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<a href="adventure_2_story.html"><button class="secondary">📖 Story</button></a>
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<h1>🚀 Adventure 2 — The Apple and the Secret of Slope</h1>
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<b>Story, video, and a short activity that zooms in on a second‑degree curve to reveal a tangent line.</b>
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<h2>🎯 What you will learn</h2>
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<h3>How to omplete a table to study acurve, and use the tangent line to understand slope at one instant.</h3>
<h3>How to find the slope of a tangent using rise/run.</h3>
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<h2>📖 1) Read / Listen: Story</h2>
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<li>📖 <strong><a href="adventure_2_story.html">Newton and Leibniz Develop Calculus</a></strong>
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<p>
First, read <strong><em>Newton and Leibniz Develop Calculus</em></strong>.
This is the Newton–Leibniz story that sets the stage: a curve can have a slope at a single instant. How do you measure change at a single instant when everything is moving? That question leads to a powerful new way of seeing curves.
</p>
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<h2>🎥 2) Watch the Video</h2>
<p>
Watch the 3Blue1Brown video <strong><a href="https://www.youtube.com/watch?v=9vKqVkMQHKk&list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr&t=22s">Open:paradox of the derivative | Chapter 2, Essence of calculus</a></strong> where a <strong> secant line</strong> slides and becomes a
<strong>tangent line</strong>. That “zooming in” is the whole secret.
In this video, you’ll explore a surprising question: how can a curve have a slope at just one single point? By zooming in closer and closer, you’ll see that curves begin to look straight, revealing a well-defined slope called the instantaneous rate of change. As you watch, focus on how this idea helps us understand motion at an exact moment — the key to defining velocity in calculus.
</p>
<h2>✏️ 3) Do the Adventure Activity</h2>
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<li>✏️ <strong><a href="adventure_2.html">Open: Adventure 2 — Activity</a></strong></li>
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<p>Capture how fast something is changing at just one moment. See how the tangent line represents the instantaneous rate of change.</p>
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<h2>📈 4) What You Will Discover</h2>
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One idea powers this Adventure:
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<li><strong>Slope = instantaneous change</strong></li>
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<strong>🧠 5) Questions to Keep in Mind</strong>
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<ul>
<li>What does it mean for a curve to have a slope “right now”?</li>
<li>Why does zooming in make the curve look like a straight line?</li>
<li>How does the slope change as t gets larger?</li>
<li>How does rise/run on the tangent match the limit idea?</li>
</ul>
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<h2>💡 6) If You Get Stuck</h2>
<p>
Pick two points close together, compute rise/run, then make them even closer.
The slope you’re chasing is exactly what the tangent line captures.
</p>
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