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<h1>✅ Adventure 6 — Solutions</h1>
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Connect graph reading, rectangle estimates, and the calculus meaning of <b>area</b>.
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<h2>✳ Functions</h2>
<ul>
<li><code>a(t)=8-2t</code></li>
<li><code>v(t)=t(8-t)=8t-t²</code></li>
<li><code>s(t)=4t²-(1/3)t³</code></li>
</ul>
<h2>1) Distance read‑offs</h2>
<ul>
<li><code>s(2)=13</code> m</li>
<li><code>s(7)=82</code> m → distance 2→7 is <code>82-13=69</code> m</li>
</ul>
<h2>2) Area under v(t)</h2>
<ul>
<li>Total distance 0→8 is the area under <code>v(t)</code>.</li>
<li>Exact integral: <code>∫₀⁸ (8t-t²)dt = 85.33</code> m</li>
</ul>
<h2>3) Rectangle estimates (0→8)</h2>
<ul>
<li>Δt = 1 : <b>84.00</b> m</li>
<li>Δt = 0.5 : <b>85.00</b> m</li>
<li>Δt = 0.25 : <b>85.25</b> m</li>
<li>These converge toward <b>85.33</b> m.</li>
</ul>
<h2>4) Velocity from acceleration area (0→8)</h2>
<ul>
<li>Signed area under <code>a(t)</code> from 0→8 is zero (positive triangle cancels negative triangle).</li>
<li>So <code>v(8)=0</code> m/s (matches the velocity graph).</li>
</ul>
<h2>5) Acceleration check (0→5)</h2>
<ul>
<li>From the velocity curve: <code>Δv = v(5)-v(0)=15</code> m/s</li>
<li>From acceleration area: area 0→4 is 16, area 4→5 is −1, net 15 m/s.</li>
</ul>
<h2>★ Calculus link</h2>
<ul>
<li><code>s'(t)=v(t)</code></li>
<li><code>v'(t)=a(t)</code></li>
<li>Integrals reverse derivatives: <code>∫v(t)dt = s(t)+C</code></li>
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