active-calculus · mattboelkins · Jul 14, 2026 · Jul 10, 2026 · Jul 12, 2026 · Jul 12, 2026
diff --git a/.gitignore b/.gitignore
@@ -18,3 +18,4 @@ codechat_config.yaml
 settings.json
 logs
 .cache
+latex/
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diff --git a/publication/publication.ptx b/publication/publication.ptx
@@ -19,8 +19,8 @@
   <common>
     <!-- List Chapters and Sections in sidebar Table of Contents -->
     <tableofcontents level="2" />
-    <exercise-divisional statement="yes" hint="no" answer="no" solution="no"/>
-    <exercise-project statement="yes" hint="no" answer="no" solution="no" />
+    <exercise-divisional statement="yes" hint="no" answer="no" solution="yes"/>
+    <exercise-project statement="yes" hint="no" answer="no" solution="yes" />
   </common>
   <webwork server="https://webwork-ptx.aimath.org" />
 </publication>
diff --git a/source/activities/act-changing-aroc-graphs.xml b/source/activities/act-changing-aroc-graphs.xml
@@ -21,7 +21,7 @@
     </introduction>
     <task workspace="2.5cm" xml:id="act-changing-aroc-graphs-task-0">
       <statement>
-        <p><m>f</m> is a function defined on <m>[-1,7]</m> such that <m>f(1) = 4</m> and <m>AV_{[1,3]} = -2</m>. 
+        <p><m>f</m> is a function defined on <m>[-1,7]</m> such that <m>f(1) = 4</m> and <m>AV_{[1,3]} = -2</m>.
 					</p>
         <sidebyside width="40%">
           <image source="images/functions-y-x-blank-axes"><description><p>ADD ALT TEXT TO THIS IMAGE</p></description></image>
@@ -30,7 +30,20 @@
       </statement>
       <hint/>
       <answer/>
-      <solution/>
+      <solution>
+        <p>
+          Since <m>AV_{[1,3]} = -2</m> and <m>f(1) = 4</m>, we need
+          <m>\frac{f(3)-4}{3-1} = -2</m>, so <m>f(3) = 0</m>.  Any graph passing through
+          <m>(1,4)</m> and <m>(3,0)</m> satisfies the conditions.  Two examples:
+          the line <m>f(x) = -2x+6</m>, or a curve such as <m>f(x) = -(x-1)^2 + 4</m>
+          (which also passes through both required points).
+        </p>
+
+        <sidebyside widths="45% 45%">
+          <image source="solution-images/Activity1.3.4ai.png"/>
+          <image source="solution-images/Activity1.3.4aii.png"/>
+        </sidebyside>
+      </solution>
     </task>
     <task workspace="2.5cm" xml:id="act-changing-aroc-graphs-task-1">
       <statement>
@@ -43,7 +56,20 @@
       </statement>
       <hint/>
       <answer/>
-      <solution/>
+      <solution>
+        <p>
+          Since <m>AV_{[0,4]} = 0.5</m> and <m>g(4) = 3</m>, we need
+          <m>\frac{3 - g(0)}{4} = 0.5</m>, so <m>g(0) = 1</m>.  Any graph passing through
+          <m>(0,1)</m> and <m>(4,3)</m> that is not always increasing on <m>(0,4)</m>
+          satisfies the conditions.  For example, a curve that dips below the secant line
+          between <m>x=0</m> and <m>x=4</m> before returning to <m>(4,3)</m>.
+        </p>
+
+        <sidebyside widths="45% 45%">
+          <image source="solution-images/Activity1.3.4bi.png"/>
+          <image source="solution-images/Activity1.3.4bii.png"/>
+        </sidebyside>
+      </solution>
     </task>
     <task workspace="3cm" xml:id="act-changing-aroc-graphs-task-2">
       <statement>
@@ -56,7 +82,15 @@
       </statement>
       <hint/>
       <answer/>
-      <solution/>
+      <solution>
+        <p>
+          These conditions are impossible to satisfy simultaneously.  Given <m>h(2) = 5</m>
+          and <m>h(4) = 3</m>, the average rate of change is determined:
+          <md>AV_{[2,4]} = \frac{h(4) - h(2)}{4-2} = \frac{3-5}{2} = -1 \ne -2.</md>
+          No function can have both the specified output values and the specified average
+          rate of change on <m>[2,4]</m>.
+        </p>
+      </solution>
     </task>
     <answer>
       <p/>

diff --git a/source/activities/act-changing-aroc-population.xml b/source/activities/act-changing-aroc-population.xml
@@ -79,7 +79,24 @@
       </statement>
       <hint/>
       <answer/>
-      <solution/>
+      <solution>
+        <p>
+          For Kent County, the population in 1990 was 500,631 and in 2010 was 602,622, so
+          <md>
+            <mrow>AV_{[1990,2010]} &amp;= \frac{602{,}622 - 500{,}631}{2010 - 1990}</mrow>
+            <mrow>&amp;= \frac{101{,}991}{20}</mrow>
+            <mrow>&amp;\approx 5099.55 \text{ people per year.}</mrow>
+          </md>
+        </p>
+        <p>
+          For Ottawa County, the population in 1990 was 187,768 and in 2010 was 263,801, so
+          <md>
+            <mrow>AV_{[1990,2010]} &amp;= \frac{263{,}801 - 187{,}768}{2010 - 1990}</mrow>
+            <mrow>&amp;= \frac{76{,}033}{20}</mrow>
+            <mrow>&amp;\approx 3801.65 \text{ people per year.}</mrow>
+          </md>
+        </p>
+      </solution>
     </task>
     <task workspace="4cm" xml:id="act-changing-aroc-population-task-1">
       <statement>
@@ -89,7 +106,11 @@
       </statement>
       <hint/>
       <answer/>
-      <solution/>
+      <solution>
+        <p>
+          The units on <m>AV_{[1990,2010]}</m> for both counties are people per year.
+        </p>
+      </solution>
     </task>
     <task workspace="4cm" xml:id="act-changing-aroc-population-task-2">
       <statement>
@@ -99,7 +120,12 @@
       </statement>
       <hint/>
       <answer/>
-      <solution/>
+      <solution>
+        <p>
+          Between 1990 and 2010, the population of Ottawa County grew by an average of
+          approximately 3801.65 people per year.
+        </p>
+      </solution>
     </task>
     <task workspace="5cm" xml:id="act-changing-aroc-population-task-3">
       <statement>
@@ -110,7 +136,19 @@
       </statement>
       <hint/>
       <answer/>
-      <solution/>
+      <solution>
+        <p>
+          For Kent County on <m>[2000,2010]</m>:
+          <md>AV_{[2000,2010]} = \frac{602{,}622 - 574{,}336}{10} = \frac{28{,}286}{10} \approx 2828.6 \text{ people per year.}</md>
+          For Ottawa County on <m>[2000,2010]</m>:
+          <md>AV_{[2000,2010]} = \frac{263{,}801 - 238{,}313}{10} = \frac{25{,}488}{10} = 2548.8 \text{ people per year.}</md>
+          Kent County had a greater average rate of change during <m>[2000,2010]</m>.
+          In both counties, the population increased in every decade from 1960 to 2010,
+          so the average rate of change was positive on every decade interval for both counties.
+        </p>
+
+        <image width="50%" source="solution-images/Kent-Pop-and-Ottawa-Pop.png"/>
+      </solution>
     </task>
     <task workspace="5cm" xml:id="act-changing-aroc-population-task-4">
       <statement>
@@ -120,7 +158,14 @@
       </statement>
       <hint/>
       <answer/>
-      <solution/>
+      <solution>
+        <p>
+          Using the most recent decade's rate of change as a guide, Ottawa County grew at
+          <m>2548.8</m> people per year from 2000 to 2010.  Projecting this rate forward
+          eight years from 2010:
+          <md>263{,}801 + 8 \cdot 2548.8 \approx 284{,}191 \text{ people.}</md>
+        </p>
+      </solution>
     </task>
     <answer>
       <p/>

diff --git a/source/activities/act-changing-aroc-trends.xml b/source/activities/act-changing-aroc-trends.xml
@@ -30,7 +30,16 @@
       </statement>
       <hint/>
       <answer/>
-      <solution/>
+      <solution>
+        <p>
+          Computing the function values: <m>q(0)=0</m>, <m>q(1)=3</m>, <m>q(2)=4</m>,
+          <m>q(3)=3</m>, <m>q(4)=0</m>.  Thus:
+          <md>AV_{[0,1]} = \frac{3-0}{1} = 3, \quad AV_{[1,2]} = \frac{4-3}{1} = 1,</md>
+          <md>AV_{[2,3]} = \frac{3-4}{1} = -1, \quad AV_{[3,4]} = \frac{0-3}{1} = -3.</md>
+          Since <m>AV_{[2,3]}</m> and <m>AV_{[3,4]}</m> are both negative, <m>q</m> is
+          decreasing on <m>[2,4]</m>.
+        </p>
+      </solution>
     </task>
     <task workspace="4cm" xml:id="act-changing-aroc-trends-task-1">
       <statement>
@@ -43,7 +52,18 @@
       </statement>
       <hint/>
       <answer/>
-      <solution/>
+      <solution>
+        <p>
+          Computing function values: <m>h(-1) = 3 - 2(0.5)^{-1} = 3 - 4 = -1</m>,
+          <m>h(1) = 3 - 2(0.5) = 2</m>, <m>h(3) = 3 - 2(0.5)^3 = \frac{11}{4} \approx 2.75</m>,
+          <m>h(5) = 3 - 2(0.5)^5 = \frac{47}{16} \approx 2.9375</m>.  Thus:
+          <md>AV_{[-1,1]} = \frac{2-(-1)}{2} = \frac{3}{2} = 1.5,</md>
+          <md>AV_{[1,3]} = \frac{\frac{11}{4} - 2}{2} = \frac{3}{8} \approx 0.375,</md>
+          <md>AV_{[3,5]} = \frac{\frac{47}{16} - \frac{11}{4}}{2} = \frac{3}{32} \approx 0.094.</md>
+          All three average rates of change are positive but decreasing, so <m>h</m> is
+          increasing on <m>[-1,5]</m>, but at a decreasing rate.
+        </p>
+      </solution>
     </task>
     <task workspace="0.7cm" xml:id="act-changing-aroc-trends-task-2">
       <statement>
@@ -63,7 +83,20 @@
       </statement>
       <hint/>
       <answer/>
-      <solution/>
+      <solution>
+        <p>On the graph of <m>q</m>, draw four line segments: one connecting <m>(0,0)</m> and <m>(1,3)</m> (slope <m>3</m>), one connecting <m>(1,3)</m> and <m>(2,4)</m> (slope <m>1</m>), one connecting <m>(2,4)</m> and <m>(3,3)</m> (slope <m>-1</m>), and one connecting <m>(3,3)</m> and <m>(4,0)</m> (slope <m>-3</m>). On the graph of <m>h</m>, draw three line segments: one connecting <m>(-1,-1)</m> and <m>(1,2)</m> (slope <m>1.5</m>), one connecting <m>(1,2)</m> and <m>(3,2.75)</m> (slope <m>0.375</m>), and one connecting <m>(3,2.75)</m> and <m>(5,2.9375)</m> (slope <m>0.094</m>). The decreasing slopes on the left graph reflect that <m>q</m> increases then decreases, while the positive but decreasing slopes on the right reflect that <m>h</m> is increasing at a decreasing rate.</p>
+
+        <sidebyside widths="30% 30% 30%">
+          <image source="solution-images/Activity1.3.3ci.png"/>
+          <image source="solution-images/Activity1.3.3cii.png"/>
+          <image source="solution-images/Activity1.3.3ciii.png"/>
+        </sidebyside>
+        <sidebyside widths="30% 30% 30%">
+          <image source="solution-images/Activity1.3.3civ.png"/>
+          <image source="solution-images/Activity1.3.3cv.png"/>
+          <image source="solution-images/Activity1.3.3cvi.png"/>
+        </sidebyside>
+      </solution>
     </task>
     <task workspace="4cm" xml:id="act-changing-aroc-trends-task-3">
       <statement>
@@ -73,7 +106,14 @@
       </statement>
       <hint/>
       <answer/>
-      <solution/>
+      <solution>
+        <p>
+          False.  Although <m>AV_{[0,3]} = \frac{q(3)-q(0)}{3} = \frac{3-0}{3} = 1 > 0</m>,
+          a positive average rate of change over an interval does not mean the function is
+          increasing everywhere on that interval.  From part (a), <m>AV_{[2,3]} = -1 &lt; 0</m>,
+          so <m>q</m> is actually decreasing on <m>[2,3]</m>.
+        </p>
+      </solution>
     </task>
     <task workspace="4cm" xml:id="act-changing-aroc-trends-task-4">
       <statement>
@@ -86,13 +126,19 @@
       </statement>
       <hint/>
       <answer/>
-      <solution/>
+      <solution>
+        <p>
+          Any linear function has constant average rate of change.  For example,
+          <m>f(x) = x</m> has <m>AV_{[a,b]} = \frac{b-a}{b-a} = 1</m> for every interval
+          <m>[a,b]</m>.
+        </p>
+      </solution>
     </task>
     <answer>
       <p/>
     </answer>
     <solution>
-      <p/>
+      <p>See solutions to individual tasks above.</p>
     </solution>
   </activity>
 </worksheet>
diff --git a/source/activities/act-changing-combining-arithmetic.xml b/source/activities/act-changing-combining-arithmetic.xml
@@ -37,7 +37,12 @@
       </statement>
       <hint/>
       <answer/>
-      <solution/>
+      <solution>
+        <p>
+          From the graphs, <m>f(0) = \frac{5}{2}</m> and <m>g(0) = 1</m>, so
+          <md>(f+g)(0) = \frac{5}{2} + 1 = \frac{7}{2}.</md>
+        </p>
+      </solution>
     </task>
     <task workspace="3.5cm" xml:id="act-changing-combining-arithmetic-task-1">
       <statement>
@@ -47,7 +52,12 @@
       </statement>
       <hint/>
       <answer/>
-      <solution/>
+      <solution>
+        <p>
+          From the graphs, <m>f(1) = 4</m> and <m>g(1) = 3</m>, so
+          <m>(g-f)(1) = 3 - 4 = -1</m>.
+        </p>
+      </solution>
     </task>
     <task workspace="3.5cm" xml:id="act-changing-combining-arithmetic-task-2">
       <statement>
@@ -57,7 +67,12 @@
       </statement>
       <hint/>
       <answer/>
-      <solution/>
+      <solution>
+        <p>
+          From the graphs, <m>f(-1) = 3</m> and <m>g(-1) = 3</m>, so
+          <m>(f \cdot g)(-1) = 3 \cdot 3 = 9</m>.
+        </p>
+      </solution>
     </task>
     <task workspace="3cm" xml:id="act-changing-combining-arithmetic-task-3">
       <statement>
@@ -67,7 +82,12 @@
       </statement>
       <hint/>
       <answer/>
-      <solution/>
+      <solution>
+        <p>
+          <m>\left(\frac{f}{g}\right)(x)</m> is undefined wherever <m>g(x) = 0</m>.
+          From the graph, <m>g(x) = 0</m> at <m>x = -2</m> and <m>x = 3</m>.
+        </p>
+      </solution>
     </task>
     <task workspace="3cm" xml:id="act-changing-combining-arithmetic-task-4">
       <statement>
@@ -77,7 +97,14 @@
       </statement>
       <hint/>
       <answer/>
-      <solution/>
+      <solution>
+        <p>
+          <m>(f \cdot g)(x) = 0</m> wherever <m>f(x) = 0</m> or <m>g(x) = 0</m>.
+          From the graphs, <m>f(x) = 0</m> at <m>x = \frac{7}{3}</m>, and
+          <m>g(x) = 0</m> at <m>x = -2</m> and <m>x = 3</m>.
+          So <m>(f \cdot g)(x) = 0</m> at <m>x \in \left\{-2,\, \frac{7}{3},\, 3\right\}</m>.
+        </p>
+      </solution>
     </task>
     <task workspace="3cm" xml:id="act-changing-combining-arithmetic-task-5">
       <statement>
@@ -87,7 +114,19 @@
       </statement>
       <hint/>
       <answer/>
-      <solution/>
+      <solution>
+        <p>
+          Yes: <m>(f-g)(x) = 0</m> when <m>f(x) = g(x)</m>.
+          One intersection is visible from the graphs at <m>x = -1</m>.
+          On the interval <m>-1 \lt x \le 1</m>, the formulas are <m>f(x) = \frac{5}{2} + x</m>
+          and <m>g(x) = -x^2 + 4</m>; setting them equal gives:
+          <md>
+            <mrow>x^2 + x - \frac{3}{2} &amp;= 0</mrow>
+            <mrow>x &amp;= \frac{-1 + \sqrt{7}}{2}.</mrow>
+          </md>
+          So <m>(f-g)(x) = 0</m> at <m>x = -1</m> and <m>x = \dfrac{\sqrt{7}-1}{2}</m>.
+        </p>
+      </solution>
     </task>
     <answer>
       <p/>
-Original file line number
+Diff line change
@@ Expand Up / @@ -18,3 +18,4 @@ codechat_config.yaml @@
     settings.json
     logs
     .cache
+    latex/