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<h1>🧩 Adventure 13 — Activity 3</h1>
<div class="subtitle">DiVA reasoning (new polynomial ATP model)</div>
<div class="chart-top"><img alt="ATP DiVA reference (curves + shading; D(t) only)" src="images/adventure_13_activity_3_diva_Student.png"/></div>
<div class="one-col">
<div class="callout">
<b>Goal:</b> Build the ATP availability curve <b>D(t)</b> using calculus — derivatives, sign tables, and a few values.
You will sketch <b>D(t)</b> on the DiVA chart using <b>critical points</b> (max/min) and an <b>inflection point</b>.
</div>
<h2>Given (simplified units)</h2>
<div class="callout">
<div><b>D(t)</b> = <span class="mono">D(t) = t^3/3 − 20t^2 + 300t + 2000</span></div>
<div><b>V(t)=D′</b> = <span class="mono">t^2 − 40t + 300</span></div>
<div><b>A(t)=V′=D″</b> = <span class="mono">2t − 40</span></div>
</div>
<h2>Part 1 — Compute the derivatives</h2>
<p><b>V(t) = D′(t) =</b> <input id="vexpr" placeholder="your expression" style="max-width:420px;" type="text"/></p>
<p><b>A(t) = V′(t) = D″(t) =</b> <input id="aexpr" placeholder="your expression" style="max-width:420px;" type="text"/></p>
<h2>Part 2 — Critical points for D(t)</h2>
<p>1) Solve <b>V(t)=0</b> to find where D(t) has a maximum or minimum.</p>
<p><b>t values:</b>
t₁ = <input id="t1" step="any" style="max-width:140px;" type="number"/>
t₂ = <input id="t2" step="any" style="max-width:140px;" type="number"/>
</p>
<p>2) Use <b>A(t)=D″(t)</b> to decide which is a max and which is a min.</p>
<p><b>D(t) has a minimum at</b> t = <input id="tmin" step="any" style="max-width:160px;" type="number"/>
<b>and a maximum at</b> t = <input id="tmax" step="any" style="max-width:160px;" type="number"/>
</p>
<p>3) Inflection point for D(t): solve <b>A(t)=0</b>.</p>
<p><b>Inflection time:</b> t = <input id="tinf" step="any" style="max-width:160px;" type="number"/></p>
<h2>Part 3 — Reasoning about V(t) and A(t)</h2>
<ul>
<li>When A(t) is <b>positive</b>, V(t) is (increasing / decreasing) → <input id="vincr" placeholder="increasing or decreasing" style="max-width:220px;" type="text"/></li>
<li>When V(t) is <b>positive</b>, D(t) is (increasing / decreasing) → <input id="dincr" placeholder="increasing or decreasing" style="max-width:220px;" type="text"/></li>
<li>Where does V(t) reach its <b>maximum</b>? (Hint: A(t)=0) → t = <input id="tvmax" step="any" style="max-width:160px;" type="number"/></li>
</ul>
<div class="callout" style="margin-top:1rem;">
<b>Important idea:</b> You can understand max/min/inflection without sketching the whole curve — if you know how the derivatives behave.
</div>
</div>
<h2>Part 4 — Sign table (structure)</h2>
<div class="small">You will use signs of <b>V</b> and <b>A</b> to decide where <b>D</b> is decreasing/increasing and where it has min/max/inflection.</div>
<table class="tbl">
<tr>
<th>Interval (seconds)</th>
<th>0–10</th><th>10</th><th>10–20</th><th>20</th><th>20–30</th><th>30</th><th>30–40</th>
</tr>
<tr>
<th>D (ATP availability)</th>
<td><input id="d_0_10"/></td>
<td><input id="d_10"/></td>
<td><input id="d_10_20"/></td>
<td><input id="d_20"/></td>
<td><input id="d_20_30"/></td>
<td><input id="d_30"/></td>
<td><input id="d_30_40"/></td>
</tr>
<tr>
<th>V = D′</th>
<td><input id="v_0_10"/></td>
<td><input id="v_10"/></td>
<td><input id="v_10_20"/></td>
<td><input id="v_20"/></td>
<td><input id="v_20_30"/></td>
<td><input id="v_30"/></td>
<td><input id="v_30_40"/></td>
</tr>
<tr>
<th>A = V′ = D″</th>
<td><input id="a_0_10"/></td>
<td><input id="a_10"/></td>
<td><input id="a_10_20"/></td>
<td><input id="a_20"/></td>
<td><input id="a_20_30"/></td>
<td><input id="a_30"/></td>
<td><input id="a_30_40"/></td>
</tr>
</table>
<h2>Table 2 — Values for V(t) and A(t)</h2>
<div class="small">Use <b>V(t)</b> and <b>A(t)</b> formulas to compute values. Then use the sign of each value to fill Table 1.</div>
<table class="tbl">
<tr><th>t</th><th>0</th><th>5</th><th>10</th><th>15</th><th>20</th><th>25</th><th>30</th><th>35</th><th>40</th></tr>
<tr><th>V(t)</th>
<td><input id="v0"/></td><td><input id="v5"/></td><td><input id="v10"/></td><td><input id="v15"/></td><td><input id="v20"/></td>
<td><input id="v25"/></td><td><input id="v30"/></td><td><input id="v35"/></td><td><input id="v40"/></td>
</tr>
<tr><th>A(t)</th>
<td><input id="a0"/></td><td><input id="a5"/></td><td><input id="a10v"/></td><td><input id="a15"/></td><td><input id="a20v"/></td>
<td><input id="a25"/></td><td><input id="a30v"/></td><td><input id="a35"/></td><td><input id="a40v"/></td>
</tr>
</table>
<h2>Critical points from derivatives</h2>
<div class="small">
<b>First derivative</b> decides if there is an extremum (max/min). <b>Second derivative</b> decides which kind.
Also, an <b>inflection</b> happens when <b>A(t)=D″(t)=0</b> (and the concavity changes).
</div>
<div class="callout">
At <b>t = 10</b> we have a
<input id="kind10" placeholder="minimum/maximum" style="width:140px"/> because V(10)=0 and A(10) is
<input id="signA10" placeholder=">0 or <0" style="width:90px"/>.
<br/><br/>
At <b>t = 30</b> we have a
<input id="kind30" placeholder="minimum/maximum" style="width:140px"/> because V(30)=0 and A(30) is
<input id="signA30" placeholder=">0 or <0" style="width:90px"/>.
<br/><br/>
At <b>t = 20</b> we have an
<input id="kind20" placeholder="inflection point" style="width:160px"/> because A(20)=0.
</div>
<h2>Table 3 — Values for D(t)</h2>
<div class="small">Compute D(t) at these times (or use your calculator). These values help anchor your sketch.</div>
<table class="tbl">
<tr><th>t</th><th>0</th><th>10</th><th>20</th><th>30</th><th>40</th></tr>
<tr><th>D(t)</th>
<td><input id="d0"/></td><td><input id="d10"/></td><td><input id="d20"/></td><td><input id="d30"/></td><td><input id="d40"/></td>
</tr>
</table>
<h2>Part 5 — Sketch the ATP Availability Curve</h2>
<ul>
<a href="ATP_DiVA_Activity_3_Blank_Student_Template.pdf" target="_blank">
Open and print the template for <b> Diva Chart for ATP DiVA: A Study of Critical Points.</b>
</a>
<li>On the <b>top graph</b> of the DiVA, mark where <b>V(t)=0</b> (min/max).</li>
<li>Mark where <b>A(t)=0</b> (inflection).</li>
<li>Use <b>Table 1</b> (signs) and <b>Table 3</b> (values) to draw a smooth, reasoned sketch.</li>
</ul>
<h2>Part 6 — Compare with the ChatGPT DiVA</h2>
<div class="callout">
My chart is very similar to the DiVA Charts produced by ChatGPT:
<label><input name="sim" type="radio"> Yes</input></label>
<label style="margin-left:12px;"><input name="sim" type="radio"> No</input></label>
<br/><br/>
Explain what is happening with ATP availability:
<br/>
<textarea style="width:100%;height:90px;"></textarea>
</div>
<h2>Part 7 — The Parting Shot</h2>
<div class="callout">
Without recalculating any values: explain how you could sketch the shape of the ATP curve using only
zeros of <b>V(t)</b>, zeros of <b>A(t)</b>, and signs.
Why is this a powerful idea in calculus?
<br/><br/>
<textarea style="width:100%;height:90px;"></textarea>
</div>
<div style="margin-top:15px;"><button id="sampleBtn" onclick="useSampleValues()">Use Sample Values</button> <button id="checkBtn" onclick="checkAnswers()">Check My Answers</button></div>
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<div class="footer">Dr. Super & Spark — Powered by ChatGPT</div>
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