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<title>Adventure 2 — Solutions</title>
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<h1>✅ Adventure 2 — Solutions</h1>
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Use this page after you try the activity. It shows the tangent picture, the rise/run idea, and the limit form using <code>h</code>.
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Tip: On the tangent segment, from t=2 to t=4 the <strong>run is 2</strong> and the <strong>rise is 12</strong>.
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<p><strong>Slope:</strong> rise/run = 12/2 = 6.</p>
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<h2>2.2 — Table of Values</h2>
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<p class="hint" style="margin-top:0;">For <code>h(t)=t^2</code>, the completed table is:</p>
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<tr><th>t</th><th>0</th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th></tr>
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<tr><th>h(t)</th><td>0</td><td>1</td><td>4</td><td>9</td><td>16</td><td>25</td></tr>
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<h2>Difference Quotient at t = 3</h2>
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<p><code>f'(3) = lim_(dt->0) ( f(3+dt) - f(3) ) / dt</code></p>
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With <code>f(t)=t^2</code>:
<code>( (3+dt)^2 - 9 ) / dt = (9 + 6dt + dt^2 - 9)/dt = 6 + dt</code>,
so <code>f'(3)=6</code>.
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<h2>General Rule</h2>
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<p><code>f'(t) = lim_(dt->0) ( f(t+dt) - f(t) ) / dt</code></p>
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<code>( (t+dt)^2 - t^2 ) / dt = (2t*dt + dt^2)/dt = 2t + dt</code>,
so <code>f'(t)=2t</code>.
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