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<title>Adventure 2 β Student Activity (Slope)</title>
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<h1>π Adventure 2 β Student Activity (Slope Only)</h1>
<div class="subtitle">
Complete the table, study the curve, and use the tangent line to understand slope at one instant.
</div>
<h2>2.1 β The Curve and the Tangent</h2>
<div class="card">
<div class="imgwrap">
<img src="images/adventure_2_image_1_student.png" alt="Quadratic with tangent at (3,9)">
</div>
<div class="inputs">
<div style="grid-column:1 / -1;">
<label><strong>Curve:</strong> h(t) = t^2</label>
<div class="hint">The marked point is (3, 9). The tangent line is drawn only from t = 2 to t = 4.</div>
</div>
</div>
</div>
<h2>2.2 β Table First</h2>
<div class="card">
<p class="hint" style="margin-top:0;">Complete the table for h(t)=t^2.</p>
<div class="tablewrap">
<table>
<thead><tr><th>t</th><th>0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th></tr></thead>
<tbody>
<tr>
<th>h(t)</th>
<td><input type="text" id="a22_0"></td>
<td><input type="text" id="a22_1"></td>
<td><input type="text" id="a22_2"></td>
<td><input type="text" id="a22_3"></td>
<td><input type="text" id="a22_4"></td>
<td><input type="text" id="a22_5"></td>
</tr>
</tbody>
</table>
</div>
</div>
<h2>2.3 β Difference Quotient (Fill In)</h2>
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<p class="hint" style="margin-top:0;"><strong>Step 1:</strong> Write the slope near t=3:</p>
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<div style="grid-column:1 / -1;">
<label for="dq1">dh(3)/dt = (h(3 + dt) β h(3)) / dt =</label>
<input id="dq1" type="text" placeholder="Fill in...">
</div>
<div style="grid-column:1 / -1;">
<label for="dq2">Substitute h(t)=t^2 and simplify until you get β6 + dtβ:</label>
<input id="dq2" type="text" placeholder="Show your simplification...">
</div>
</div>
<p class="hint"><strong>Step 2:</strong> As dt β 0, the slope at t = 3 is:</p>
<div class="inputs" style="margin-top:0.4rem;">
<div style="grid-column:1 / -1;">
<label for="slope3">Slope at t=3 =</label>
<input id="slope3" type="text" placeholder="Slope =">
</div>
</div>
</div>
<h2>2.4 β The General Rule</h2>
<div class="card">
<p class="hint" style="margin-top:0;">Repeat for an unknown t:</p>
<div class="inputs">
<div style="grid-column:1 / -1;">
<label for="gen1">(h(t + dt) β h(t)) / dt =</label>
<input id="gen1" type="text" placeholder="Fill in...">
</div>
<div style="grid-column:1 / -1;">
<label for="gen2">As dt β 0, dh/dt =</label>
<input id="gen2" type="text" placeholder="dh/dt =">
<div class="hint">This is the βsecret of slopeβ for h(t)=t^2.</div>
</div>
</div>
</div>
</div>
<div style="display:flex; gap:12px; margin-top:20px; margin-bottom: 20px; margin-left: 10px;">
<button id="sampleBtn" onclick="sampleA2()">βοΈ Sample Values</button>
<button id="checkBtn" onclick="checkA2()">β
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a22_1.value = 1;
a22_2.value = 4;
a22_3.value = 9;
a22_4.value = 16;
a22_5.value = 25;
dq1.value = "( (3+dt)^2 - 9 ) / dt";
dq2.value = "6 + dt";
slope3.value = "6";
gen1.value = "( (t+dt)^2 - t^2 ) / dt";
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