Essence of Calculus — 3Blue1Brown

🎯 Project Positioning "Essence of Calculus" is a series of calculus courses launched by 3Blue1Brown (Grant Sanderson), driven by visual animations. It aims to allow learners to truly experience and "invent" the core ideas of calculus through intuitive animations and geometric thinking.

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📚 Content Structure (12 Chapters, Approximately 3 Hours)

1. The essence of calculus

• Overview of course motivation: How to "discover" calculus from images, including the reconstruction of the circle area formula.

2. Derivative paradox

• The essence of derivatives: Revealing the paradox of "instantaneous rate of change" and "limits."

3. Geometric derivatives

• Deriving derivative formulas (such as the derivative of power functions) through geometric decomposition, making the formulas "inventable."

4. Visualization of chain & product rule

• Visually demonstrating the chain rule and product rule, understanding derivative combinations from the perspective of graphical transformations.

5. Euler’s number e

• Exploring why exponential functions are special, with e defined as the unique base whose derivative is itself.

6. Implicit differentiation

• Implicit differentiation: Understanding how to find dy/dx when two variables are related.

7. Limits, L'Hôpital’s, ε–δ

• Rigorously defining limits, L'Hôpital's rule, and ε–δ, building the axiomatic foundation of calculus.

8. Integration & Fundamental Theorem

• Explaining the meaning of integration and its inverse relationship with derivatives (the Fundamental Theorem of Calculus).

9. Area vs. slope

• Exploring the deep connection between "area" and "slope."

10. Higher-order derivatives

• Second and third derivatives and their physical/geometric meanings.

11. Taylor series

• Discussing Taylor series as polynomial approximations of arbitrary functions locally.

12. Alternative derivative visualized

• Introducing an alternative visualization of derivatives, paving the way for subsequent higher-order calculus.

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🧩 Course Features

• Visual Priority: Animations and geometric visuals are used throughout, giving abstract symbols intuitive meaning.
• Recreating the "Invention" Path of Calculus: Not just learning formulas, but also understanding why these rules exist.
• Approximation vs. Limit Unified: Emphasizing approximating by treating Δx as "small but non-zero," then obtaining precision through limits.
• Balancing Theory and Intuition: Providing both rigorous definitions and intuitive diagrams and common derivations.

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🧠 Target Audience

• Learners who already have a high school calculus foundation.
• Those who want to break through the intuition and understanding bottlenecks behind formulas, and deepen their perception of derivatives, integrals, limits, Taylor expansions, and the chain rule.

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⏱️ Recommended Learning Method

• Watch the official YouTube tutorials (completely free).
• Supplement with traditional textbook exercises or mathematical analysis courses to consolidate ε–δ concepts and more formal proofs.
• For each chapter, it is recommended to watch the animation while simultaneously understanding with notes + graphics ("reason like a mathematician").