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<title>Adventure 13 – Critical Points & DiVA Reasoning (Home)</title>
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<a href="index.html"><button>🧭 Hub</button></a>
<a href="adventure_13_story.html"><button class="secondary">📜 Story</button></a>
<a href="adventure_13_activity1_student.html"><button class="secondary">✅ Activity 1</button></a>
<a href="adventure_13_activity3_student.html"><button class="secondary">✅ Activity 2</button></a>
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<h1>Adventure 13 – Critical Points & DiVA Reasoning</h1>
<div class="subtitle">Maximum • Minimum • Inflection Point • Using derivatives to “read” a function</div>
<div class="card">
<h2>🎥 Watch First (optional)</h2>
<ul>
<li>🎥 <strong><a href="https://https://www.youtube.com/watch?v=V239IIoGVT8" target="_blank" rel="noopener">
Relative Extrema, Local Maximum and Minimum, First Derivative Test, Critical Points- Calculus</a></strong></li>
<p>Learn how calculus finds the “turning points” of a curve — where a function reaches a local maximum, local minimum, or changes direction. This video shows how critical points and the first derivative test let us understand the shape and behavior of a graph without plotting every point.</p>
<li><b>Reminder:</b> A maximum/minimum happens when the <b>first derivative changes sign</b>. An inflection point happens when the <b>second derivative changes sign</b>.</li>
</ul>
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<h2>📜 Read the Story (or Listen)</h2>
<p><b>ATP: The Curve Behind Every Movement</b> — a story about how a biological curve can be studied using the same ideas as motion graphs. ATP is the tiny energy molecule that powers every movement in your body — from running and jumping to simply breathing. In this story, you will see how the rise and fall of ATP can be studied using the same calculus ideas we use for motion graphs, critical points, and curves.</p>
<p><a href="adventure_13_story.html">Open the story page</a></p>
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<div class="card">
<h2>🧩 What you will do</h2>
<ol>
<li><b>Activity 1:</b> Study four “training” functions: <code>x²</code>, <code>−x²</code>, <code>x³</code>, <code>−x³</code>. Compute <code>f</code>, <code>f′</code>, <code>f″</code> values and identify max/min/inflection.</li>
<li><b>Activity 2:</b> Compute derivatives and find critical points. Then use a DiVA chart to draw <code>A(t)</code> and <code>V(t)</code> and draw the curve for <code>D(t)</code> based on the these and where the derivatives change signs (without finding many point for drawing full curves for <code>D(t)</code>).</li>
</ol>
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