🏠 Adventure 11 — Taylor Series

• Derivatives give the building blocks of a function at a point.
• Taylor series uses those pieces to reconstruct the entire function.
• Special functions like e^x, sin(x), and cos(x) reveal deep patterns—and connect through Euler’s identity.

Next, on YouTube watch 3Blue1Brown — Essence of Calculus, Chapter 5

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^x is so special.

• 📖 Halley’s Comet, Newton’s Falling Moon & the Secret Behind Taylor Series
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.

• ✏️ Open: Adventure 11 — Activity Page
Fill in tables to construct e^x, sinx, and cosx step by step from derivatives. Then combine them to uncover one of the most beautiful results in math: Euler’s identity.

• 📈 Open: Adventure 11 — Charts (PDF)