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<title>Adventure 11 — Free Fall: Gravity and the Birth of Calculus</title>
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<a href="index.html"><button class="secondary">🧭 Hub</button></a>
<a href="adventure_11_activity_student.html"><button class="secondary">✅ Activity</button></a>
<a href="adventure_11_story.html"><button class="secondary">📖 Story</button></a>
<span id="score" class="subtitle" style="margin:0; padding:0;"></span>
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<h1>🏠 Adventure 11 — Taylor Series</h1>
<div class="subtitle">Taylor series as a ‘function builder’: use derivatives to construct e<sup>x</sup>, sin(x), and cos(x), then connect to Euler’s identity.</div>
<h2>🎯 What you will learn</h2>
<div class="callout">
<ul>
<b>
<li>Derivatives give the building blocks of a function at a point.</li>
<li>Taylor series uses those pieces to reconstruct the entire function.</li>
<li>Special functions like e<sup>x</sup>, sin(x), and cos(x) reveal deep patterns—and connect through Euler’s identity.</li>
</b>
</ul>
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<h2>🎥 Watch the Video</h2>
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<p>
Next, on YouTube watch <strong><a href="https://www.youtube.com/watch?v=m2MIpDrF7Es&list=PLZHQObOWTQDMsr9K-rj53DwVRMYO3t5Yr&index=5" target="_blank" rel="noopener">3Blue1Brown — Essence of Calculus, Chapter 5</a></strong>
</p>
What's so special about Euler's number e? Watch how the number e and its series come from simple patterns in change. This video shows how repeating derivatives build powerful functions—and why e<sup>x</sup> is so special.
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<h2>📖 1) Read / Listen: Story</h2>
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<ul>
<li>📖 <strong><a href="adventure_11_story.html">Halley’s Comet, Newton’s Falling Moon & the Secret Behind Taylor Series</a></strong></li>Halley bets on a comet. Newton reveals a deeper secret: the Moon is always falling, and motion can be understood through tiny steps. From this idea comes something powerful—using small pieces to build entire curves.
</ul>
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<h2>✏️ 2) Do the Adventure Activity</h2>
<div class="callout">
<ul>
<li>✏️ <strong><a href="adventure_11_activity_student.html">Open: Adventure 11 — Activity Page</a></strong>
</li>
Fill in tables to construct e<sup>x</sup>, sinx, and cosx step by step from derivatives. Then combine them to uncover one of the most beautiful results in math: Euler’s identity.
</ul>
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<h2>📈 3) Charts - for printing</h2>
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<ul>
<li>📈 <strong><a href="images/adventure_11_charts.pdf" target="_blank" rel="noopener">Open: Adventure 11 — Charts (PDF)</a></strong></li>
</ul>
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<div class="footer">Dr. Super & Spark — Powered by ChatGPT</div>
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