis}[xmin=-2.05,xmax=2.05,
               ymin=-1.95,ymax=1.95,]
-              \addplot+[smooth,infinite] coordinates {(-1.91,-1.57) (-1.82,-1.61) (-1.68,-1.61) (-1.7,-1.49) (-1.74,-1.38)(-1.76,-1.28) (-1.64,-1.26) (-1.54,-1.29) (-1.36,-1.35) (-1.2,-1.42)(-1.09,-1.46) (-0.989,-1.49) (-0.86,-1.5) (-0.714,-1.45)(-0.566,-1.35) (-0.327,-1.18) (-0.15,-1.08)(0.107,-0.947) (0.216,-0.895) (0.332,-0.832) (0.429,-0.755) (0.464,-0.623) (0.429,-0.52) (0.308,-0.335)(0.142,-0.144) (-0.071,0.0714) (-0.239,0.261) (-0.336,0.45) (-0.336,0.622) (-0.278,0.757) (-0.17,0.884) (-0.046,0.975)(0.0714,1.03) (0.216,1.07) (0.48,1.09) (0.714,1.07) (0.929,1.05)(1.15,1.07) (1.24,1.13) (1.27,1.25) (1.28,1.37) (1.29,1.49)(1.36,1.58) (1.47,1.57) (1.59,1.52) (1.68,1.48) (1.79,1.43)(1.89,1.39) (1.98,1.36) (1.98,1.6)};
+              \addplot+[smooth,infinite] coordinates {(-2.03,-1.513)(-1.988,-1.536)(-1.972,-1.543)(-1.879,-1.586)(-1.786,-1.621)(-1.68,-1.572)(-1.707,-1.464)(-1.749,-1.357)(-1.739,-1.261)(-1.621,-1.264)(-1.5,-1.299)(-1.348,-1.357)(-1.179,-1.428)(-1.074,-1.467)(-0.9643,-1.494)(-0.8524,-1.495)(-0.6789,-1.429)(-0.5,-1.303)(-0.3182,-1.179)(-0.1408,-1.073)(0,-1.)(0.1397,-0.9317)(0.2397,-0.8826)(0.3496,-0.8214)(0.4457,-0.7329)(0.4619,-0.6071)(0.4189,-0.5)(0.276,-0.2954)(0.07143,-0.07178)(-0.07143,0.07183)(-0.2684,0.3031)(-0.3426,0.4855)(-0.3214,0.6721)(-0.2591,0.7857)(-0.1429,0.9072)(0,1.)(0.1009,1.042)(0.2857,1.083)(0.5178,1.089)(0.7793,1.065)(0.9789,1.05)(1.179,1.08)(1.248,1.145)(1.275,1.275)(1.279,1.393)(1.3,1.514)(1.393,1.582)(1.5,1.558)(1.607,1.514)(1.712,1.467)(1.804,1.426)(1.856,1.405)};
               \end{axis}
 
               \end{tikzpicture}
@@ -532,7 +532,7 @@
           The tangent lines have been added to the graph of the function in <xref ref="fig_implicit6"/>.
         </p>
 
-        <figure xml:id="fig_implicit6" vshift="-1">
+        <figure xml:id="fig_implicit6" vshift="3">
           <caption>A graph of the implicitly defined curve <m>\sin\mathopen{}\left(x^2y^2\right)\mathclose{}+y^3=x+y</m> and certain tangent lines</caption>
           <!-- START figures/fig_implicit6.tex -->
           <image width="47%">
@@ -550,9 +550,9 @@
             <latex-image label="img_implicit6">
 
               \begin{tikzpicture}
-              \begin{axis}[xmin=-1.95,xmax=1.99,
+              \begin{axis}[xmin=-2.05,xmax=2.05,
               ymin=-1.95,ymax=1.95,]
-              \addplot+[infinite,smooth] coordinates {(-1.91,-1.57) (-1.82,-1.61) (-1.68,-1.61) (-1.7,-1.49) (-1.74,-1.38)(-1.76,-1.28) (-1.64,-1.26) (-1.54,-1.29) (-1.36,-1.35) (-1.2,-1.42)(-1.09,-1.46) (-0.989,-1.49) (-0.86,-1.5) (-0.714,-1.45)(-0.566,-1.35) (-0.327,-1.18) (-0.15,-1.08)(0.107,-0.947) (0.216,-0.895) (0.332,-0.832) (0.429,-0.755) (0.464,-0.623) (0.429,-0.52) (0.308,-0.335)(0.142,-0.144) (-0.071,0.0714) (-0.239,0.261) (-0.336,0.45) (-0.336,0.622) (-0.278,0.757) (-0.17,0.884) (-0.046,0.975)(0.0714,1.03) (0.216,1.07) (0.48,1.09) (0.714,1.07) (0.929,1.05)(1.15,1.07) (1.24,1.13) (1.27,1.25) (1.28,1.37) (1.29,1.49)(1.36,1.58) (1.47,1.57) (1.59,1.52) (1.68,1.48) (1.79,1.43)(1.89,1.39) (1.98,1.36) (1.98,1.6)};
+              \addplot+[infinite,smooth] coordinates {(-2.03,-1.513)(-1.988,-1.536)(-1.972,-1.543)(-1.879,-1.586)(-1.786,-1.621)(-1.68,-1.572)(-1.707,-1.464)(-1.749,-1.357)(-1.739,-1.261)(-1.621,-1.264)(-1.5,-1.299)(-1.348,-1.357)(-1.179,-1.428)(-1.074,-1.467)(-0.9643,-1.494)(-0.8524,-1.495)(-0.6789,-1.429)(-0.5,-1.303)(-0.3182,-1.179)(-0.1408,-1.073)(0,-1.)(0.1397,-0.9317)(0.2397,-0.8826)(0.3496,-0.8214)(0.4457,-0.7329)(0.4619,-0.6071)(0.4189,-0.5)(0.276,-0.2954)(0.07143,-0.07178)(-0.07143,0.07183)(-0.2684,0.3031)(-0.3426,0.4855)(-0.3214,0.6721)(-0.2591,0.7857)(-0.1429,0.9072)(0,1.)(0.1009,1.042)(0.2857,1.083)(0.5178,1.089)(0.7793,1.065)(0.9789,1.05)(1.179,1.08)(1.248,1.145)(1.275,1.275)(1.279,1.393)(1.3,1.514)(1.393,1.582)(1.5,1.558)(1.607,1.514)(1.712,1.467)(1.804,1.426)(1.856,1.405)};
 
               \addplot [tangentline,domain=-.5:.5] {-x};
               \addplot [tangentline,domain=-.5:.5] {0.5*x+1};
@@ -617,7 +617,7 @@
           (It turns out that all normal lines to a circle pass through the center of the circle.)
         </p>
 
-        <figure xml:id="fig_implicit7" vshift="0">
+        <figure xml:id="fig_implicit7" vshift="2.5">
           <caption>The unit circle with its tangent line at <m>(1/2,\sqrt{3}/2)</m></caption>
           <!-- START figures/fig_implicit7.tex -->
           <image width="47%">
@@ -755,7 +755,7 @@
           as shown in <xref ref="fig_implicit9"/>.
         </p>
 
-        <figure xml:id="fig_implicit9" vshift="0">
+        <figure xml:id="fig_implicit9" vshift="1">
           <caption>An astroid, traced out by a point on the smaller circle as it rolls inside the larger circle</caption>
           <!-- START figures/fig_implicit9.tex -->
           <image width="47%">
@@ -807,7 +807,7 @@
           is shown in <xref ref="fig_implicit8"/>.
         </p>
 
-        <figure xml:id="fig_implicit8" vshift="-1">
+        <figure xml:id="fig_implicit8" vshift="0">
           <caption>An astroid with a tangent line</caption>
           <!-- START figures/fig_implicit8.tex -->
           <image width="47%">
@@ -904,22 +904,21 @@
       It is well-defined for <m>x \gt 0</m> and we might be interested in finding equations of lines tangent and normal to its graph.
       How do we take its derivative?
 
-                <idx><h>logarithmic differentiation</h></idx>
-                <idx><h>derivative</h>
-                     <h>logarithmic</h>
-                </idx>
+      <idx><h>logarithmic differentiation</h></idx>
+      <idx><h>derivative</h><h>logarithmic</h></idx>
     </p>
-    <aside xml:id="aside-derivative-implicit-00" vshift="3">
+    <aside xml:id="aside-derivative-implicit-00" vshift="-7">
       <p>
-        In calculus the expression <m>0^0</m> is also considered well-defined and equal to <m>1</m>.
-        This is easily confused with a limit of the <em>form</em>
+        There are several reasons why the expression <m>0^0</m> can 	be <em>defined</em> to equal <m>1</m>. However, this is easily confused with a <em>limit of the form</em>
         <m>0^0</m>, which is indeterminate.
-        We skirt the issue here.
+			</p>
+			<p>
+        When implicitly deriving <m>y=x^x</m>, we obtain <m>\ln(x)</m> which is not defined when <m>x=0</m>. Therefore we restrict the domain here to <m>x>0</m>.
       </p>
     </aside>
 
 
-    <figure xml:id="fig_logdiffa" vshift="-2">
+    <figure xml:id="fig_logdiffa" vshift="-1">
       <caption>A plot of <m>y=x^x</m></caption>
       <!-- START figures/fig_logdiffa.tex -->
       <image width="47%">
@@ -1004,7 +1003,7 @@
           graphs <m>y=x^x</m> along with this tangent line.
         </p>
 
-        <figure xml:id="fig_implicit10" vshift="-4">
+        <figure xml:id="fig_implicit10" vshift="1.5">
           <caption>A graph of <m>y=x^x</m> and its tangent line at <m>x=1.5</m></caption>
           <!-- START figures/fig_implicit10.tex -->
           <image width="47%">
diff --git a/ptx/sec_deriv_interpret.ptx b/ptx/sec_deriv_interpret.ptx
index d0eddf631..7df2db21c 100644
--- a/ptx/sec_deriv_interpret.ptx
+++ b/ptx/sec_deriv_interpret.ptx
@@ -205,8 +205,8 @@
       For instance,
       <m>s</m> could measure the height of a projectile or the distance an object has traveled.
     </p>
-    <aside vshift="0">
-      <title>Convention with <m>s</m></title>
+    <aside vshift="0.5">
+      <title>Conventional notation</title>
       <p>
         Using <m>s(t)</m> to represent position is a fairly common mathematical convention.
         It is also common to use <m>s</m> to represent arc length.
@@ -379,7 +379,7 @@
           by considering the slopes of the respective tangent lines.
         </p>
 
-        <figure xml:id="fig_xsquaredwithgrid" vshift="-4">
+        <figure xml:id="fig_xsquaredwithgrid" vshift="-3.5">
           <caption>A graph of <m>f(x)=x^2</m> and tangent lines at <m>x=1</m> and <m>x=3</m></caption>
           <!-- START figures/fig_xsquaredwithgrid.tex -->
           <image width="47%">
@@ -508,7 +508,7 @@
           <m>x=2</m> and <m>x=3</m> to help better visualize the <m>y</m> value of <m>\fp</m> at those points.
         </p>
 
-        <figure xml:id="fig_fwithderivdots" vshift="-5">
+        <figure xml:id="fig_fwithderivdots" vshift="-4.5">
           <caption>Graphs of <m>f</m> and <m>\fp</m> in <xref ref="ex_der_meaning4"/></caption>
           <!-- START figures/fig_fwithderivdots.tex -->
           <image width="47%">
@@ -582,7 +582,7 @@
           It is often useful to leave the tangent line in point-slope form.
         </p>
 
-        <figure xml:id="fig_fwithderivzoom3" vshift="-3">
+        <figure xml:id="fig_fwithderivzoom3" vshift="-1.5">
           <caption>Zooming in on <m>f</m> and its tangent line at <m>x=3</m> for the function given in <xref ref="ex_der_meaning4">Examples</xref> and <xref ref="ex_der_meaning5"/></caption>
           <!-- START figures/fig_fwithderivzoom3.tex -->
           <image width="47%">
@@ -1122,10 +1122,10 @@
               <p>
                 Is <m>T(8)</m> likely greater than or less than 0?
                 Why?
-              </p>
+							</p>
               <p>
                 <var form="essay"/>
-              </p>
+							</p>
             </statement>
             <solution>
               <p>
@@ -1137,10 +1137,12 @@
         </webwork>
       </exercise>
 
+			<pagebreak-latex/>
+
       <exercisegroup cols="2" xml:id="exset-deriv-interpret-graph">
         <introduction>
           <p>
-            Graphs of functions <m>f</m> and <m>g</m> are given.
+						Graphs of functions <m>f</m> and <m>g</m> are given.
             Identify which function is the derivative of the other.
           </p>
         </introduction>
@@ -1178,7 +1180,8 @@
                 math_ev3(g).' is the derivative of '.math_ev3(f).'.'
               ],0);
             </pg-code>
-            <statement>
+						<statement>
+						<p> Is f the derivative of g? </p>
               <image>
                 <shortdescription>f(x) is line with slope 2<var name="a"/> and x intercept <var name="h"/>. g(x) is a parabola with vertex at (<var name="h"/>,<var name="k"/>) </shortdescription>
                 <description>
diff --git a/ptx/sec_deriv_intro.ptx b/ptx/sec_deriv_intro.ptx
index e295c9a28..9979d5fcc 100644
--- a/ptx/sec_deriv_intro.ptx
+++ b/ptx/sec_deriv_intro.ptx
@@ -74,12 +74,12 @@
         \frac{f(2.5)-f(2)}{2.5-2} = \frac{50-86}{0.5} =-72\,\text{ft/s}
       </me>.
     </p>
-    <aside vshift="0">
+    <aside vshift="1">
       <title>Units in Calculations</title>
       <p>
-        In the above calculations,
-        we left off the units until the end of the problem.
-        You should always be sure that you label your answer with the correct units.
+        In our calculations of the difference quotients, we did not label the 
+				units until giving the final answer. Be sure to always label your answer
+				with the correct units.
         For example, if <m>g(x)</m> gave you the cost
         (in $)
         of producing <m>x</m> widgets,
@@ -125,7 +125,7 @@
       <quantity><mag>-64</mag><unit base="foot"/><per base="second"/></quantity>.
     </p>
 
-    <figure xml:id="table_falling" vshift="0">
+    <figure xml:id="table_falling" vshift="1">
       <caption>Approximating the instantaneous velocity with average velocities over a small time period <m>h</m></caption>
       <tabular>
         <row bottom="medium">
@@ -395,6 +395,7 @@
 
     <p>
       Some examples will help us understand these definitions.
+			<enlarge-page skipsize="2"/>
     </p>
 
     <example xml:id="ex_derv_point1">
@@ -482,7 +483,7 @@
           along with the tangent lines at <m>x=1</m> and <m>x=3</m>.
         </p>
 
-        <figure xml:id="fig_tangent1" vshift="3">
+        <figure xml:id="fig_tangent1" vshift="3.5">
           <caption>A graph of <m>f(x) = 3x^2+5x-7</m> and its tangent lines at <m>x=1</m> and <m>x=3</m></caption>
           <!-- START figures/fig_tangent1.tex -->
           <image width="47%">
@@ -594,7 +595,7 @@
           <em>look</em> perpendicular.
         </p>
 
-        <figure xml:id="fig_normal1" vshift="0">
+        <figure xml:id="fig_normal1" vshift="1">
           <caption>A graph of <m>f(x)=3x^2+5x-7</m>, along with its normal line at <m>x=1</m></caption>
           <!-- START figures/fig_normal1.tex -->
           <image>
@@ -754,7 +755,7 @@
           The graph seems to imply the approximation is rather good.
         </p>
 
-        <figure xml:id="fig_tangentsinx" vshift="2">
+        <figure xml:id="fig_tangentsinx" vshift="2.5">
           <caption><m>f(x) = \sin(x)</m> graphed with an approximation to its tangent line at <m>x=0</m></caption>
           <!-- START figures/fig_tangentsinx.tex -->
           <image>
@@ -910,6 +911,7 @@
 
     <p>
       Examples will help us understand this definition.
+			<enlarge-page skipsize="2"/>
     </p>
 
     <example xml:id="ex_deriv1">
@@ -1009,7 +1011,7 @@
             \text{Product/sum limit rules}</mrow>
             <mrow>\amp = \sin(x)\cdot 0 + \cos(x) \cdot 1 \amp \amp  \text{Applied }
             <xref ref="thm_special_limits"/></mrow>
-            <mrow>\amp = \cos(x). \amp \amp \text{(Are you surprised?)}</mrow>
+            <mrow>\amp = \cos(x). \amp \amp </mrow>
           </md>
         </p>
 
@@ -1173,7 +1175,7 @@
           So <m>f(x) = \abs{x}</m> is differentiable everywhere except at <m>0</m>.
         </p>
 
-        <figure xml:id="fig_absolutevalueprime" vshift="2">
+        <figure xml:id="fig_absolutevalueprime" vshift="3">
           <caption>A graph of the derivative of <m>f(x) = \abs{x}</m></caption>
           <!-- START figures/fig_absolutevalueprime.tex -->
           <image width="47%">
@@ -1331,7 +1333,7 @@
           for a graph of this derivative function.
         </p>
 
-        <figure xml:id="fig_piecewisecosx1" vshift="5">
+        <figure xml:id="fig_piecewisecosx1" vshift="3.5">
           <caption>A graph of <m>\fp(x)</m> in <xref ref="ex_diff_piecewise"/>.
           </caption>
           <!-- START figures/fig_piecewisecosx1.tex -->
@@ -1511,7 +1513,13 @@
           we conclude <m>g</m> is differentiable on its domain of <m>[0,\infty)</m>.
         </p>
 
-        <figure xml:id="fig_diff_closed1" vshift="0">
+        <p>
+          We state (without proof) that <m>\gp(x) = 3\sqrt{x}/2</m>.
+          Note that <m>\lim_{x\to0^+}\gp(x) = 0</m>;
+          again, this limit is easier to evaluate than the limit of the difference quotient.
+        </p>
+
+<figure xml:id="fig_diff_closed1" vshift="-0.5">
           <caption>A graph of <m>y=x^{1/2}</m> and <m>y=x^{3/2}</m> in <xref ref="ex_diff_closed1"/></caption>
           <image width="47%">
             <shortdescription>
@@ -1545,13 +1553,7 @@
             </latex-image>
           </image>
         </figure>
-
-        <p>
-          We state (without proof) that <m>\gp(x) = 3\sqrt{x}/2</m>.
-          Note that <m>\lim_{x\to0^+}\gp(x) = 0</m>;
-          again, this limit is easier to evaluate than the limit of the difference quotient.
-        </p>
-
+				
         <p>
           The two functions are graphed in <xref ref="fig_diff_closed1"/>.
           Note how <m>f(x) = \sqrt{x}</m> seems to <q>go vertical</q>
diff --git a/ptx/sec_deriv_inverse_function.ptx b/ptx/sec_deriv_inverse_function.ptx
index ba1ef67d0..31681abf1 100644
--- a/ptx/sec_deriv_inverse_function.ptx
+++ b/ptx/sec_deriv_inverse_function.ptx
@@ -65,7 +65,7 @@
     whatever we know about <m>f</m> can quickly be transferred into knowledge about <m>g</m>.
   </p>
 
-  <figure xml:id="fig_inverse1" vshift="-3">
+  <figure xml:id="fig_inverse1" vshift="3">
     <caption>A function <m>f</m> along with its inverse <m>f^{-1}</m>. (Note how it does not matter which function we refer to as <m>f</m>; the other is <m>f^{-1}</m>.)</caption>
       <!-- START figures/fig_inverse1.tex -->
     <image width="47%">
@@ -127,7 +127,7 @@
     This then tells us that <m>g'(b)=1/\fp(a)</m>.
   </p>
 
-  <figure xml:id="fig_inverse2" vshift="-5">
+  <figure xml:id="fig_inverse2" vshift="0">
     <caption>Corresponding tangent lines drawn to <m>f</m> and <m>f^{-1}</m></caption>
       <!-- START figures/fig_inverse2.tex -->
     <image width="47%">
@@ -191,6 +191,10 @@
         <cell><m>(1.5,1)</m> lies on <m>g</m></cell>
       </row>
       <row>
+        <cell> </cell>
+        <cell> </cell>
+      </row>
+			<row>
         <cell>
           <p>
             Slope of tangent line to <m>f</m> at <m>x=1</m> is <m>3</m>
@@ -203,6 +207,10 @@
         </cell>
       </row>
       <row>
+        <cell> </cell>
+        <cell> </cell>
+      </row>
+			<row>
         <cell><m>\fp(1) = 3</m></cell>
         <cell><m>g'(1.5) = 1/3</m></cell>
       </row>
@@ -629,8 +637,6 @@
     intended to be a reference for further work.
   </p>
 
-  <pagebreak-latex/>
-  
   <theorem xml:id="thm_deriv_glossary" hstretch="420">
     <title>Glossary of Derivatives of Elementary Functions</title>
     <statement>
diff --git a/ptx/sec_deriv_prodquot.ptx b/ptx/sec_deriv_prodquot.ptx
index e3ff3b7cd..9d7f75e88 100644
--- a/ptx/sec_deriv_prodquot.ptx
+++ b/ptx/sec_deriv_prodquot.ptx
@@ -85,7 +85,7 @@
         it helps validate its truth.
       </p>
 
-      <figure xml:id="fig_5xsquaredsinx" vshift="10">
+      <figure xml:id="fig_5xsquaredsinx" vshift="3">
         <caption>A graph of <m>y = 5x^2\sin(x)</m> and its tangent line at <m>x=\pi/2</m></caption>
           <!-- START figures/fig_5xsquaredsinx.tex -->
         <image width="47%">
@@ -156,7 +156,7 @@
       then do some regrouping as shown.
     </p>
 
-    <aside xml:id="aside-derivative-add-zero" vshift="0">
+    <aside xml:id="aside-derivative-add-zero" vshift="0.9">
       <p>
         Adding <m>0</m> in some clever form is a common mathematical proof technique.
       </p>
@@ -164,7 +164,7 @@
     
     <p>
       <md>
-        <mrow>\amp\lzoo{x}{f(x)g(x)} = \lim_{h\to0} \frac{f(x+h)g(x+h)-f(x)g(x)}{h}</mrow>
+        <mrow>\lzoo{x}{f(x)g(x)} \amp= \lim_{h\to0} \frac{f(x+h)g(x+h)-f(x)g(x)}{h}</mrow>
         <mrow>\amp \text{ (now add 0 to the numerator) }</mrow>
         <mrow>\amp = \lim_{h\to0} \frac{f(x+h)g(x+h)-f(x)g(x+h)+f(x)g(x+h)-f(x)g(x)}{h}</mrow>
         <mrow>\amp \text{ (regroup) }</mrow>
@@ -460,7 +460,7 @@
         is graphed in <xref ref="fig_tanx"/>.
       </p>
 
-      <figure xml:id="fig_tanx" vshift="0">
+      <figure xml:id="fig_tanx" vshift="3">
         <caption>A graph of <m>y=\tan(x)</m> along with its tangent line at <m>x=\pi/4</m></caption>
         <!-- START figures/fig_tanx.tex -->
         <image width="47%">
diff --git a/ptx/sec_differentials.ptx b/ptx/sec_differentials.ptx
index 9daf11e27..ea9cbc9b2 100644
--- a/ptx/sec_differentials.ptx
+++ b/ptx/sec_differentials.ptx
@@ -159,20 +159,22 @@
     Clearly, <m>f(c) = \ell(c)</m>.
     Let <m>\dx</m> be a small number,
     representing a small change in the <m>x</m>-value.
-    We assert that:
+    We assert that
     <me>
       f(c+\dx) \approx \ell(c+\dx)
     </me>,
     since the tangent line to a function approximates well the values of that function near <m>x=c</m>.
-    This tangent line approximation is used frequently enough in applications that we give it a name.
+    In fact, this tangent line is the linear function that best approximates the value of <m>f(x)</m> for <m>x</m> near <m>c</m>. Because of its value in applications, we give it a name.
   </p>
 
   <definition xml:id="def-linearization">
     <statement>
       <p>
-        The function <m>\ell(x)</m> is often referred to as the <term>linearization</term>,
-        or <term>linear approximation</term> of <m>f</m> at <m>c</m>.
-        It is the linear function that best approximates the value of <m>f(x)</m> when <m>x</m> is close to <m>c</m>.
+        Let <m>f</m> be differentiable on an interval <m>I</m> containing <m>c</m>. The function <m>\ell(x) = \fp(c)(x-c)+f(c)</m> is the <term>linearization</term>,
+        or <term>linear approximation</term>, of <m>f</m> at <m>c</m>.
+        <!--
+				It is the linear function that best approximates the value of <m>f(x)</m> when <m>x</m> is close to <m>c</m>.
+				-->
         <idx><h>linearization</h></idx>
         <idx><h>approximation</h><h>tangent line</h></idx>
         <idx><h>approximation</h><h>linear</h></idx>
@@ -253,7 +255,7 @@
     It is helpful to organize our new concepts and notations in one place.
   </p>
 
-  <aside vshift="0">
+  <aside vshift="3.5">
     <title>Differentials and linearization</title>
     <p>
       The relationship between the differential and the linearization given in
@@ -408,15 +410,26 @@
     studied in introductory Differential Equations courses.
   </p>
 
-  <aside vshift="0">
-    <title>PID controllers</title>
+  <aside vshift="3">
+    <title>Applications of Differentials</title>
     <p>
-      Another place differentials are used is in a PID controller,
+      Differentials are used within <q>Proportional Integral Derivative</q> (PID) controllers which use both integral and differential calculus to accurately control a process.
+		</p>
+		<p>
+			For instance, consider the task of steering a self<ndash/>driving car. If the vehicle drifts to the left, how much should the wheel be turned to the right to correct the path? Too little and the problem may not be corrected at all; too much and the car will overshoot its target and extreme corrective action will need to be taken. PID controllers consider the <em>rate</em> at which the drift and corrective measures take effect.
+		</p>
+		<p>
+		Common, everyday applications of PID controllers are in automotive cruise-control and maintaining espresso machine temperature.	
+		</p>	
+			<!--
+			Another place differentials are used is in a PID controller,
       which stands for <q>Proportional Integral Derivative</q>.
-      A PID controller uses concepts of both derivative and integral calculus to very accurately control a process
+      A PID controller uses concepts of both derivative and integral calculus to very accurately control a process.
+		</p>
+			
       (such as maintaining a stable temperature on an espresso machine).
-    </p>
-  </aside>
+			-->
+    </aside>
   <p>
     We use differentials once more to approximate the value of a function.
     Even though calculators are very accessible,
diff --git a/ptx/sec_graph_concavity.ptx b/ptx/sec_graph_concavity.ptx
index bbd7fd02f..5037ae996 100644
--- a/ptx/sec_graph_concavity.ptx
+++ b/ptx/sec_graph_concavity.ptx
@@ -56,6 +56,26 @@
       </statement>
     </definition>
 
+    <aside vshift="4.5">
+      <title>Alternate Terminology</title>
+      <p>
+        We often state that <q><m>f</m> is concave up</q>
+        instead of <q>the graph of <m>f</m> is concave up</q> for simplicity.
+      </p>
+    </aside>
+
+    <aside vshift="1">
+      <title>Geometric Concavity</title>
+      <p>
+        Geometrically speaking,
+        a function is concave up if its graph lies below its secant line segments and above its tangent lines; see <xref ref="fig-concavity-defn1"/> and <xref ref="fig_concavity1"/>.
+			</p>
+			<p>
+        A function is concave down if its graph lies above its secant line segments and below its tangent lines; see <xref ref="fig-concavity-defn2"/> and <xref ref="fig_concavity2"/>.
+      </p>
+    </aside>
+
+
     <p>
       Geometrically, the condition in <xref ref="eqn-concave-up">Equation</xref>
       states that a graph is concave up if the midpoint of the secant line from <m>(a,f(a))</m>
@@ -130,13 +150,6 @@
     </figure>
 
 
-    <aside vshift="1">
-      <title>Loose Language</title>
-      <p>
-        We often state that <q><m>f</m> is concave up</q>
-        instead of <q>the graph of <m>f</m> is concave up</q> for simplicity.
-      </p>
-    </aside>
 
     <p>
       Consider a function <m>f</m> such that <m>f</m> is continuous on <m>[a,b]</m> and differentiable on <m>(a,b)</m>.
@@ -204,9 +217,9 @@
       </statement>
     </theorem>
 
-    <aside vshift="9">
+    <aside vshift="6">
       <p>
-        As with <xref ref="thm_incr_decr"/>, <xref ref="thm-concave-derivative"/>
+        <xref ref="thm-concave-derivative"/>
         lets us conclude that the graph of a function is concave up (or down)
         on a <em>closed</em> interval, assuming that the function is continuous on that interval.
         Again, we follow the convention that when a problem asks us to give the intervals on which the graph is concave up or down,
@@ -237,7 +250,7 @@
       value of <m>\fp</m>.
     </p>
 
-    <figure xml:id="fig_concavity1" vshift="3">
+    <figure xml:id="fig_concavity1" vshift="0">
       <caption>A function <m>f</m> with a concave up graph. Notice how the slopes of the tangent lines, when looking from left to right, are increasing. (The slope values pictured are <m>-12, -6, 6 </m>  and <m>12</m>).</caption>
       <!-- START figures/fig_concavity1.tex -->
       <image width="47%">
@@ -304,7 +317,7 @@
       value of <m>\fp</m>.
     </p>
 
-    <figure xml:id="fig_concavity2" vshift="-1">
+    <figure xml:id="fig_concavity2" vshift="-2.5">
       <caption>A function <m>f</m> with a concave down graph. Notice how the slopes of the tangent lines, when looking from left to right, are decreasing.</caption>
       <!-- START figures/fig_concavity2.tex -->
       <image width="47%">
@@ -346,7 +359,7 @@
       then the <em>rate</em> of decrease is decreasing.
       The function is decreasing at a faster and faster rate.
     </p>
-    <aside vshift="-6">
+    <aside vshift="-7">
       <title>Concavity Depravity</title>
       <p>
         A mnemonic for remembering what concave up/down means is:
@@ -464,14 +477,6 @@
       </sbsgroup>
     </figure>
 
-    <aside vshift="3">
-      <title>Geometric Concavity</title>
-      <p>
-        Geometrically speaking,
-        a function is concave up if its graph lies above its tangent lines and below secant line segments.
-        A function is concave down if its graph lies below its tangent lines and above secant line segments.
-      </p>
-    </aside>
     <p>
       If knowing where a graph is concave up/down is important,
       it makes sense that the places where the graph changes from one to the other is also important.
@@ -495,7 +500,7 @@
       shows a graph of a function with inflection points labeled.
     </p>
 
-    <figure xml:id="fig_concavity4" vshift="1">
+    <figure xml:id="fig_concavity4" vshift="2">
       <caption>A graph of a function with its inflection points marked. The intervals where concave up/down are also indicated.</caption>
       <!-- START figures/fig_concavity4.tex -->
       <image width="47%">
@@ -597,7 +602,7 @@
           <m>\fpp(x)\lt 0</m> and concave up when <m>\fpp(x)\gt 0</m>.
         </p>
 
-        <figure xml:id="fig_concline1" vshift="4">
+        <figure xml:id="fig_concline1" vshift="7">
           <caption>A number line determining the concavity of <m>f</m> in <xref ref="ex_conc1"/></caption>
           <!-- START figures/fig_concline1.tex -->
 
@@ -633,7 +638,7 @@
           <!-- figures/fig_concline1.tex END -->
         </figure>
 
-        <figure xml:id="fig_conc1" vshift="-1">
+        <figure xml:id="fig_conc1" vshift="2">
           <caption>A graph of <m>f(x)</m> used in <xref ref="ex_conc1"/></caption>
           <!-- START figures/fig_conc1.tex -->
           <image width="47%">
@@ -758,7 +763,7 @@
           </dl>
         </p>
 
-        <figure xml:id="fig_concline2" vshift="10">
+        <figure xml:id="fig_concline2" vshift="8">
           <caption>Number line for <m>f</m> in <xref ref="ex_conc2"/></caption>
 
           <!-- START figures/fig_concline2.tex -->
@@ -836,7 +841,7 @@
           The inflection in <m>f</m> occurs where <m>\fpp</m> changes sign.
         </p>
 
-        <figure xml:id="fig_conc2" vshift="7">
+        <figure xml:id="fig_conc2" vshift="5">
           <caption>A graph of <m>f(x)</m> and <m>\fpp(x)</m> in <xref ref="ex_conc2"/></caption>
           <!-- START figures/fig_conc2.tex -->
           <image width="47%">
@@ -909,7 +914,7 @@
           Find the point at which sales are decreasing at their greatest rate.
         </p>
 
-        <figure xml:id="fig_conc3" vshift="1">
+        <figure xml:id="fig_conc3" vshift="2">
           <caption>A graph of <m>S(t)</m> in <xref ref="ex_conc3"/>, modeling the sale of a product over time</caption>
           <!-- START figures/fig_conc3.tex -->
           <image width="47%">
@@ -971,7 +976,7 @@
           so the decline in sales is <q>leveling off.</q>
         </p>
 
-        <figure xml:id="fig_conc3b" vshift="5">
+        <figure xml:id="fig_conc3b" vshift="3">
           <caption>A graph of <m>S(t)</m> in <xref ref="ex_conc3"/>, along with <m>S'(t)</m></caption>
           <!-- START figures/fig_conc3b.tex -->
           <image width="47%">
@@ -1005,7 +1010,7 @@
           <!-- figures/fig_conc3.tex END -->
         </figure>
       </solution>
-      <!-- <solution component="video" vshift="2">
+      <!-- <solution component="video" vshift="0">
         <title>Video solution</title>
         <video width="98%" youtube="yjcSXaGkL5Y" label="vid_graphs_concav_ex_3" component="video"/>
       </solution> -->
@@ -1025,7 +1030,7 @@
       as shown in <xref ref="fig_concavity5"/>.
     </p>
 
-    <figure xml:id="fig_concavity5" vshift="1.5">
+    <figure xml:id="fig_concavity5" vshift="-.5">
       <caption>A graph of <m>f(x) = x^4</m>. Clearly <m>f</m> is always concave up, despite the fact that <m>\fpp(x) = 0</m> when <m>x=0</m>. In this example, the <em>possible</em> point of inflection <m>(0,0)</m> is not a point of inflection.</caption>
       <!-- START figures/fig_concavity5.tex -->
       <image width="47%">
@@ -1066,7 +1071,7 @@
       See <xref ref="fig_concavity6"/> for a visualization of this.
     </p>
 
-    <figure xml:id="fig_concavity6" vshift="-1.5">
+    <figure xml:id="fig_concavity6" vshift="-4">
       <caption>Demonstrating the fact that relative maxima occur when the graph is concave down and relative minima occur when the graph is concave up</caption>
       <!-- START figures/fig_concavity6.tex -->
       <image width="47%">
@@ -1169,7 +1174,7 @@
           These results are confirmed in <xref ref="fig_conc4"/>.
         </p>
 
-        <figure xml:id="fig_conc4" vshift="-6">
+        <figure xml:id="fig_conc4" vshift="-2">
           <caption>A graph of <m>f(x)</m> in <xref ref="ex_conc4"/>. The second derivative is evaluated at each critical point. When the graph is concave up, the critical point represents a local minimum; when the graph is concave down, the critical point represents a local maximum.</caption>
           <!-- START figures/fig_conc4.tex -->
           <image width="47%">
@@ -1208,6 +1213,7 @@
       </solution>
     </example>
 
+<!--
     <aside vshift="-14">
       <title>Use Wisely</title>
       <p>
@@ -1220,8 +1226,19 @@
         <xref ref="thm_first_der" text="title"/> does not.
       </p>
     </aside>
+-->		
 
-    <p>
+      <p>
+        The second derivative test can only be used on a function that is twice differentiable at <m>c</m>.
+        For functions that are not twice differentiable at <m>c</m>,
+        you will need to use the <xref ref="thm_first_der" text="title"/>.
+        If you've already determined the sign diagram for <m>\fp</m>,
+        the <xref ref="thm_first_der" text="title"/> is usually
+        easier to apply, and it applies in cases when
+        <xref ref="thm_first_der" text="title"/> does not.
+      </p>
+			
+			<p>
       We have been learning how the first and second derivatives of a function relate information about the graph of that function.
       We have found intervals of increasing and decreasing,
       intervals where the graph is concave up and down,
diff --git a/ptx/sec_graph_extreme_values.ptx b/ptx/sec_graph_extreme_values.ptx
index e9ef5bc35..b3b20e80c 100644
--- a/ptx/sec_graph_extreme_values.ptx
+++ b/ptx/sec_graph_extreme_values.ptx
@@ -51,7 +51,7 @@
     </statement>
   </definition>
 
-  <aside xml:id="aside-graph-extreme-terminology" vshift="0">
+  <aside xml:id="aside-graph-extreme-terminology" vshift="2.5">
     <p>
       <em>Note:</em> The extreme values of a function are
       <q><m>y</m></q> values,
@@ -240,7 +240,7 @@
         Approximate the extreme values of <m>f</m>.
       </p>
 
-      <figure xml:id="fig_extval1" vshift="0">
+      <figure xml:id="fig_extval1" vshift="-.5">
         <caption>A graph of <m>f(x) = 2x^3-9x^2</m> as in <xref ref="ex_extval1"/></caption>
 
           <!-- START figures/fig_extval1.tex -->
@@ -300,12 +300,28 @@
     In short, a <q>relative max</q>
     is a <m>y</m>-value that's the largest <m>y</m>-value <q>nearby.</q>
   </p>
-
+<enlarge-page skipsize="3"/>
   <figure xml:id="vid_graphs_extreme_val_rel_extrema" component="video" vshift="-1">
     <caption>Video presentation of <xref ref="def_rel_ext"/></caption>
     <video youtube="nBZTnxn0vqQ" label="vid_graphs_extreme_val_rel_extrema"/>
   </figure>
 
+  <aside xml:id="aside-graph-extreme-vocabulary2" vshift="-2.5">
+    <title>Alternative Vocabulary</title>
+    <p>
+      The terms <em>local minimum</em>
+      and <em>local maximum</em>
+      are often used as synonyms for <em>relative minimum</em>
+      and <em>relative maximum</em>.
+    </p>
+
+    <p>
+      As it makes intuitive sense that an absolute maximum is also a relative maximum,
+      <xref ref="def_rel_ext"/>
+      allows a relative maximum to occur at an interval's endpoint.
+    </p>
+  </aside>
+
   <definition xml:id="def_rel_ext">
     <title>Relative Minimum and Relative Maximum</title>
     <statement>
@@ -344,21 +360,6 @@
     </statement>
   </definition>
 
-  <aside xml:id="aside-graph-extreme-vocabulary2" vshift="1">
-    <title>Alternative Vocabulary</title>
-    <p>
-      The terms <em>local minimum</em>
-      and <em>local maximum</em>
-      are often used as synonyms for <em>relative minimum</em>
-      and <em>relative maximum</em>.
-    </p>
-
-    <p>
-      As it makes intuitive sense that an absolute maximum is also a relative maximum,
-      <xref ref="def_rel_ext"/>
-      allows a relative maximum to occur at an interval's endpoint.
-    </p>
-  </aside>
 
   <p>
     We briefly practice using these definitions.
@@ -374,7 +375,7 @@
         At each of these points, evaluate <m>\fp</m>.
       </p>
 
-      <figure xml:id="fig_extval2" vshift="-1.5">
+      <figure xml:id="fig_extval2" vshift="-.5">
         <caption>A graph of <m>f(x) = (3x^4-4x^3-12x^2+5)/5</m> as in <xref ref="ex_extval2"/></caption>
           <!-- START figures/fig_extval2.tex -->
         <image width="47%">
@@ -431,7 +432,7 @@
         At each of these points, evaluate <m>\fp</m>.
       </p>
 
-      <figure xml:id="fig_extval3" vshift="-1.5">
+      <figure xml:id="fig_extval3" vshift="-1">
         <caption>A graph of <m>f(x) = (x-1)^{2/3}+2</m> as in <xref ref="ex_extval3"/></caption>
           <!-- START figures/fig_extval3.tex -->
         <image width="47%">
@@ -507,7 +508,7 @@
     </statement>
   </definition>
 
-  <aside xml:id="aside-graph-extreme-vocabulary3" vshift="1.25">
+  <aside xml:id="aside-graph-extreme-vocabulary3" vshift="1.75">
     <p>
       In this text we use <q>critical number</q> and <q>critical value</q>
       interchangeably. Other textbooks reserve the term <em>critical value</em>
@@ -542,9 +543,7 @@
     as illustrated in <xref ref="fig_extreme4"/>.
   </p>
 
-  <pagebreak-latex/>
-  
-  <figure xml:id="fig_extreme4" vshift="-2">
+<figure xml:id="fig_extreme4" vshift="2">
     <caption>A graph of <m>f(x)=x^3</m> which has a critical value of <m>x=0</m>, but no relative extrema</caption>
       <!-- START figures/fig_extreme4.tex -->
     <image width="47%">
@@ -570,6 +569,10 @@
       <!-- figures/fig_extreme4.tex END -->
   </figure>
 
+  <pagebreak-latex/>
+  
+  
+
   <p>
     <xref ref="thm_extreme_val"/>
     states that a continuous function on a closed interval will have both an absolute maximum and an absolute minimum.
@@ -628,7 +631,7 @@
         graphed in <xref ref="fig_extval4"/>.
       </p>
 
-      <figure xml:id="fig_extval4" vshift="0.5">
+      <figure xml:id="fig_extval4" vshift="0">
         <caption>A graph of <m>f(x) = 2x^3+3x^2-12x</m> on <m>[0,3]</m> as in <xref ref="ex_extval4"/></caption>
           <!-- START figures/fig_extval4.tex -->
         <image width="47%">
@@ -681,7 +684,7 @@
         the maximum value is <m>45</m> and the minimum value is <m>-7</m>.
       </p>
 
-      <figure xml:id="table_ext4" vshift="-5.5">
+      <figure xml:id="table_ext4" vshift="0">
         <caption>Finding the extreme values of <m>f(x)= 2x^3+3x^2-12x</m> in <xref ref="ex_extval4"/></caption>
         <tabular>
           <row bottom="medium">
@@ -1783,7 +1786,8 @@
         <introduction>
           <p>
             Find the extreme values of the function on the given interval.
-          </p>
+          <enlarge-page skipsize="2"/>
+					</p>
         </introduction>
         <exercise label="ex-graph-extreme-values-find-1">
           <webwork xml:id="webwork-ex-graph-extreme-values-find-1">
diff --git a/ptx/sec_graph_incr_decr.ptx b/ptx/sec_graph_incr_decr.ptx
index 43ebe2ae3..877322db5 100644
--- a/ptx/sec_graph_incr_decr.ptx
+++ b/ptx/sec_graph_incr_decr.ptx
@@ -24,7 +24,7 @@
     We formally define these terms here.
   </p>
 
-  <figure xml:id="fig_incr0" vshift="3">
+  <figure xml:id="fig_incr0" vshift="1.5">
     <caption>A graph of a function <m>f</m> used to illustrate the concepts of <em>increasing</em> and <em>decreasing</em></caption>
     <!-- START figures/fig_incr0.tex -->
     <image width="47%">
@@ -83,9 +83,9 @@
     </statement>
   </definition>
 
-  <aside vshift="-9">
+  <aside vshift="-4">
     <p>
-      <alert>Caution:</alert> the definition we give in <xref ref="def_incr_decr"/>
+      <alert>Note:</alert> the definition we give in <xref ref="def_incr_decr"/>
       is not the one you will find in formal mathematics textbooks.
       Such texts define a function to be increasing on <m>I</m> if,
       for every <m>a\lt b</m> in <m>I</m>, <m>f(a)\leq f(b)</m>.
@@ -265,7 +265,8 @@
     <m>c</m> between <m>a</m> and <m>b</m> where <m>\fp(c) = 0</m>.
     (It turns out that this is still true even if <m>\fp</m> is not continuous on <m>[a,b]</m>.)
     This leads us to the following method for finding intervals on which a function is increasing or decreasing.
-  </p>
+  <enlarge-page skipsize="2"/>
+	</p>
 
   <insight xml:id="idea_incr_decr">
     <title>Finding Intervals on Which <m>f</m> is Increasing or Decreasing</title>
@@ -381,7 +382,7 @@
         This is shown in <xref ref="fig_incrline1"/>.
       </p>
 
-      <figure xml:id="fig_incrline1" vshift="3">
+      <figure xml:id="fig_incrline1" vshift="2">
         <caption>Number line for <m>f</m> in <xref ref="ex_incr1"/></caption>
 
         <image width="47%">
@@ -460,7 +461,7 @@
         <xref ref="fig_incrline1a"/> summarizes our work.
       </p>
 
-      <figure xml:id="fig_incrline1a" vshift="3">
+      <figure xml:id="fig_incrline1a" vshift="8">
         <caption>Completed number line for <m>f</m> in <xref ref="ex_incr1"/></caption>
 
         <!-- START figures/fig_incrline1.tex -->
@@ -533,7 +534,7 @@
         <m>\fp\lt 0</m> when <m>f</m> is decreasing.
       </p>
 
-      <figure xml:id="fig_incr1" vshift="-2">
+      <figure xml:id="fig_incr1" vshift="4">
         <caption>A graph of <m>f(x)</m> in <xref ref="ex_incr1"/>, showing where <m>f</m> is increasing and decreasing</caption>
         <!-- START figures/fig_incr1.tex -->
         <image width="47%">
@@ -850,7 +851,7 @@
         As previously stated, <m>x=1</m> is a vertical asymptote of <m>f</m>.
       </p>
 
-      <figure xml:id="fig_incrline2" vshift="6">
+      <figure xml:id="fig_incrline2" vshift="8">
         <caption>Number line for <m>f</m> in <xref ref="ex_incr2"/></caption>
 
         <!-- START figures/fig_incrline2.tex -->
@@ -937,7 +938,7 @@
         again demonstrating that <m>f</m> is increasing when <m>\fp\gt 0</m> and decreasing when <m>\fp\lt 0</m>.
       </p>
 
-      <figure xml:id="fig_incr2" vshift="2">
+      <figure xml:id="fig_incr2" vshift="4">
         <caption>A graph of <m>f(x)</m> in <xref ref="ex_incr2"/>, showing where <m>f</m> is increasing and decreasing</caption>
         <!-- START figures/fig_incr2.tex -->
         <image width="47%">
@@ -1027,44 +1028,7 @@
         breaking the number line into four subintervals as shown in <xref ref="fig_incrline3"/>.
       </p>
 
-      <p>
-        <dl>
-          <li>
-            <title>Interval 1: <m>(\infty,-1)</m></title>
-            <p>
-              We choose <m>p=-2</m>; we can easily verify that <m>\fp(-2)\lt 0</m>.
-              So <m>f</m> is decreasing on <m>(-\infty,-1)</m>.
-            </p>
-          </li>
-          <li>
-            <title>Interval 2: <m>(-1,0)</m></title>
-            <p>
-              Choose <m>p=-1/2</m>.
-              Once more we practice finding the sign of <m>\fp(p)</m> without computing an actual value.
-              We have <m>\fp(p) = (8/3)p^{-1/3}(p-1)(p+1)</m>;
-              find the sign of each of the three terms at the chosen value of <m>p</m>.
-              <me>
-                \fp(p) = \frac{8}{3} \cdot \underbrace{p^{-\frac{1}{3}}}_{\lt 0}\cdot \underbrace{(p-1)}_{\lt 0}\underbrace{(p+1)}_{\gt 0}
-              </me>.
-            </p>
-
-            <p>
-              We have a <q>negative <times/> negative <times/> positive</q>
-              giving a positive number;
-              <m>f</m> is increasing on <m>(-1,0)</m>.
-            </p>
-          </li>
-          <li>
-            <title>Interval 3: <m>(0,1)</m></title>
-            <p>
-              We do a similar sign analysis as before,
-              using <m>p</m> in <m>(0,1)</m>.
-              <me>
-                \fp(p) = \frac{8}{3} \cdot \underbrace{p^{-\frac{1}{3}}}_{\gt 0}\cdot \underbrace{(p-1)}_{\lt 0}\underbrace{(p+1)}_{\gt 0}
-              </me>.
-            </p>
-
-            <figure xml:id="fig_incrline3" vshift="2">
+            <figure xml:id="fig_incrline3" vshift="1">
               <caption>Number line for <m>f</m> in <xref ref="ex_incr3"/></caption>
       
               <!-- START figures/fig_incrline3.tex -->
@@ -1143,6 +1107,45 @@
               <!-- figures/fig_incrline3.tex END -->
             </figure>
 
+
+      <p>
+        <dl>
+          <li>
+            <title>Interval 1: <m>(\infty,-1)</m></title>
+            <p>
+              We choose <m>p=-2</m>; we can easily verify that <m>\fp(-2)\lt 0</m>.
+              So <m>f</m> is decreasing on <m>(-\infty,-1)</m>.
+            </p>
+          </li>
+          <li>
+            <title>Interval 2: <m>(-1,0)</m></title>
+            <p>
+              Choose <m>p=-1/2</m>.
+              Once more we practice finding the sign of <m>\fp(p)</m> without computing an actual value.
+              We have <m>\fp(p) = (8/3)p^{-1/3}(p-1)(p+1)</m>;
+              find the sign of each of the three terms at the chosen value of <m>p</m>.
+              <me>
+                \fp(p) = \frac{8}{3} \cdot \underbrace{p^{-\frac{1}{3}}}_{\lt 0}\cdot \underbrace{(p-1)}_{\lt 0}\underbrace{(p+1)}_{\gt 0}
+              </me>.
+            </p>
+
+            <p>
+              We have a <q>negative <times/> negative <times/> positive</q>
+              giving a positive number;
+              <m>f</m> is increasing on <m>(-1,0)</m>.
+            </p>
+          </li>
+          <li>
+            <title>Interval 3: <m>(0,1)</m></title>
+            <p>
+              We do a similar sign analysis as before,
+              using <m>p</m> in <m>(0,1)</m>.
+              <me>
+                \fp(p) = \frac{8}{3} \cdot \underbrace{p^{-\frac{1}{3}}}_{\gt 0}\cdot \underbrace{(p-1)}_{\lt 0}\underbrace{(p+1)}_{\gt 0}
+              </me>.
+            </p>
+
+
             <p>
               We have two positive factors and one negative factor;
               <m>\fp(p)\lt 0</m> and so <m>f</m> is decreasing on <m>(0,1)</m>.
@@ -1173,7 +1176,7 @@
         highlighting once more that <m>f</m> is increasing when <m>\fp\gt 0</m> and is decreasing when <m>\fp\lt 0</m>.
       </p>
 
-      <figure xml:id="fig_incr3" vshift="-3">
+      <figure xml:id="fig_incr3" vshift="2">
         <caption>A graph of <m>f(x)</m> in <xref ref="ex_incr3"/>, showing where <m>f</m> is increasing and decreasing</caption>
         <!-- START figures/fig_incr3.tex -->
         <image width="47%">
diff --git a/ptx/sec_graph_mvt.ptx b/ptx/sec_graph_mvt.ptx
index bb559278d..32ef8f232 100644
--- a/ptx/sec_graph_mvt.ptx
+++ b/ptx/sec_graph_mvt.ptx
@@ -237,7 +237,7 @@
     Rolle's Theorem guarantees at least one; there may be more.
   </p>
 
-  <figure xml:id="fig_mvt3" vshift="4">
+  <figure xml:id="fig_mvt3" vshift="1">
     <caption>A graph of <m>f(x) = x^3-5x^2+3x+5</m>, where <m>f(a) = f(b)</m>. Note the existence of <m>c</m>, where <m>a\lt c\lt b</m>, where <m>\fp(c)=0</m>.</caption>
       <!-- START figures/fig_mvt3.tex -->
     <image width="47%">
@@ -299,7 +299,7 @@
       Let <m>f</m> be differentiable on <m>(a,b)</m> where <m>f(a)=f(b)</m>.
       We consider two cases.
     </p>
-    <case>
+    <case><title>Case 1</title>
       <p>
         Consider the case when <m>f</m> is constant on <m>[a,b]</m>;
         that is, <m>f(x) = f(a) = f(b)</m> for all <m>x</m> in <m>[a,b]</m>.
@@ -307,7 +307,7 @@
         showing there is at least one value <m>c</m> in <m>(a,b)</m> where <m>\fp(c)=0</m>.
       </p>
     </case>
-    <case>
+    <case><title>Case 2</title>
       <p>
         Now assume that <m>f</m> is not constant on <m>[a,b]</m>.
         The Extreme Value Theorem guarantees that <m>f</m> has a maximal and minimal value on <m>[a,b]</m>,
@@ -431,7 +431,7 @@
         Note how these lines are parallel (<ie/>, have the same slope) to the secant line.
       </p>
 
-      <figure xml:id="fig_mvt4" vshift="0">
+      <figure xml:id="fig_mvt4" vshift="3">
         <caption>Demonstrating the Mean Value Theorem in <xref ref="ex_mvt2"/></caption>
           <!-- START figures/fig_mvt4.tex -->
         <image width="47%">
diff --git a/ptx/sec_graph_sketch.ptx b/ptx/sec_graph_sketch.ptx
index d0af4683d..35fb31006 100644
--- a/ptx/sec_graph_sketch.ptx
+++ b/ptx/sec_graph_sketch.ptx
@@ -1000,7 +1000,7 @@
     (in fact, this is the method used for many graphs in this text).
     However, in regions where the graph is very <q>curvy,</q>
     this can generate noticeable sharp edges on the graph unless a large number of points are used.
-    High quality computer algebra systems,
+    High quality computer algebra systems (<init>CAS</init>),
     such as <pubtitle>Mathematica</pubtitle> and <pubtitle>Sage</pubtitle>,
     use special algorithms to plot lots of points only where the graph is <q>curvy.</q>
   </p>
@@ -1009,7 +1009,7 @@
     In <xref ref="fig_sage_sinx"/>,
     two graph of <m>y=\sin(x)</m> is given,
     generated by <pubtitle>Sage</pubtitle> and <pubtitle>Mathematica</pubtitle>.
-    The small points represent each of the places where each <init>CAS</init> sampled the function.
+    The small points represent each of the places where each <init>CAS</init>  sampled the function.
     Notice how at the <q>bends</q>
     of <m>\sin(x)</m>, lots of points are used;
     where <m>\sin(x)</m> is relatively straight, fewer points are used.
@@ -1019,7 +1019,7 @@
   </p>
 
   <figure xml:id="fig_sage_sinx">
-    <caption>CAS plots of <m>y=\sin(x)</m> illustrating the sample points</caption>
+    <caption><init>CAS</init> plots of <m>y=\sin(x)</m> illustrating the sample points</caption>
     <sidebyside widths="47% 47%" valign="bottom" margins="0%">
       <figure>
         <caption><pubtitle>Sage</pubtitle> output</caption>
diff --git a/ptx/sec_lhopitals_rule.ptx b/ptx/sec_lhopitals_rule.ptx
index fe63bf38b..bdc088b09 100644
--- a/ptx/sec_lhopitals_rule.ptx
+++ b/ptx/sec_lhopitals_rule.ptx
@@ -50,7 +50,7 @@
   </introduction>
 
   <subsection xml:id="sec_lhopital_statement">
-    <title>L'Hospital's Rule with indeterminate forms <m>0/0</m> and <m>\infty/\infty</m></title>
+    <title>L'Hospital's Rule with Indeterminate Forms <m>0/0</m> and <m>\infty/\infty</m></title>
     <theorem xml:id="thm_LHR">
       <title>L'Hospital's Rule, Part 1</title>
       <statement>
@@ -123,7 +123,9 @@
             </li>
           </ol>
         </p>
+			<pagebreak-latex/>	
       </statement>
+			
       <solution>
         <p>
           <ol>
diff --git a/ptx/sec_limit_analytically.ptx b/ptx/sec_limit_analytically.ptx
index 9d38baf19..9f9a33109 100644
--- a/ptx/sec_limit_analytically.ptx
+++ b/ptx/sec_limit_analytically.ptx
@@ -199,7 +199,7 @@
                 <mrow>\amp = 3\bigl(\lim_{x\to 2}x\bigr)^2-5\lim_{x\to 2}(x) +7</mrow>
                 <mrow>\amp = 3\cdot 2^2 - 5\cdot 2+7</mrow>
                 <mrow>\amp = 9</mrow>
-              </md>
+              </md>.
             </p>
           </li>
         </ol>
@@ -345,10 +345,7 @@
 
           <li xml:id="li_root_limit"><m>\lim_{x\to c}\sqrt[n]{x} = \sqrt[n]{c}</m></li>
         </ol>
-        (<xref ref="li_root_limit"/> follows from the
-        <xref ref="identity-limit-rule" text="title"/> and
-        <xref ref="root-limit-rule" text="title"/> rules.)
-      </p>
+			</p>
     </statement>
   </theorem>
 
@@ -478,7 +475,33 @@
     That is what the Squeeze Theorem states.
     This is illustrated in <xref ref="fig-squeeze-theorem"/>.
   </p>
-  <figure xml:id="fig-squeeze-theorem" vshift="-1">
+  
+  <theorem xml:id="thm_sqz">
+    <title>Squeeze Theorem</title> <statement>
+      <p>
+        Let <m>f</m>,
+        <m>g</m> and <m>h</m> be functions on an open interval <m>I</m>
+        containing <m>c</m> such that for all <m>x</m> in <m>I</m>,
+        <me>
+          f(x)\leq g(x) \leq h(x)
+        </me>.
+
+              <idx><h>limit</h><h>Squeeze Theorem</h></idx>
+              <idx><h>Squeeze Theorem</h></idx>
+
+        If
+        <me>
+          \lim_{x\to c} f(x) = L = \lim_{x\to c} h(x)
+        </me>,
+        then
+        <me>
+          \lim_{x\to c} g(x) = L
+        </me>.
+      </p>
+    </statement>
+  </theorem>
+	
+	<figure xml:id="fig-squeeze-theorem" vshift="2">
     <caption>An illustration of the Squeeze Theorem</caption>
     <image width="47%">
       <description>
@@ -519,32 +542,7 @@
       </latex-image>
     </image>
   </figure>
-
-  <theorem xml:id="thm_sqz">
-    <title>Squeeze Theorem</title> <statement>
-      <p>
-        Let <m>f</m>,
-        <m>g</m> and <m>h</m> be functions on an open interval <m>I</m>
-        containing <m>c</m> such that for all <m>x</m> in <m>I</m>,
-        <me>
-          f(x)\leq g(x) \leq h(x)
-        </me>.
-
-              <idx><h>limit</h><h>Squeeze Theorem</h></idx>
-              <idx><h>Squeeze Theorem</h></idx>
-
-        If
-        <me>
-          \lim_{x\to c} f(x) = L = \lim_{x\to c} h(x)
-        </me>,
-        then
-        <me>
-          \lim_{x\to c} g(x) = L
-        </me>.
-      </p>
-    </statement>
-  </theorem>
-
+	
   <figure xml:id="vid_limit_analytic_squeeze_thm" component="video" vshift="-2">
     <caption>Explaining the Squeeze Theorem</caption>
     <video youtube="8Tv-GRQdAVA" label="vid_limit_analytic_squeeze_thm"/>
diff --git a/ptx/sec_limit_continuity.ptx b/ptx/sec_limit_continuity.ptx
index 4e7a7b944..5feac2c0c 100644
--- a/ptx/sec_limit_continuity.ptx
+++ b/ptx/sec_limit_continuity.ptx
@@ -99,7 +99,7 @@
         Give the interval(s) on which <m>f</m> is continuous.
       </p>
 
-      <figure xml:id="fig_continuous1" vshift="1">
+      <figure xml:id="fig_continuous1" vshift="2">
         <caption>A graph of <m>f</m> in <xref ref="ex_contint1"/></caption>
         <!-- START figures/fig_continuous1.tex -->
         <image width="47%">
@@ -196,7 +196,37 @@
         Give the intervals on which <m>f</m> is continuous.
       </p>
 
-      <figure xml:id="fig_continuous2" vshift="0">
+      
+    </statement>
+    <solution>
+      <p>
+        We examine the three criteria for continuity.
+        <ol>
+          <li>
+            <p>
+              The limits <m>\lim\limits_{x\to c} f(x)</m>
+              do not exist at the jumps from one <q>step</q> to the next,
+              which occur at all integer values of <m>c</m>.
+              Therefore the limits exist for all <m>c</m> except when
+              <m>c</m> is an integer.
+            </p>
+          </li>
+          <li>
+            <p>
+              The function is defined for all values of <m>c</m>.
+            </p>
+          </li>
+          <li>
+            <p>
+              The limit <m>\lim\limits_{x\to c} f(x) = f(c)</m> for all values of
+              <m>c</m> where the limit exist,
+              since each step consists of just a line.
+            </p>
+          </li>
+        </ol>
+      </p>
+
+			<figure xml:id="fig_continuous2" vshift="0">
         <caption>A graph of the step function in <xref ref="ex_contint2"/></caption>
         <!-- START figures/fig_continuous2.tex -->
         <image width="47%">
@@ -234,34 +264,6 @@
         </image>
         <!-- figures/fig_continuous2.tex END -->
       </figure>
-    </statement>
-    <solution>
-      <p>
-        We examine the three criteria for continuity.
-        <ol>
-          <li>
-            <p>
-              The limits <m>\lim\limits_{x\to c} f(x)</m>
-              do not exist at the jumps from one <q>step</q> to the next,
-              which occur at all integer values of <m>c</m>.
-              Therefore the limits exist for all <m>c</m> except when
-              <m>c</m> is an integer.
-            </p>
-          </li>
-          <li>
-            <p>
-              The function is defined for all values of <m>c</m>.
-            </p>
-          </li>
-          <li>
-            <p>
-              The limit <m>\lim\limits_{x\to c} f(x) = f(c)</m> for all values of
-              <m>c</m> where the limit exist,
-              since each step consists of just a line.
-            </p>
-          </li>
-        </ol>
-      </p>
 
       <p>
         We conclude that <m>f</m> is continuous everywhere except at integer values of <m>c</m>.
@@ -281,7 +283,7 @@
     by considering the appropriate one-sided limits at the endpoints.
   </p>
 
-  <aside vshift="0">
+  <aside vshift="-7">
     <p>
       In this text, when we use the term <q>closed interval</q>,
       we mean an interval of the form <m>[a,b]</m>,
@@ -705,7 +707,7 @@
         and <xref ref="thm_continuous_functions" text="global"/> as appropriate.
       </p>
 
-      <figure xml:id="fig_continuous3" vshift="-1.5">
+      <figure xml:id="fig_continuous3" vshift="0">
         <caption>A graph of <m>f(x)=\sqrt{x-1}+\sqrt{5-x}</m></caption>
         <!-- START figures/fig_continuous3.tex -->
         <image width="47%">
@@ -789,7 +791,7 @@
     </solution>
   </example>
 
-
+<enlarge-page skipsize="4"/>
   <paragraphs>
     <title>Classification of discontinuities</title>
     <p>
diff --git a/ptx/sec_limit_def.ptx b/ptx/sec_limit_def.ptx
index 3e71ee97b..78201c695 100644
--- a/ptx/sec_limit_def.ptx
+++ b/ptx/sec_limit_def.ptx
@@ -806,10 +806,10 @@
         We can then set <m>\delta</m> to be the minimum of
         <m>\abs{\ln(1-\varepsilon)}</m> and <m>\ln(1+\varepsilon)</m>; <ie/>,
         <me>
-          \delta = \min\{\abs{\ln(1-\varepsilon)}, \ln(1+\varepsilon)\} = \ln(1+\varepsilon)
-        </me>.
+          \delta = \min\{\abs{\ln(1-\varepsilon)}, \ln(1+\varepsilon)\} = \ln(1+\varepsilon).\quad\text{(See marginal note.)}
+        </me>
       </p>
-      <aside xml:id="aside-limit-def-min-explain" vshift="2">
+      <aside xml:id="aside-limit-def-min-explain" vshift="1">
         <p>
         Recall <m>\ln 1= 0</m> and
         <m>\ln x\lt 0</m> when <m>0\lt x\lt 1</m>.
diff --git a/ptx/sec_limit_infty.ptx b/ptx/sec_limit_infty.ptx
index 97c0ca93e..6139bd721 100644
--- a/ptx/sec_limit_infty.ptx
+++ b/ptx/sec_limit_infty.ptx
@@ -29,7 +29,7 @@
       </me>.
     </p>
 
-    <figure xml:id="fig_oneoverxsquared" vshift="0">
+    <figure xml:id="fig_oneoverxsquared" vshift="2.5">
       <caption>Graphing <m>f(x) = 1/x^2</m> for values of <m>x</m> near 0</caption>
       <!-- START figures/fig_oneoverxsquared.tex -->
       <image width="47%">
@@ -145,6 +145,8 @@
       We define one-sided limits that approach infinity in a similar way.
     </p>
 
+    <pagebreak-latex/>
+
     <definition xml:id="def_onesided_limit_of_infinity">
       <title>One-Sided Limits of Infinity</title>
       <statement>
@@ -201,7 +203,7 @@
           Find <m>\lim\limits_{x\to 1}\frac1{(x-1)^2}</m> as shown in <xref ref="fig_inflim1"/>.
         </p>
 
-        <figure xml:id="fig_inflim1" vshift="6">
+        <figure xml:id="fig_inflim1" vshift="-2">
           <caption>Observing infinite limit as <m>x\to 1</m> in <xref ref="ex_inflim1"/></caption>
           <!-- START figures/fig_nolimit2.tex -->
           <image width="47%">
@@ -280,8 +282,8 @@
           as shown in <xref ref="fig_oneoverx"/>.
         </p>
 
-        <figure xml:id="fig_oneoverx" vshift="4">
-          <caption>Evaluating <m>\lim\limits_{x\to 0}\frac1x</m></caption>
+        <figure xml:id="fig_oneoverx" vshift="2">
+          <caption>Evaluating <m>\lim\limits_{x\to 0}\frac1x</m> in <xref ref="ex_inflim2"/></caption>
           <!-- START figures/fig_oneoverx.tex -->
           <image width="47%">
             <description>
@@ -391,7 +393,7 @@
           Thus the vertical asymptotes are at <m>x=\pm2</m>.
         </p>
 
-        <figure xml:id="fig_multipleasymptotes" vshift="-4">
+        <figure xml:id="fig_multipleasymptotes" vshift="2">
           <caption>Graphing <m>f(x) = \frac{3x}{x^2-4}</m></caption>
           <!-- START figures/fig_multipleasymptotes.tex -->
           <image width="47%">
@@ -470,7 +472,7 @@
       rather, a hole exists in the graph at <m>x=1</m>.
     </p>
 
-    <figure xml:id="fig_noasy" vshift="-4.5">
+    <figure xml:id="fig_noasy" vshift="-1">
       <caption>Graphically showing that <m>f(x) = \frac{x^2-1}{x-1}</m> does not have an asymptote at <m>x=1</m></caption>
       <!-- START figures/fig_noasy.tex -->
       <image width="47%">
diff --git a/ptx/sec_limit_intro.ptx b/ptx/sec_limit_intro.ptx
index edab7e8ae..048d413a9 100644
--- a/ptx/sec_limit_intro.ptx
+++ b/ptx/sec_limit_intro.ptx
@@ -130,7 +130,7 @@
       for this function simply by letting <m>x=0</m>.
     </p>
 
-    <figure xml:id="fig_sinx_over_x" vshift="0">
+    <figure xml:id="fig_sinx_over_x" vshift="-2">
       <caption><m>\sin(x)/x</m> near <m>x=0</m></caption>
       <!-- START figures/sinx_over_x_2.tex -->
       <image width="47%">
@@ -232,7 +232,7 @@
       we are only concerned with the values of the function when <m>x</m> is <em>near</em> 1.
     </p>
 
-    <figure xml:id="table_sinx_1" vshift="0">
+    <figure xml:id="table_sinx_1" vshift="1">
       <caption>Values of <m>\sin(x)/x</m> with <m>x</m> near <m>1</m></caption>
       <tabular>
         <row bottom="medium">
@@ -282,7 +282,7 @@
       only on the behavior of the function <em>near</em> <m>0</m>.
     </p>
 
-    <figure xml:id="table_sinx_2" vshift="-3">
+    <figure xml:id="table_sinx_2" vshift="-2.5">
       <caption>Values of <m>\sin(x)/x</m> with <m>x</m> near <m>0</m></caption>
       <tabular>
         <row bottom="medium">
@@ -761,7 +761,8 @@
         </sidebyside>
       </solution>
     </example>
-
+		
+		<enlarge-page skipsize="2"/>
     <example xml:id="ex_no_limit2">
       <title>The Function Grows Without Bound</title>
       <statement>
@@ -1055,7 +1056,7 @@
           </sidebyside>
         </figure>
 
-        <figure xml:id="table_nolimit3c" vshift="-1">
+        <figure xml:id="table_nolimit3c" vshift="2">
           <caption>Observing that <m>f(x)=\sin(1/x)</m> has no limit as <m>x\to0</m> in <xref ref="ex_no_limit3"/></caption>
           <tabular>
             <row bottom="medium">
@@ -1169,7 +1170,7 @@
       See <xref ref="fig_diffquot1"/>.
     </p>
 
-    <figure xml:id="fig_diffquot1" vshift="-2">
+    <figure xml:id="fig_diffquot1" vshift="3">
       <caption>Interpreting a difference quotient as the slope of a secant line</caption>
       <!-- START figures/fig_diff_quot1.tex -->
       <image width="47%">
@@ -1360,7 +1361,7 @@
       The table gives us reason to assume the value of the limit is about <m>8.5</m>.
     </p>
 
-    <figure xml:id="table_diff_quot_smallh" vshift="4">
+    <figure xml:id="table_diff_quot_smallh" vshift="2">
       <caption>The difference quotient evaluated at values of <m>h</m> near <m>0</m></caption>
       <tabular>
         <row bottom="medium">
diff --git a/ptx/sec_limit_onesided.ptx b/ptx/sec_limit_onesided.ptx
index dfe88f70b..59fd7e15e 100644
--- a/ptx/sec_limit_onesided.ptx
+++ b/ptx/sec_limit_onesided.ptx
@@ -525,7 +525,7 @@
         We can confirm our analytic result by consulting the graph of <m>f</m> shown in <xref ref="fig_onesidedb"/>.
         Note the open circles on the graph at <m>x=0</m>, <m>1</m> and <m>2</m>, where <m>f</m> is not defined.
       </p>
-      <figure xml:id="fig_onesidedb" vshift="6">
+      <figure xml:id="fig_onesidedb" vshift="4">
         <caption>A graph of <m>f</m> from <xref ref="ex_onesideb"/></caption>
         <!-- START figures/fig_one_sidedlimit2.tex -->
         <image width="47%">
@@ -596,7 +596,7 @@
         </ol>
       </p>
 
-      <figure xml:id="fig_onesidedc" vshift="4">
+      <figure xml:id="fig_onesidedc" vshift="2">
         <caption>Graphing <m>f</m> in <xref ref="ex_onesidec"/></caption>
         <!-- START figures/fig_one_sidedlimit3.tex -->
         <image width="47%">
diff --git a/ptx/sec_newton.ptx b/ptx/sec_newton.ptx
index ac9b45454..c50459a31 100644
--- a/ptx/sec_newton.ptx
+++ b/ptx/sec_newton.ptx
@@ -227,7 +227,7 @@
     </p>
   </insight>
 
-  <aside vshift="0">
+  <aside vshift="2">
     <title>Newton's Method is not Infallible</title>
     <p>
       The sequence of approximate values may not converge,
@@ -317,7 +317,7 @@
         and shows that Newton's Method can be robust enough that we do not have to make a very accurate initial approximation.
       </p>
 
-      <figure xml:id="fig_newt2" vshift="0">
+      <figure xml:id="fig_newt2" vshift="2">
         <caption>A graph of <m>f(x) = x^3-x^2-1</m> in <xref ref="ex_newt2"/></caption>
         <image width="47%">
           <shortdescription>
@@ -425,8 +425,8 @@
           xmin=-1.1,xmax=1.1%
           ]
           \addplot+ [infinite,domain=-.1:1] {cos(deg(x))-x};
-          \addplot+ [infinite,domain=-1:1] {cos(deg(x))};
-          \addplot+ [infinite,domain=-1:1] {x};
+          \addplot+ [infinite,domain=-1:1,dashed] {cos(deg(x))};
+          \addplot+ [infinite,domain=-1:1,dashed] {x};
           \addplot [guideline] coordinates {(0.739,0) (0.739,0.739)};
           \end{axis}
           \end{tikzpicture}
diff --git a/ptx/sec_optimization.ptx b/ptx/sec_optimization.ptx
index be9e93832..479960f12 100644
--- a/ptx/sec_optimization.ptx
+++ b/ptx/sec_optimization.ptx
@@ -253,9 +253,9 @@
         <li>
           <p>
             Create equations relevant to the context of the problem,
-            using the information given. (One of these should describe the
+            using the information given. One of these should describe the
             quantity to be optimized.
-            We'll call this the <em>fundamental equation.</em>)
+            We'll call this the <em>fundamental equation.</em>
           </p>
         </li>
 
@@ -264,8 +264,7 @@
             If the fundamental equation defines the quantity to be optimized as
             a function of more than one variable,
             reduce it to a single variable function using substitutions derived
-            from the other equations
-            (we'll call these <term>constraint</term> equations).
+            from the other equations, which we call the <term>constraint</term> equations.
           </p>
         </li>
 
@@ -284,9 +283,15 @@
 
         <li>
           <p>
-            Identify the values of all relevant quantities of the problem.
+            Identify the values of all relevant quantities of the problem and write a full sentence conclusion.
           </p>
         </li>
+				
+				<li>
+					<p>
+					
+					</p>
+				</li>
       </ol>
     </p>
   </insight>
@@ -933,7 +938,7 @@
           <statement>
             <p>
               A rancher has <m><var name="$total"/></m> feet of fencing in which to construct
-              adjacent, equally sized rectangular pens.
+              adjacent, equally sized rectangular pens as shown below.
               What dimensions should these pens have to maximize the enclosed area?
             </p>
             <image width="33%">
@@ -1032,7 +1037,7 @@
               <var name="$hU" width="20"/>
             </p>
             <p>
-              The <q>#10 can</q>is a standard sized can used by the restaurant
+              The <q>#10 can</q> is a standard sized can used by the restaurant
               industry that holds about
               <m><var name="$V"/></m> with a diameter of
               <m>6\,\frac{3}{16}\,\text{in}</m>
diff --git a/ptx/sec_related_rates.ptx b/ptx/sec_related_rates.ptx
index 221cae607..919c7a7f1 100644
--- a/ptx/sec_related_rates.ptx
+++ b/ptx/sec_related_rates.ptx
@@ -38,14 +38,24 @@
 
   </p>
 
+<!--
   <remark>
     <p>
       This section relies heavily on implicit differentiation,
       so referring back to <xref ref="sec_imp_deriv"/> may help.
     </p>
   </remark>
+-->
 
-  <p>
+	<aside vshift="1">
+    <title>Note</title>
+		<p>
+      This section relies heavily on implicit differentiation,
+      so referring back to <xref ref="sec_imp_deriv"/> may help.
+    </p>
+	</aside>	
+	
+	<p>
     We demonstrate the concepts of related rates through examples.
   </p>
 
@@ -124,49 +134,36 @@
 
     <p>
       <ol>
-        <li>
-          <p>
-            Read the problem carefully and identify the quantities that are
-            changing with time.
-            (There may be many quantities that change with time,
-            try to identify which variables are important to your goal and only
-            focus on these quantities.)
-          </p>
-        </li>
-
-        <li>
+        <li xml:id="li_rrmodel1">
           <p>
-            Draw a diagram
-            (if applicable)
-            and assign mathematical variables to each quantity that is changing
-            with time.
-            (If you are given a particular value of a quantity that is also
-            changing with time, do not include these values on your diagram.
-            We will call these <q>instantaneous values</q> of the variable.)
-          </p>
+            Understand the problem. Clearly identify the quantity whose rate of change you need to determine. Make a sketch if helpful.
+					</p>
         </li>
 
         <li xml:id="li_rrmodel">
-
           <p>
-            Relate the important variables using a mathematical model.
-            (Typical models are known formulas for area,
-            perimeter, the Pythagorean Theorem or Trigonometric Ratios.)
-            It may be necessary to use more than one technique
-            (such as similar triangles)
-            to reduce your model down to one that only involves the variables
-            of interest.
+            Identify other quantities relevant to the context of the problem 
+						and create an equation that relates them to the quantity identified 
+						in <xref ref="li_rrmodel1" text="local">Step</xref>. If values for
+						certain quantities are already known, do not substitute these values
+						into your equation yet. These <q> instantaneous values</q> will be
+						used in <xref ref="li_rrmodel4" text="local">Step</xref>. 
           </p>
+					<p>
+						<em>(continued<m>\ldots</m></em>
+						<pagebreak-latex/>
+					</p>
         </li>
 
         <li>
           <p>
             Implicitly differentiate both sides of the equation found in
-            <xref ref="li_rrmodel" text="local">Step</xref> with respect to <m>t</m>.
+            <xref ref="li_rrmodel" text="local">Step</xref> with respect 
+						to <m>t</m>.
           </p>
         </li>
 
-        <li>
+        <li xml:id="li_rrmodel4">
           <p>
             Substitute in the known values of rates and known instantaneous
             values of the variables.
@@ -175,7 +172,8 @@
 
         <li>
           <p>
-            Solve for the unknown rate.
+            Solve for the unknown rate identified in 
+						<xref ref="li_rrmodel1" text="local">Step</xref>.
           </p>
         </li>
 
@@ -417,7 +415,7 @@
         intersection of their two roads.
       </p>
 
-      <figure xml:id="fig_rr3" vshift="2">
+      <figure xml:id="fig_rr3" vshift="0">
         <caption>A sketch of a police car (at bottom) attempting to measure the
           speed of a car (at right) in <xref ref="ex_rr3"/></caption>
         <image width="47%">
@@ -509,7 +507,26 @@
         The radar measurement is <m>\lz{C}{t} = 20</m>.
         We want to find <m>\lz{B}{t}</m>.
       </p>
+			
+			<aside vshift="1">
+    <title>Practicality</title>
+    <p>
+      <xref ref="ex_rr3"/>
+      is both interesting and impractical.
+      It highlights the difficulty in using radar in a nonlinear fashion,
+      and explains why <q>in real life</q>
+      the police officer would follow the other driver to determine their speed,
+      and not pull out pencil and paper.
+    </p>
 
+    <p>
+      The principles here are important, though.
+      Many automated vehicles make judgments about other moving objects based on
+      perceived distances,
+      radar-like measurements and the concepts of related rates.
+    </p>
+  </aside>
+	
       <p>
         We have values for everything except <m>\lz{B}{t}</m>.
         Solving for this we have:
@@ -534,24 +551,7 @@
     </solution>
   </example>
 
-  <aside vshift="0">
-    <title>Practicality</title>
-    <p>
-      <xref ref="ex_rr3"/>
-      is both interesting and impractical.
-      It highlights the difficulty in using radar in a nonlinear fashion,
-      and explains why <q>in real life</q>
-      the police officer would follow the other driver to determine their speed,
-      and not pull out pencil and paper.
-    </p>
-
-    <p>
-      The principles here are important, though.
-      Many automated vehicles make judgments about other moving objects based on
-      perceived distances,
-      radar-like measurements and the concepts of related rates.
-    </p>
-  </aside>
+  
 
   <example xml:id="ex_rr4">
     <title>Studying related rates</title>
@@ -966,7 +966,7 @@
                 The officer is traveling due north at <m><var name="$speed"/>\,\text{mph}</m>
                 and is <m><var name="$f"/></m> mile from the intersection,
                 while the other car is <m>1</m> mile from the intersection
-                traveling west and the radar reading is <m>-<var name="$rate"/>\,\text{mph}</m>?
+                traveling west and the radar reading is <m>-<var name="$rate"/>\,\text{mph}</m>.
               </p>
               <p>
                 <var name="$a" width="20"/>
@@ -980,7 +980,7 @@
                 The officer is traveling due north at <m><var name="$speed"/>\,\text{mph}</m>
                 and is <m>1</m> mile from the intersection,
                 while the other car is <m><var name="$f"/></m> mile from the
-                intersection traveling west and the radar reading is <m>-<var name="$rate"/>\,\text{mph}</m>?
+                intersection traveling west and the radar reading is <m>-<var name="$rate"/>\,\text{mph}</m>.
               </p>
               <p>
                 <var name="$b" width="20"/>
@@ -1116,7 +1116,7 @@
           </pg-code>
           <introduction>
             <p>
-              An F-22 aircraft is flying at
+              An F-22 aircraft is flying at <m><var name="$spd"/>\,\text{mph}</m>
               <var name="$speedU"/>
               with an elevation of <m>100\,\text{ft}</m>
               on a straight-line path that will take it directly over an
@@ -1124,7 +1124,7 @@
               (note the lower elevation here).
             </p>
             <p>
-              How fast must the gun be able to turn to accurately track the
+              How fast (in radians per second) must the gun be able to turn to accurately track the
               aircraft when the plane is:
             </p>
           </introduction>
diff --git a/xsl/apex-latex-print-style.xsl b/xsl/apex-latex-print-style.xsl
index 8004cc00a..0c35b2eda 100644
--- a/xsl/apex-latex-print-style.xsl
+++ b/xsl/apex-latex-print-style.xsl
@@ -335,7 +335,7 @@ https://tex.stackexchange.com/questions/605955/can-i-avoid-indentation-of-margin
   </xsl:if>
 </xsl:template>
 
-<!-- fix this one damn display math that doesn't fit for 2 side printing -->
+<!-- fix this one darn display math that doesn't fit for 2 side printing -->
 <xsl:template match="md">
   <xsl:if test="(@hskip) and ($b-latex-two-sides)">
     <xsl:text>&#xa;&#xa;\noindent\hskip-</xsl:text>